Corrected Finite Element and Finite Difference ... - System Simulation

Corrected Finite Element and Finite Difference Methods for Elliptic Problems with Singularities Korrigierte Finite-Element- und Finite-Differenzen-Methoden für elliptische Probleme mit Singularitäten Der Technischen Fakultät der Friedrich-Alexander-Universität Erlangen-Nürnberg zur Erlangung des Doktorgrades Dr.-Ing. vorgelegt von Ayate Behairy aus Kairo, Ägypten

Als Dissertation genehmigt von der Technischen Fakultät der Friedrich-AlexanderUniversität Erlangen-Nürnberg Tag der mündlichen Prüfung: 27.4.2018 Vorsitzende des Promotionsorgans: Prof. Dr. Reinhard Lerch Gutachter: Prof. Dr. Ulrich Rüde Prof. Dr. Günther Greiner

Contents 1

Introduction 1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Goals and contributions . . . . . . . . . . . . . . . . . . . . . . . 1.3 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2

Scientific Background 3 2.1 Mathematical fundamentals . . . . . . . . . . . . . . . . . . . . . 3 2.2 Literature review of corner singularities . . . . . . . . . . . . . . 8 2.3 Other possible types of singularities . . . . . . . . . . . . . . . . 13

3

Introduction to the correction technique 3.1 Energy-correction technique in the literature . . . . . . . . . . . . 3.1.1 Modified Galerkin approximation . . . . . . . . . . . . . 3.1.2 Main results: the pollution effect and the optimal convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.3 Construction of the correction and the modification parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Main idea of our work . . . . . . . . . . . . . . . . . . . . . . . 3.3 Multigrid method . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Finite element method . . . . . . . . . . . . . . . . . . . . . . . 3.5 Finite difference method . . . . . . . . . . . . . . . . . . . . . .

4

5

Correction technique using finite element discretization 4.1 Modified finite element approximation . . . . . . . . . . . . . . 4.2 The modification parameter and the scaling relation between the energy defect functions . . . . . . . . . . . . . . . . . . . . . . 4.3 Modification of the stiffness matrix . . . . . . . . . . . . . . . . 4.4 Defect functions . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 L-shaped domain . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Slit domain with a fixed tip at the origin . . . . . . . . . . . . . 4.7 Re-entrant corner domains in 3D with the third co-ordinate being constant in the solution . . . . . . . . . . . . . . . . . . . . . . 4.8 Analytic energy . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 1 2

14 15 16 18 19 20 24 26 29

30 . 30 . . . . .

32 40 50 52 61

. 65 . 65

Periodicity of the modification parameter 76 5.1 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . 78

i

5.2

Analytic energy . . . . . . . . . . . . . . . . . . . . . . . . . . .

83

6

Correction technique using finite difference discretization 87 6.1 Modification of the 5-point stencil . . . . . . . . . . . . . . . . . 87 6.2 L-shaped domain . . . . . . . . . . . . . . . . . . . . . . . . . . 90 6.3 Slit domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

7

Conclusion

107 Zusammenfassung

In dieser Arbeit betrachten wir ein Korrekturschema für Finite-Elemente und FiniteDifferenzen-Verfahren angewandt auf zwei-dimensionale elliptische Probleme definiert auf Gebieten mit einer einspringenden Ecke. Durch das Korrekturschema wird die sub-optimale Konvergenz aufgrund der Störungen durch nicht glatte sowie singuläre Ränder verbessert. Eine Konvergenz zweiter Ordnung kann durch das Korrekturschema basierend auf lokalen Veränderungen der Energieberechnung erreicht werden. Die globale Störung durch die Singularität an der einspringenden Ecke wird dadurch verhindert. Die Konvergenzrate der Finite-Differenzen und Finite-Elemente-Lösung wird durch die Störung der Funktion an der Randbedingung beeinflusst. Wir können dieses Problem theoretisch beheben, indem wir das Definitionsgebiet verkleinern, während wir die Spitze der einspringenden Ecke festhalten. Dieses Ergebnis wurde numerisch bestätigt, indem das Korrekturschema auf andere Funktionen als die Energie angewandt wurde. Diese Funktionen sind die diskretisierte Energie, der Diskretisierungsfehler und die punktweise Näherungslösung. Außerdem kann eine Konvergenz zweiter Ordnung der Finite-Elemente-Lösung bezüglich der L2 -norm sowie der L∞ -norm beobachtet werden, wenn ein bestimmtes Gebiet um die Singularität ausgeschlossen wird. Dieses bestimmte Gebiet hängt nicht vom Verfeinerungslevel ab. Als Beispiele für Gebiete mit einspringender Ecke betrachten wir das L-förmige Gebiet und das Schlitz-Gebiet. Das Verfahren behält seine Gültigkeit allerdings auch für beliebige Gebiete mit einspringender Ecke. Für generelle Schlitzdomains mit einem Schlitz parallel zur x-Achse ist das Verhältnis zwischen der Spitze des Schlitzes und des Modifikationsparameters periodisch. Die Periode ist die Schrittweite. Wir betrachten ein Problem bei dem die Spitze des Schlitzes kein Gitterpunkt ist. Es wird angenommen, dass die Spitze auf jedem Verfeinerungslevel den gleichen relativen Abstand zum Gitterpunkt hat. In diesem Fall zeigt der gewählte Modifikationsparameter eine Konvergenz zweiter Ordnung für die Defektfunktion ii

und die L2 - sowie die L∞ -norm des Diskretisierungsfehlers ausgeschlossen einer Umgebung um den singulären Punkt, der nicht von der Schrittweite abhängt.

iii

Abstract We consider in this thesis a correction technique using finite element and finite difference methods applied to elliptic problems defined on re-entrant corner domains in two dimensions. The correction technique is used to avoid the sub-optimal convergence resulting from the pollution effect that is caused by the singular data and the non-smoothness of the boundary with an angle of the re-entrant corner between π and 2π. A second order convergence of the finite difference and the finite element solution results from the correction technique that is based on a local modification of the definition of the energy. The global pollution resulting from the singularity at the re-entrant corner is eliminated by this local modification. The rate of convergence of the finite difference and the finite element solution is affected by a perturbation in the function defined on the boundary. We can overcome this effect theoretically by shrinking the domain to smaller domains similar to the original one fixing the tip of the re-entrant corner. This result is confirmed numerically using the correction technique by modifying functions other than the energy. These functions are the discrete energy, the discretization error, and the approximate solution measured point-wise. Also, we get a second order convergence of the finite element solution in L2 norm and L∞ -norm if we exclude from the domain a fixed neighborhood of the singularity. This fixed neighborhood does not depend on the refinement level. As examples of re-entrant corner domains, we consider the L-shaped domain and the slit domain, though our methods are valid for a general reentrant corner domain. For a general slit domain with a slit line parallel to the x-axis, the relation between the tip of the slit domain and the modification parameter is periodic with its period being the step-size. We consider a problem in which the tip of the slit domain is not a grid node. This tip is assumed to change its position with the refinement such that it is always in a fixed relative position at each grid level from a grid node. Then the determined modification parameters exhibit second-order convergence rates of the defect functions and the L2 - and L∞ -norms of the discretization error excluding from the domain a neighborhood of the singular point not depending on the step-size.

iv

1 1.1

Introduction Motivation

The class of scalar elliptic partial differential equations gains importance in many engineering problems. In such problems, the boundary of the domain can be nonsmooth. If the domain is a bounded polygonal domain with re-entrant corners, then the representation of the general solution has singular function components. Scalar elliptic partial differential equations whose solutions have singular function components occur in a wide range of applications, e.g., in fracture mechanics, thermoelasticity and heterogeneous porous media flow. We consider in this thesis the Laplace equation defined in a bounded polygonal re-entrant corner domain such that the solution is singular and the function defined on the boundary is the local representation of the general solution in a vicinity of the corner singularity. The slit domain is a re-entrant corner domain. We consider in this thesis the Laplace equation defined in a slit domain whose tip is inside a discrete element of the approximating network and does not conform to the discrete system. This means that the tip of the slit domain is not accurately represented. The tip of the slit domain is considered to be in a fixed relative distance at each refinement from a grid node and this distance depends on the step-size. As a motivation to this problem, we refer to the crack-growth problems. In many industrial situations and engineering applications, it is of great interest to compute the growth of a crack in a two-dimensional domain.

1.2

Goals and contributions

We follow in this thesis the correction technique [48, 94, 104] for dealing with corner singularities. One of the advantages of the correction technique is that it can be easily implemented by modifying existing codes. In this technique, the structure of the system matrix remains unchanged and only a finite number of the entries is changed. The convergence rates exhibited based on the correction technique are the same as the convergence rates that result in the regular case that does not suffer from singularities. We use as a discretization method the finite element discretization with the simplest finite element and a uniform triangulation, called the Friedrichs-Keller triangulation. We use also the central finite difference method based on the fivepoint stencil using a uniform triangulation. The two methods are equivalent and 1

the numerical results are the same. The linear system of equations, resulting from each of the two discretizations, is solved using the multigrid method [32]. The author implemented the finite difference discretization of the considered problem and the multigrid solver using a C-code. For the implementation of the standard finite element method and the multigrid solver, the iFEM Matlab software [38] is used. The methods of the correction technique considered in this thesis are implemented by the author for the two considered discretizations; finite element and finite difference. What distinguishes our method from other correction techniques is that we use methods in which it is not necessary to know the analytic solution and we do not have to go through difficult or impossible computations. What is new in this thesis is that we use the correction technique for dealing with perturbations added to the boundary by shrinking the domain toward the singularity. This approach is proved theoretically to be valid in section 4.2 and it is confirmed numerically in section 4.5. We use the correction technique for dealing with a special case of a crack problem such that the domain considered is a slit domain with a moving tip. The tip of the slit domain in our problem is not accurately represented at each refinement step. This means that the tip of the slit domain is not necessarily a grid node. The advantage of using this method for this kind of problem is that there is no remeshing needed to accurately represent the tip of the slit domain. We get the same convergence rates as for the regular case, where there is no singularity. An interesting numerical result is provided about one property of the so-called modification parameter, that is defined in definition 4.1.3. It is found numerically in section 5 that the relation between the tip of the slit domain and the modification parameter is periodic with the period being the step-size.

1.3

Outline

This thesis begins with a scientific background of the corner singularities in section 2. In section 2.1, we set the mathematical fundamentals about the methods of discretizations and the type of the problem considered such that the exact form of the problem is stated in section 3.2. Section 2.2 and section 2.3 provide a literature review of dealing with corner singularities and other kinds of singularities different from the corner singularities. An introduction to the correction technique is given in section 3. In section 3.1, a summary of the correction technique as provided and studied in [48] is given. We follow this approach in our work. Section 3.2 is a summary and an 2

introduction to the work done in this thesis such that the main model problem is stated. What is new and what is gained compared to previous approaches is clarified in section 3.2. Section 3.4 presents preliminaries of the finite element method considered for the discretization of the main problem. In section 3.5, we refer to the equivalence between the finite element method and the finite difference method applied in this thesis. The most important result is provided in section 4; namely section 4.2, where we study the correction technique using the finite element method. We provide in section 4.2 a method for dealing with perturbations added to the boundary by zooming in toward smaller auxiliary domains. In section 4.3 and section 4.4, we define the different methods of the correction technique using different styles in applying this technique. Our results in section 4.5 and section 4.6 are in a good agreement with the results published in the literature [48]. An interesting theoretical result, that is confirmed numerically in section 4.5 (figure 4.12), is proved. In section 5, we consider a special case of the main model problem such that the domain considered is a slit domain with a moving tip. We deal with the problem that we consider in section 5, using the correction technique. In section 6, the same problem as in section 4, is considered such that the correction technique is applied using the finite difference method based on the five-point stencil. We get the same results; meaning that we get the same modification parameters and the same convergence rates.

2 2.1

Scientific Background Mathematical fundamentals

Solving partial differential equations accurately is one of the central problems in computational mathematics and engineering. For the treatment of partial differential equations, numerical methods have been developed for the cases where the analytic solution is not available. The process of solving the partial differential equation begins by the discretization, that is the approximating network. Elliptic equations are typified by the Laplace equation ∆u = 0, and its non-homogeneous counterpart, the Poisson equation −∆u = f , 3

where we use the notation ∆=

∂2 ∂2 + ∂ x2 ∂ y2

for the Laplace operator. We begin by considering the Poisson equation with homogeneous Dirichlet boundary conditions −∆u = f in Ω, u = 0 on ∂ Ω,

(2.1)

where f ∈ L2 (Ω); the Lebesgue space of square integrable functions with norm || · ||L2 (Ω) and Ω be a plane domain bounded by a smooth curve. Then, the solution u to problem (2.1) is smooth if the data, f , is smooth [58,76]. Weyl’s lemma states that if f is analytic in any sub-domain Ω0 of Ω, then the solution u is analytic in Ω0 . If ∂ Ω is analytic and f is analytic in Ω, then the solution u is analytic in Ω [99]. For the discretization methods we consider in this thesis, the central finite difference method based on the five-point stencil. Also, the standard finite element method with linear trial and test functions is used. For the standard finite element method, we consider the Ritz case in which the test space and the subspace of trial functions are the same [99]. A finite difference discretization of the domain Ω that is bounded by a smooth curve could be done using the Shortley-Weller approximation [80], so that the boundary grid nodes lay on the boundary curve. If we consider the square Ω = (0, 1) × (0, 1) as an example of a polygonal domain, then a uniform Cartesian grid can be generated as xi = ih, yi = ih, i = 0, . . . , N, h =

1 , N

(2.2)

where h is the step-size and (N + 1)2 is the number of grid nodes. For a polygonal domain with a uniform discretization as in (2.2), a central finite difference method can be used for the solution of (2.1). The construction of the solution of (2.1) could by given by finite difference methods using the five-point difference equation −ui+1, j + 2ui j − ui−1, j −ui, j+1 + 2ui j − ui, j−1 + = fi, j , h2 h2 where [ fi j ]1≤i, j≤N is the vector associated to the right-hand side f such that fi j = f (xi , y j ), ∀i, j, assuming that f is continuous in order that fi j is well-defined. The 4

solution u is associated with the vector [ui j ]1≤i, j≤N such that ui j is the approximation of ui j := u(xi , y j ), that is the solution u at the grid node (xi , y j ). If the boundary of Ω is an analytic curve, f is analytic in Ω and the discretization is chosen such that all the boundary nodes are on the boundary curve (e.g. as in the Shortley-Weller approximation), then the truncation error of the associated discrete approximations is of order 2 [58,76,95,104]. For the rectangular domain, the best estimate of the convergence rates for the five-point scheme has order less than 2 [76, 99]. In a square, the estimate of the error u − u is given by [99] max |ui j − ui j | ≤ Ch2 | ln h| max | f |, i, j

for a constant C. In this case, we say that the convergence of the error in this considered norm is optimal up to a log factor. Optimality up to a log factor is called in the literature [70] quasi-optimality. In section 4, we use the standard finite element method and therefore we need to define the spaces H 1 (Ω) and H01 (Ω) as ∂u ∂u , ∈ L2 (Ω)}, ∂x ∂y 1 1 H0 (Ω) = {v ∈ H (Ω) : v = 0 on ∂ Ω}.

H 1 (Ω) = {u ∈ L2 (Ω) :

The weak solution u ∈ H01 (Ω) of (2.1) is given as ∀v ∈ H01 (Ω)

a(u, v) = l(v),

(2.3)

where the standard bi-linear form a(·, ·) is defined as Z

∇v · ∇w dΩ,

a(v, w) =

(2.4)

Ω

R

and l(v) = Ω f v dΩ. The energy of the solution u is defined to be a(u, u). A triangulation TΩ of Ω is a partition of the domain Ω into sub-domains that are usually either triangles or rectangles and these sub-domains are called elements [15]. We consider the partition of Ω into triangular elements. Then, S {T : T ∈ TΩ } is a polygon. Since the boundary ∂ Ω is a smooth curve, then S there will be a nonempty "skin"; Ω \ {T : T ∈ TΩ } [99]. The triangulation TΩ is called conforming if the intersection of two non-disjoint, non-identical elements is either a common vertex or a common edge [11]. We define hT as the diameter of the circumscribing circle for T ∈ TΩ and we set the step-size h = max{hT : 5

T ∈ TΩ }. A conforming triangulation is called regular if there exists a constant c independent of h such that each triangular element T ∈ TΩ contains a circle of radius ch and is contained in a circle of radius h/c [21]. The condition of quasiuniformity is that there exists a constant δ > 0 independent of h such that for all T ∈ TΩ , hT /h ≥ δ [10, 12]. Consider a family {T`Ω }` of regular triangulations obtained by a uniform refinement of a coarse triangulation T0Ω of the domain Ω such that this refinement is obtained by subsequently dividing the triangles into four congruent ones, where ` is the refinement level. The triangulations T`Ω are quasi-uniform by construction [48]. The standard finite element space of continuous piece-wise linear functions vanishing at the boundary is given by V ` := {v ∈ H01 (Ω) : v|T ∈ P1 (T ), ∀T ∈ T`Ω }, where P1 (T ) is the space of piece-wise linear functions defined on the triangular element T . The finite element solution u` ∈ V ` is given as a(u` , v` ) = l(v` ), ∀v` ∈ V ` . The energy of the standard finite element solution u` is a(u` , u` ). The analytic energy is a(u, u). Then, the error in energy is given by a(u − u` , u − u` ). The order of accuracy of the finite element solution in the L2 -norm is 2 and it is 1 in the H 1 -norm. The error in energy is of order 2 [99]. In this thesis, we assume that Ω is a polygonal domain. If Ω is a polygonal re-entrant corner domain, then the regularity of the solution is affected by the maximal interior angle θ (see figure 2.1 for an example of a polygonal re-entrant corner domain with a single re-entrant corner, where (r, ϕ) represents the polar coordinates such that r is the distance to the singular corner). In the presence of re-entrant corners with θ > π, the solution has singular components of type θ r π [45, 73, 81]. Because of these singular solution components, the standard finite element method on a uniform triangulation shows reduced convergence rates of 2 1 the error of order 2π θ in the L -norm. The convergence order in the H -norm is the same order as for the best approximation [48]. In the presence of the corner singularities, the so called pollution effect is a phenomenon that refers to the lack of accuracy of the finite element solution globally and even far from the singularity [22, 48, 91, 94]. In the context of finite elements, the weighted L2 -spaces play a role in the error estimates. Therefore, we introduce the definition of the weighted L2 -space, 6

r θ ϕ ϕ =0

Figure 2.1: A polygonal domain with a single re-entrant corner. L2,ζ (Ω), for every ζ ∈ R as L2,ζ (Ω) = {u ∈ L2 (Ω) : rζ u ∈ L2 (Ω)} with the norm || · ||0,ζ = ||rζ u||L2 (Ω) . The weighted L2 -spaces have the advantage that the weight weakens the norm only around the singularity if ζ > 0, letting it behave like the usual L2 -norm far from the location of the singularity [1]. It is shown in [48] that the convergence rates of the error in the weighted L2 -norm is of order 2π θ under certain condition on ζ as clarified in more details in section 3.1.2. This order of convergence rates cannot be improved in general because of the pollution effect. The reason for this pollution effect is that the singular components are not smooth, and this results in this sub-optimal behavior and high accuracy can not be expected [20]. The error in weighted L2 -norms can be small only with a sufficient approximation of the energy and this holds for a wide class of Galerkin-type methods. The order of convergence of the standard finite element method in any weighted L2 -space is less than 2 if θ > π. This pollution effect is a result of insufficient approximation of the energy on quasiuniform meshes. Improving the approximation of the energy is shown in [48] to be a necessary and sufficient condition to recover the full convergence order for finite element methods. This is under certain conditions on ζ and the stress intensity factors as we clarify in details in section 3.1 and section 3.1.2. Various approaches to improve the approximation of the energy have been considered in the literature. A class of approaches use methods based on geometric or adaptive local mesh refinement [2, 5, 7, 9, 102]. Decreasing the size of elements close to the singularity allows to approximate the singular components appropriately. Another approach depends on augmenting the finite element 7

space with suitable singular functions [60, 99] as in the dual singular function method [20] that has attracted interest [23, 29, 34, 78] because of its direct representation of the stress intensity factors, that are the coefficients of the singular functions. Another set of strategies is based on suitable modifications of the discrete problems. In a Petrov-Galerkin method, singular functions are used in [9] as multiplicative factors, and the authors showed optimal convergence in weighted norms. The authors in [22] eliminate the pollution effect by using an extrapolation procedure, i.e., by a linear combination of finite element solutions on different mesh levels. It is possible to use the finite difference method for the re-entrant corner domains that can be discretized uniformly as in (2.2). For the finite difference solution, the presence of singular components of the solution shows reduced convergence rates as well. Consider the Laplace equation with a non-homogeneous boundary defined on the L-shaped domain, where θ = 3π 2 . Assume that the function defined on the boundary is continuous on ∂ Ω and analytic on every edge of ∂ Ω. Then using a five-point stencil, it is shown in [25, 104] that the error mea∞ sured at a fixed point is only of order 2π θ , while the L -norm of the error is even π worse; namely of order θ .

2.2

Literature review of corner singularities

Corner singularities are studied in the literature in different aspects. Various types of problems are considered using different methods and discretizations. In this section, we present an overview of some of the previous work done in the literature dealing with corner singularities that we consider in this thesis. We summarize in this literature review different kinds of problems using discretizations and methods different from the methods that we consider in this thesis. At the end of this section, we present one of the techniques that we follow in this thesis and that is considered in the literature. This technique deals with corner singularities and is called the correction technique. A standard approach. One of the standard approaches to deal with the corner singularities is the singular function method (SFM) [54, 99]. This method is a modified finite element method that improves the approximation of the singular solution. In the singular function method, the basis of the finite element space is extended by adding singular functions that are defined locally near the singularity. Applying the singular function method has some difficulties. One of

8

these difficulties is the evaluation of the inner products that involve the singular functions. Another difficulty is the inversion of the stiffness matrix. The added singular functions destroy the band structure of the stiffness matrix and the condition number is greatly increased. Correctly ordering the unknowns can avoid these difficulties [53,99]. By using the singular function method, the approximate stress intensity factors may lead to bad convergence of the approximate solution near the corner [20, 47]. The pollution effect. The pollution effect, that is caused by the corner singularities and that destroys the accuracy of the standard finite element methods for scalar elliptic boundary value problems on polygonal domains, is described in many article in the literature [18, 20, 22]. A model situation is provided in [22], where the Poisson equation (2.1) is considered with a non-homogeneous smooth function defined on the boundary such that the domain Ω is a polygonal domain with re-entrant corners. For the discretization of the considered problem, a standard finite element method with piece-wise linear elements is used. A regular triangulation T` is considered such that for each triangular element T ∈ T` , hT /ρT ≤ σ , for some constant σ independent of the step-size h. Here, for T ∈ T` , ρT is defined as ρT = sup{diam(B) : B is a ball contained in T } and hT is defined to be the diameter of the circumscribing circle for T [12, 41]. The triangulation also satisfies that the diameters hT approach zero with the refinement. For some technical reasons, the author imposed some conditions on the triangulations to be satisfied such that they are stretching invariant, where we refer to [22] for more details about the conditions imposed. In [22], the pollution effect is described in terms of asymptotic error expansion with respect to fractional powers of the mesh-size parameter. An extrapolation technique is used for reducing this pollution effect and the full order of accuracy is recovered for both the solution and the corresponding stress intensity factors. In this thesis, we use as a discretization method the finite element method with piecewise linear elements. Also, we assume that Ω is a general polygonal domain with re-entrant corners. The author in [18] describes this pollution effect with respect to different norms based on the validity of a singular decomposition of the solution, and proposes and analyses several procedures to recover the full order of convergence for both the Galerkin solution and the stress intensity factors. For approximating elliptic equations on domains with re-entrant corners, a finite element method, called the dual singular function method (DSFM) is pro-

9

posed in [20]. Problem (2.1) is considered where Ω is a bounded re-entrant corner domain and for simplicity, the author assumed that ∂ Ω is smooth outside the corner. The triangulation considered is a conforming regular triangulation. The dual singular function method, proposed in [20], uses the singular functions of the problem in the trial space finding good approximations of the stress intensity factors. We also consider in this thesis a conforming regular triangualtion. The convergence rate. An analysis of the convergence rates of the singular functions and the stress intensity factors are considered in [28, 29, 31]. In [29], the Poisson equation (2.1) is considered in a bounded polygonal domain Ω ⊂ R2 with at least one re-entrant corner such that the largest re-entrant angle θ of the domain satisfies π < θ < 2π. In [31], the Poisson equation (2.1) is considered in a bounded polygonal domain Ω ⊂ R2 with cracks; i.e. re-entrant corners with θ = 2π. In [29, 31], the singular function representation of the unique solution u ∈ H01 (Ω) is considered as a sum of a singular part and a smooth part U that is more regular than the solution u. The singular part is a sum of singular functions with real coefficients, that are called in elasticity problems stress intensity factors. The finite element method is used with piece-wise linear finite element space V ` ⊂ H01 (Ω). A quasi-uniform triangulation is considered with uniform refinements. The authors implement a multigird method using the singular function representation of the solution u and therefore solving for the smooth part U instead of solving for the general solution u. In this case, the right-hand side of the discrete equation contains the right-hand side f of (2.1), the singular functions and their coefficients. The stress intensity factors are computed at each grid level ` using the approximate solution of u obtained in the ` − 1st level and therefore the right-hand side of the discrete equation changes from level to level. The convergence rate is improved because the computation of the smooth part U instead of the general solution u and since U has better regularity than u. Let ε > 0 be arbitrarily small. It is shown that if f ∈ L2 (Ω), then the convergence rate for the stress intensity factors is of order 1 + θπ − ε, π < θ ≤ 2π. The convergence rate of the error in the H 1 -norm is of order 1 if π < θ < 2π, and it is of order 1 − ε and if θ = 2π. If f ∈ H 1 (Ω) is more regular, then the smooth part U is also in a more regular space. In this case the multigrid method is modified and the triangulation T` satisfies additional condition, that is the uniform band condition. A uniform band in a triangulation is a collection of triangles between two parallel lines such that any two triangles sharing a common side form a parallelogram, and the boundary of the band consists of the two parallel lines and parts 10

of ∂ Ω. A triangulation is said to satisfy the uniform band condition if it can be divided completely into uniform bands. For any polygonal domain whose vertices all have rational coordinates, it is possible to find a triangulation that satisfies the uniform band condition. The uniform band condition is preserved by the uniform subdivision. The multigrid method is modified such that the inter-grid transfer ` operator I`−1 : V `−1 −→ V ` of the multigrid method is defined using the quadratic Lagrange finite element space Q` ⊂ H01 (Ω) associated with T` as follows. Let `−2 −→ V `−1 and I` `−2 −→ V ` be the nodal interpolation operaI`−1 `−2 : Q `−2 : Q tors. Since Q`−2 and V `−1 share the same nodal points, then I`−1 `−2 is an isomor` ` −1 phism. The inter-grid transfer operator I`−1 is defined as I`−1 = I``−2 ◦ (I`−1 `−2 ) . In [28], a generalization to the case that f ∈ H m (Ω), m ≥ 2 is considered using quasi-uniform triangulation such that the V ` is the Pm+1 Lagrange finite element space on T` [30, 41]. It is shown that the convergence rate of the stress intensity factors is of order m + 1 − ε for m ≥ 1 and the convergence rate of the error is of order 1 if m = 1 and it is of order m + 1 − ε if m ≥ 2. For m = 1, the order of the convergence rate is valid for π < θ ≤ 2π. If m = 1, it is shown that the maximum error of the multigrid solution over the vertices of the triangulation in this case is of order 2 − ε. In [28], where f ∈ H m (Ω), m ≥ 2, the domain Ω is considered as a polygonal domain with re-entrant corners (π < θ < 2π), but the method considered can be adapted to domains with cracks. The advantage of the multigrid methods proposed in [28, 29, 31] is that they can be easily implemented by modifying existing multigrid codes. In these methods, the knowledge of the singular function representation is necessary. We use the multigrid method in this thesis as a solver of the linear system of equations that result from the discretization methods. The convergence rate can be improved by the method of local grid refinement [6, 26]. We now propose the papers [6, 26] that optimize the discretization locally in a vicinity of the corner singularity as a tool to overcome the pollution effect. For this method, the generation of the grids is non-trivial. It is shown in [6] that a proper refinement of the elements around the corners results in the same rate of convergence as for the domains with smooth boundary. Optimizing mesh-sizes near the singularity. A criterion for optimizing meshsizes near singularities was provided in [26] developing fast multigrid solvers for creating the nonuniform grids and their solutions. The Poisson equation is studied as an example with either algebraic singularities in the forcing terms or reentrant corner singularities. The multigrid methods are used such that extra finer 11

levels cover increasingly narrower neighborhoods of the singularity by local refinements. Different kind of problems. Next, we summarize some papers [55, 65] dealing with different kind of problems as the Stokes equation defined in re-entrant corner domains. In the context of the hp-version of the finite element method, [55] provides an analysis of the stable Galerkin formulation and a stabilized Galerkin least squares formulation for the Stokes problem. Exponential rates of convergence are established theoretically under realistic assumptions on the input data using the two formulations. These results are confirmed numerically on an L-shaped domain in which the solution exhibits corner singularities. In [65], modifications to standard low order finite element approximations of the Stokes system are proposed to improve the quality of the approximation and the process of the parallel algebraic solution. In [65], the operator itself, and not the approximation spaces, is modified so that the fundamental physical properties as mass and energy conservation are ensured. The representation of the energy in discrete systems is improved by special local a-priori correction techniques at re-entrant corners that suppress the global pollution effect. It is shown that an aposteriori correction to the finite element flux causes local mass conservation and therefore artifacts in coupled multi-physics transport problems are avoided. The correction technique. The correction technique that is discussed in details in section 3.1, for dealing with scalar elliptic problems with corner singularities, have been considered in many articles [6, 64, 65, 91, 104]. It is shown in [6] that proper refinement of the elements around corners leads to the rate of convergence that is the same as it would be on a domain with smooth boundary. Energy corrected finite element methods provide an attractive technique for dealing with elliptic problems in domains with re-entrant corners. Since the re-entrant corner domains are not smooth, this results in reduced convergence rates for finite element approximations on families of graded meshes and singular data. The authors in [48] show that it is possible to regain a full order of convergence by a local modification of the bi-linear form in a vicinity of the singularity and thus to overcome the pollution effect. The authors show that the stress intensity factors can be computed with optimal accuracy. Nested Newton strategies for energy corrected finite element methods are provided in [94], where it is shown that a local modification of the stiffness matrix at the re-entrant corner recovers 12

optimal convergence rates in weighted L2 -norms. The authors prove that a unique modification parameter exists asymptotically and this unique modification parameter is the limit of level dependent modification parameters such that each of them is defined to be the root of the error in energy. Three nested Newton-type algorithms are proposed using only one Newton step per refinement level showing local and global convergence to this asymptotic modification parameter. The basic idea for energy correction schemes was originally introduced for finite difference approaches [104], where a correction of the standard five-point difference method is considered for the Poisson equation on a special polygonal domain with given boundary values. The correction is considered at few points in the neighborhood of the corners such that the order of convergence at interior points is the same as in the case of a smooth boundary. The authors provide improved error bounds for the usual method in the neighborhood of corners. In [91], correction techniques on equidistant meshes for model problems as Laplace’s equation with non-smooth boundary data and Poisson’s equation with singular source terms are studied and used to recover the high accuracy. The author demonstrates theoretically and experimentally that it is possible to use the corrections to gain a second-order convergence as for smooth solutions. The correction techniques are applied as well within the multigrid method and it is shown that the high accuracy of the computation of the singular solutions can be obtained using the combination with the Richardson or τ-extrapolation. An energy corrected finite element methods of several scalar elliptic problems with singularities in 2D are considered in [64]. The authors outline older theoretical developments in energy corrected techniques and show numerically that it is possible to recover optimal convergence orders in weighted Sobolev norms using local and easy to implement modifications of the discrete operators.

2.3

Other possible types of singularities

There are other kinds of singularities considered in the literature in some areas like fluid, structure and applied mechanics. These types of singularities are different from the corner singularities that we consider in this thesis. Here, we give some overview about some previous work done in this regard. The authors in [8] introduce an approach called an auxiliary mapping technique, in the framework of the p-version of the finite element method, that deals with elliptic problems with singularities and yields an exponential rate of convergence. This technique can deal with elliptic problems on unbounded domains in R2 as shown in [8]. 13

Extrapolation methods for computing accurate solutions of elliptic problems with singular functions are provided in [71]. In practical applications, generalized functions occur as source terms in partial differential equations; for example, point loads and dipoles that are source terms for electrostatic potentials. It is not possible to use standard techniques to analyze the accuracy of such computations since there is no global smoothness. The solution tends to infinity at the singularity and consequently the error norms do not converge. To overcome these difficulties, other metrics are used in [71] to measure the accuracy and the convergence of the numerical solution. To obtain the same asymptotic accuracy and efficiency as for regular and smooth solutions, minor modifications are made to the discretization and solver. Then, it is possible to avoid adaptive refinement or making use of the analytic knowledge of the singularity. The core point of this technique is considering a representation of the singular sources constructed by an appropriate smoothing and this representation depends on the mesh-size. The point-wise accuracy is proved to be of the same order as in the regular case. It is shown that when approaching the singularity, the error coefficient deteriorate and the error estimates break down. Consequently, this method can be used for computing the global solution accurately by excluding a small neighborhood of the singular points. These approaches can be integrated with a multigrid technique using methods like the Richardson and the τ-extrapolation to improve the accuracy. Since the finite elements for elliptic partial differential equations with Dirac measures as source terms converge sub-optimal in classical norms due to the fact that the solution is not in H 1 , a standard remedy is to use graded meshes, then quasi-optimality, i.e., optimal up to a log-factor, for low order finite elements can be recovered, e.g., in L2 -norm. In [70], quasi-optimality is shown for the lowest order case. For the higher order finite elements optimal a priori estimates on a family of quasi-uniform meshes in a L2 -semi-norm are shown. The semi-norm is defined as a L2 -norm on a fixed sub-domain that excludes the locations of the Delta source terms.

3

Introduction to the correction technique

Let Ω be a bounded polygonal re-entrant corner domain in R2 . For the ease of presentation, we assume that Ω has a single re-entrant corner with an angle π < θ < 2π as illustrated in figure 2.1, though the methods proposed in this thesis, in section 4 and section 6 will naturally generalize to problems with several re14

entrant corners.

3.1

Energy-correction technique in the literature

We describe mainly in this section the energy-correction technique and the results of [48]. The numerical solution of the Poisson equation (2.1) is considered. The main target of [48] is to overcome the pollution effect and recover a second order convergence of the error in weighted L2 -norms. It is well known (see, e.g., [29, 45, 46, 81]) that for the Poisson equation (2.1), the solution has singular components of type kπ kπϕ , k ∈ N. (3.1) uk (r, ϕ) = r θ sin θ This yields a reduction of the convergence rates using the standard finite element method on quasi-uniform meshes [20, 99]. The weighted Sobolev space H 2,ζ (Ω), ζ ∈ R, is defined as [48, 73, 81] ∂ 2u ζ ∂ 2u ζ ∂ 2u , ,r ,r ∂ x2 ∂ y2 ∂ x∂ y ∂u ∂u rζ −1 , rζ −1 , rζ −2 u ∈ L2 (Ω)}. ∂x ∂y

H 2,ζ (Ω) = {u ∈ L2 (Ω) : rζ

If f ∈ L2,ζ (Ω), for some ζ < 1 with 1 − ζ 6= kπ θ , ∀k ∈ Z, then the unique 1 solution u ∈ H0 (Ω) permits the representation [22, 48, 73, 81] (see lemma 2.1 in [48]), u= (3.2) ∑ αk η(r)uk +U, 0 32 π, a symmetry assumption on the mesh is required such that the mesh is symmetric in a neighborhood of the singularity. A sufficient condition for this is given to be (S) the coarse mesh T0Ω is locally symmetric around the singularity (see figure 3.2). 17

3.1.2

Main results: the pollution effect and the optimal convergence

It is demonstrated in [48] that it is possible to overcome the pollution effect using a simple local modification of the bi-linear form. Theorem 2.2 in [48] illustrates that an accurate representation of the error in energy is a necessary condition for fully reducing the error in weighted L2 -norms. Therefore, it is necessary to compute a good approximation of the energy in order to gain a fast convergence of the error in weighted L2 -norms. This holds for a wide class of Galerkin-type approximations as shown in the proof of the theorem. Theorem 3.1.1 (Theorem 2.2 in [48]: necessary condition). Let f ∈ L2,−ζ (Ω) for some ζ > −1 and assume that the modified Galerkin approximation satisfies ||u − R`m u||0,ζ = O(h2 ), where h = h` . Then, |a(u, u) − a` (R`m u, R`m u)| = O(h2 ).

(3.5)

The term a(u, u) − a` (R`m u, R`m u) is referred to as the energy defect function and it will be used frequently later on. Theorem 2.3 in [48] is a consequence of this result that illustrates that the pollution effect is a global phenomenon and shows that even if the error is measured in weighted norms, the standard Galerkin approximation is sub-optimal. This means that it is not possible to use the weighted norms as a tool to overcome the pollution effect for the standard Galerkin approximation. Theorem 3.1.2 (Theroem 2.3 in [48]: pollution effect). Let u be the solution of (2.1) with f ∈ L2,−ζ (Ω) for some ζ > −1, and assume that the stress intensity factor α1 6= 0 (see (3.1)) when ζ > θπ − 1. Then, ||u − R` u||0,ζ ||∇(u − R` u)||2L2 (Ω) h2π/θ , where h = h` is the step-size at the grid level `. The main result is given in theorem 2.4 of [48], where using an appropriate correction a` (·, ·) in the definition of the Galerkin method and a right-hand side f , that is sufficiently regular in a vicinity of the singular point, it is possible to recover a second order convergence in weighted norms even if the mesh is quasi-uniform. Theorem 3.1.3 (Theorem 2.4 in [48]: sufficient condition). Let f ∈ L2,−ζ (Ω) for some 1 − π/θ < ζ < 1, and assume that (C1)-(C3) are valid and that (S) holds if 3 ` 2 π < θ < 2π. If, in addition, the correction a (·, ·) satisfies a(u˜1 − R`m u˜1 , u˜1 − R`m u˜1 ) − a` (R`m u˜1 , R`m u˜1 ) = O(h2 ), 18

where h = h` , then convergence rates of the optimal order hold, i.e., ||∇(u − R`m u)||0,ζ h || f ||0,−ζ and ||u − R`m u||0,ζ h2 || f ||0,−ζ . Theorem 2.2 and theorem 2.4 in [48] give a condition that is necessary and sufficient for the correction technique to get rid of the pollution effect, and this condition is a full convergence of the energy defect. It is proved that it is possible to use a simple correction of the form (3.3) to satisfy the conditions in theorem 2.4 using a suitably chosen modification parameter 0 ≤ γ ≤ 12 with the condition `

that Ω is sufficiently large. If the domain Ω has several re-entrant corners, then all the results in [48] hold and the correction is defined to be the sum of the local corrections for the individual singular points. 3.1.3

Construction of the correction and the modification parameter

It is shown in [48] that it is possible to construct a correction a` (·, ·) of the form (3.3) satisfying conditions (C1)-(C3) and the condition of the full convergence of the energy defect. Lemma 5.1 in [48] shows that a suitable parameter κ in (3.4), on which the domain Ω` depends, and that is independent of the step-size, exist. Lemma 5.2 in [48] proves the existence of a unique modification parameter 0 ≤ γ ` ≤ 21 depending on the step-size and being the root of the energy defect function; that is a(u˜1 − R`m u˜1 , u˜1 − R`m u˜1 ) − a` (R`m u˜1 , R`m u˜1 ) = 0, where u˜1 = η(r)u1 . The precise value of γ ` in lemma 5.2 in [48] is not practically required as proved in lemma 5.3 in [48], that shows that by choosing a modification parameter 0 ≤ γ ≤ 12 such that |γ − γ ` | = O(h2−2π/θ ), where h = h` , a full convergence of the energy defect is recovered. According to [48], the optimal value γ ` converge to an asymptotic value γ0 with |γ ` − γ0 | = O(h2−2π/θ ), where h = h` . With a correction a` (·, ·) of the form (3.3) and a level dependent fixed value of the modification parameter, it is possible to obtain the second order convergence. A similar observation was originally introduced by Zenger and Gietl [104] who recovered a second order convergence using a modified finite difference method with a fixed value of γ. 19

In the numerical results, the authors considered the Poisson problem −4u = 0

in Ω,

u = u1

on ∂ Ω,

considering Ω being a L-shaped domain with θ = 3π 2 or a slit domain with θ = 2π. By the usual procedure, this problem can be turned into (2.1) and the correction technique is applied and a second order convergence is recovered. The numerical results begin by considering symmetric meshes for both the L-shaped and the slit domains. The authors of [48] provide some examples of other kinds of meshes as the non-symmetric locally refined mesh and an example of a domain with several re-entrant corners.

3.2

Main idea of our work

According to [73], the general solution of (2.1) can be represented locally in the form u = ∑K k=1 αk uk , where αk ∈ R and K ∈ N. Therefore, we will study the pollution effect for the different singular components in the form K

g=

∑ αk uk ,

(3.6)

k=1

where αk ∈ R, K ∈ N. As a model problem we thus consider the Poisson problem with homogeneous right hand side, f = 0, but non-homogeneous boundary conditions −∆u = 0 in Ω, u = g on ∂ Ω.

(3.7)

In this work, we use an energy-correction technique, following [48, 94, 95, 104]. This energy-correction technique is presented in section 3.1. The advantage of this method is that it does not lead to major changes in the stiffness matrix and consequently the linear solvers can be applied without significant changes. We consider in section 4 a modification of the standard finite element method on quasi-uniform meshes, where the definition of the energy is locally modified in a neighborhood of the finite elements connected to the singularity. This is introduced formally in definition 4.1.2. This modification is referred to as energy correction and it does not alter the approximation space nor the structure of the stiffness matrix. The mesh structure, the matrix sparsity pattern, and all but a small number of the entries of the stiffness matrix remain unchanged. Hence, it is cheap and easy to implement into existing codes. 20

The authors in [48, 70] show that such a strictly local modification of the bilinear form of the finite element system in a vicinity of the singularity results in the full-order of convergence even in weighted L2 -norms. Optimal convergence rates are also recovered in L2 -norm on Ω \ Bε , where Bε stands for a ball centred at the tip τ of the re-entrant corner domain Ω having a fixed radius ε > 0 independent of the mesh-size. The energy correction is performed using a modification parameter that is determined as the root of the energy defect function [48] as presented in section 3.1 and as we will define it in definition 4.1.3. This thesis presents new numerical techniques to determine suitable modification parameters. As a first contribution, it studies the accuracy with which the modification parameter can be determined under perturbations, i.e., when the dominating singular function u1 is only weakly presented in the problem to be computed. In this case it is possible that no sufficiently accurate modification parameter γ can be found. We show that full accuracy can be recovered when we zoom in towards the singularity by choosing a suitable small auxiliary domain (see section 4.1 and section 4.2). The authors in [48, 94] evaluate the values of the modification parameters in terms of the energy defect function while we propose and use alternative defect functions. This is useful in situations, where the analytic knowledge of the solution or the exact computation of the energy is not available or requires more computational work. We use beside the energy defect function, in particular the discretization error measured point-wise. The different defect functions lead to modification parameters that do not necessarily have the same value, however they have asymptotically the same accuracy of the finite element solution. On the other hand, we use in section 4.3 two styles of the correction technique based on either the edges or the finite elements in a neighborhood of the singularity. Although the two different styles result in different modification parameters (even asymptotically), they produce the same quality of correction. The correction technique discussed in this work can be used if the domain Ω is a general re-entrant corner domain. The main finding in the numerical results in this thesis is to determine appropriate modification parameters and then show the efficiency of these modification parameters by solving the problem and computing the convergence rates of different defect functions and of the L2 - and the L∞ -norms of the discretization error measured in Ω \ S such that S is a fixed rectangle not depending on the step-size. We follow in this norm measure [70], where we exclude from the domain a fixed rectangle, instead of a ball with fixed radius, for simplicity. The modification parameter is appropriate to the correction technique if the convergence rates are of 21

second order [48, 95]. In this thesis, we determine the modification parameter as the root of a defect function using the Bisection method. The Bisection method is based on the fact that if F(x) is a continuous function on an interval [a, b] ⊂ R and F(a)F(b) < 0, then there exists c ∈ (a, b) such that F(c) = 0. Algorithm 3.2.1. Given a continuous function F(x) on an interval [a, b] ⊂ R and F(a)F(b) < 0. The Bisection algorithm is given as Do 1. c =

a+b 2 .

2. if F(a)F(c) < 0, then b = c else a = c while (F(c) > some tolerance). Different nested Newton strategies that can be used in determining the modification parameter are presented in [94]. These algorithms can be more efficient than the Bisection method. As a solver of the system of equations that result from the modified Galerkin approximation, we use the multigrid method, that is as a fast iterative solver is considered to be very efficient. The multigrid method is described in section 3.3. In section 4 and section 6, we provide the definition of the defect functions that are used to determine the modification parameter γ ` as the root of the defect function; noticing that these defect functions do not necessarily lead to the same modification parameter γ ` . In section 6, we use the finite difference discretization based on the the 5-point stencil, while in section 4, we use the standard finite element discretization based on the Friedrichs-Keller triangulation (see figure 3.1). In both cases, we use the multigrid method to solve the resulting system of equations with the Jacobi method as the smoother (see algorithm 3.2.2). For the ordering of the grid nodes, we consider the Red-Black ordering. In this ordering, the grid node (i, j) is colored red if i + j is even, and it is colored black if i + j is odd. Then, the values of the solution at the red grid nodes update the values at the black grid nodes. Algorithm 3.2.2. Jacobi method. for i = 1, . . . , n` for j = 1, . . . , n` ui j = ((h` )2 fi j + ui, j+1 + ui, j−1 + ui+1, j + ui−1, j )/4. 22

end end

Figure 3.1: Friedrichs-Keller triangulation [69] (page 64). We first begin by considering problem (3.7) with g = u1 (K = 1 and α1 = 1 in (3.6)). We determine the values of modification parameters that exhibit convergence rates of the different defect functions of order 2 in agreement with [48, 94–96]. We compute as well the convergence rates of the L2 -norm and the L∞ -norm of the discretization error measured in Ω \ S, that are of second order, where we exclude from the domain Ω a rectangle S that contains the node of singularity (0, 0) (see [48, 70, 94]). We determine the values of the modification parameter γ ` , that is dependent on the grid level `, using the bisection method. For determining the asymptotic value of the modification parameter as h` → 0, we use the Richardson extrapolation to find the extrapolated values γ `∗ . The Richardson extrapolation will be defined later in (4.40). The values of the modification parameter γ ` that we compute are in a good agreement with the values (γ0 = 0.13802 for the Lshaped domain and γ0 = 0.28187 for the slit domain) determined in [48] as we see in section 4 in figure 4.8 and figure 4.13 using a symmetric triangulation. Since the values of the modification parameter γ ` determined at the fine grid levels are valid for the correction at the coarser levels, we use for the correction at each grid level ` a fixed value that is the extrapolated modification parameter γ `∗ determined at the finest grid level. We observe in our numerical results that any 23

perturbation of the function g = u1 defined on the boundary causes a perturbation of the values of the modification parameter γ ` . A large perturbation of the values of the modification parameters means that the resulting convergence rates of the error norms of the different defect functions are not of second order. We provide the most important result in section 4.1 and section 4.2, where we prove theoretically in theorem 4.2.4 that if g = ∑K k=1 αk uk , for αk ∈ R, then by shrinking the domain Ω = Ω1 to smaller similar domains Ωs and determining the ` , corresponding modification parameters γs` , then we find that lims−→∞ γs` = γ1,1 ` is the modification parameter determined for the original domain such that γ1,1 Ω = Ω1 and considering g = u1 (K = 1 in (3.6) with α1 = 1). Theorem 4.2.4 is confirmed by our numerical results in figure 4.12, where the perturbation done to the function g = u1 is g = u1 + α2 u2 (K = 2 in (3.6) with α1 = 1) and α2 = {1, 2, 5, 10, 100}. Our methods are valid for any bounded polygonal re-entrant corner domain Ω, although we use the L-shaped and the slit domain as examples in our numerical results. As a simple application to 2D, we provide in section 4.7 the solution of the Poisson equation in 3D using cylindrical coordinates and a solution with the third coordinate being constant and we can compute appropriate modification parameters, that are different from the ones determined in 2D, recovering second order convergence. We consider the L-shaped and the slit domains as specific examples. In section 5, we consider the Poisson equation with a singular solution u1 defined on a slit domain with an arbitrary tip τ and a slit line being parallel to the x-axis. We show numerically that the relation between the modification parameter γ ` and the tip τ of the slit domain is periodic with a period being the step-size h` . Therefore, the modification parameters determined if the tip of the slit domain is an inner grid node and the ones determined if the tip of the slit domain is (0, 0) are almost the same. We assume that the tip of the slit domain is not a grid node and is in a fixed relative distance from a grid node. This relative distance is considered to depend on h` , at each grid level. Then, we show that it is possible to determine an appropriate modification parameter recovering a second order convergence of the different defect functions and of the L2 - and the L∞ -norms of the discretization error measured in Ω \ S.

3.3

Multigrid method

Now, we describe briefly the multigrid method [32] that we use for solving the system of equation A` u` = f ` resulting from the discretization of problem (3.7).

24

As a first step, a smoothing iteration is applied to this system of equations with an initial guess with components 0 jj π , 0 ≤ j ≤ n` , 1 ≤ j0 ≤ n` − 1. v j = sin n These components are called Fourier modes. Here, n` is the number of grid nodes and j0 is called the wave number or frequency. The high-frequency modes are more oscillatory than the low-frequency modes. After applying the smoothing iteration, the error decreases rapidly and it begins to decrease very slowly after some iterations. The smoothing iteration causes a quick elimination of the highfrequency modes of the error and this is the reason of the initial decrease. The slow decrease of the error is a result of the existence of the low-frequency modes, that are less oscillatory or more smooth. After applying the smoothing iteration and due to the existence of the low-frequency modes, using coarse grids becomes a remedy. On the coarse grid, the smooth error becomes higher in frequency. This means that the low frequency modes on the fine grid become high frequency modes on the coarse grid. Then, relaxation or applying the smoothing iteration becomes more effective. The transfer operator I``−1 from the fine grid to the coarse ` grid is called restriction and the transfer operator I`−1 from the coarse grid to the fine grid is called interpolation. For more details about the construction of the transfer operators, we refer to [32]. Next, we present the algorithms of the twogrid method (TGM) and the multigrid method (MGM), that is a recursive two-grid method. Algorithm 3.3.1 (Two-grid (TG) method). (u` ← T GM(A` , f ` , u0 )) Step 1. Pre-smoothing. ◦ Relax on A` u` = f ` on Ω` with an initial guess u0 and compute u` . Step 2. Coarse-grid correction. 1. Compute the residual r` = f ` − A` u` . 2. Restrict the residual to the coarser grid r`−1 = I``−1 r` . 3. Solve the coarse-grid residual equation A`−1 e`−1 = r`−1 directly on Ω`−1 . ` e`−1 . 4. Interpolate the error to the fine grid e` = I`−1

5. Correct the fine-grid solution u` ← u` + e` . 25

Step 3. Post-smoothing. ◦ Relax on A` u` = f ` on Ω` with an initial guess u` . In the two-grid method, the residual equation is solved directly at the coarsegrid using a direct solver as the Gaussian elimination. In the multigrid method, the residual equation is solved using the two-grid method at each coarse grid such that this coarse grid is not the coarsest one. At the coarsest grid level, the residual equation is solved directly. Algorithm 3.3.2 (Multigrid (MG) method). (u` ← MGM(A` , f ` , µ, u0 )) Step 1. Pre-smoothing. ◦ Relax on A` u` = f ` on Ω` with an initial guess u0 and compute u` . Step 2. Coarse-grid correction. 1. Compute the residual r` = f ` − A` u` . 2. Restrict the residual to the coarser grid r`−1 = I``−1 r` . 3. Solve the coarse-grid residual equation A`−1 e`−1 = r`−1 . If Ω`−1 is the coarsest grid, solve e`−1 = (A`−1 )−1 r`−1 , else set initial guess e`−1 = 0 and call e`−1 = MGM(A`−1 , f `−1 , µ, e`−1 ) µ times. ` e`−1 . 4. Interpolate the error to the fine grid e` = I`−1

5. Correct the fine-grid solution u` ← u` + e` . Step 3. Post-smoothing. ◦ Relax on A` u` = f ` on Ω` with an initial guess u` .

3.4

Finite element method

The weak solution of the Poisson equation (3.7) is defined by: find u ∈ HE1 (Ω) such that ∀v ∈ H01 (Ω), aΩ (u, v) = 0, (3.8)

26

where the standard bi-linear form aΩ (·, ·) := a(·, ·) is defined in (2.4) and the solution space HE1 (Ω) is defined by [50] HE1 (Ω) = {u ∈ H 1 (Ω) : u = g on ∂ Ω}, such that the function g is defined in (3.6). Let T0Ω denote a regular triangulation of the domain Ω that is locally symmetric around the singular point (cf. figure 3.2), and let {T`Ω }` , as defined in section 2, be a sequence of uniform refinements of the coarse mesh T0Ω that are obtained by subsequently dividing the triangles into four congruent ones (see figure 3.2), where ` is the refinement level. As presented in section 3.1, the main result in [48] (theorem 2.4), on which our methods are based, requires the condition that the mesh is symmetric in a neighborhood of the singular point, when the angle θ ≥ 3π 2 . Therefore, we assume ` that the triangulations {TΩ }` have the symmetry property locally. The symmetry property is not necessary, if θ < 3π 2 . ` ⊂ H 1 (Ω) is Using the Galerkin finite element method, we assume that VΩ,0 0 `

a finite dimensional vector space with a nodal basis {φΩ1 , . . . , φΩN } of continuous functions that are piece-wise linear polynomials in the triangulation T`Ω and that vanish at the boundary ∂ Ω. We extend this basis by defining additional functions φΩN

` +1

N ` +N∂`

, . . . , φΩ

`

N ` +N∂`

N +1 , and selecting fixed coefficients uΩ , . . . , uΩ

N ` +N∂` j uΩ ∑ j=N ` +1

, so that

j φΩ

the function interpolates the boundary data g on ∂ Ω at the nodal values. The Galerkin approximation of (3.8) is given by: find u` ∈ VΩ` such that ∀v` ∈ VΩ` , aΩ (u` , v` ) = 0. (3.9) The finite element solution u`Ω ∈ VΩ` is uniquely associated with the vector ` u`Ω = [uiΩ ]N i=1 of real coefficients such that u`Ω

N`

=

∑

j uΩ

j φΩ +

N ` +N∂`

∑

j

j

uΩ φΩ .

(3.10)

j=N ` +1

j=1

Following [48, 94], the energy-corrected Galerkin approximation then reads: find u` ∈ VΩ` such that ∀v` ∈ VΩ` , aΩ (u` , v` ) − aΩ` (u` , v` ) = 0, 27

(3.11)

where Ω` is the union of triangular elements adjacent to the re-entrant corner. This is explained in more details in definition 4.1.1 and figure 4.1. Additionally, we study the sensitivity of the correction method when the data are perturbed and the exact singular function u1 is not available. It is shown in section 4.2 that the effect of such perturbations can be reduced by choosing suitably small auxiliary sub-domains on which the modification parameter is computed. As illustrated in figure 3.2, where we considered the L-shape domain as a demonstrating example of Ω, we define a sequence of domains {Ωs , s = 1, 2, 3, . . .} such that Ω1 = Ω and for s > 1, Ωs+1 = F(Ωs ) where F is a scaling map r (3.12) F(r, ϕ) = ( , ϕ). 2 We redefine problem (3.7) such that the domain is Ω = Ωs , s ∈ N instead of Ω. The weak formulation on Ωs is then given by: find u ∈ HE1 (Ωs ) such that ∀v ∈ H01 (Ωs ) as (u, v) = 0, (3.13) where as (·, ·) := aΩs (·, ·). 1 2s

1 2s−1

L τ

L τ

1 2s

1 2s−1

T`s+1

T`s

Figure 3.2: Graphical illustrations of local domains for the initial and once refined meshes T`s and T`s+1 of the domains Ωs and Ωs+1 for some s, for an L-shaped domain Ω. The bisection axis L illustrates the mirror symmetry property. For each s ∈ N, and each level `, let T`s := T`Ωs be a uniform refinement of S T0Ωs . At a fixed refinement level `, T`s+1 = {F(T ) : T ∈ T`s } and for all s, the 28

triangulations T`s have the same number of triangular elements and nodes. For the ease of notations, we use the subscript s instead of Ωs . The finite element ` solution u`s ∈ Vs` := VΩ` s is uniquely associated with the vector u`s = [uis ]N i=1 of real coefficients such that N`

u`s

=

∑

usj

φsj +

N ` +N∂`

∑

usj φsj .

(3.14)

j=N ` +1

j=1

`

The finite element approximation of (3.13) is given by: find u1s , . . . , uN s such that ` for i = 1, . . . , N , N`

∑

usj as (φsj , φsi ) = −

N ` +N∂`

∑

usj as (φsj , φsi ).

(3.15)

j=N ` +1

j=1

This results in the sequence of linear systems A`s u`s = fs` , s = 1, 2, . . . ,

(3.16)

with the stiffness matrices `

i, j j i ` A`s = [ai,s j ]N i, j=1 , as = as (φs , φs ), i, j = 1, . . . , N ,

(3.17)

and the right hand side vectors N`

fs` = [ fsi ]i=1 , fsi = −

N ` +N∂`

∑

usj as (φsj , φsi ), i = 1, . . . , N ` .

(3.18)

j=N ` +1

3.5

Finite difference method

In section 6, we consider the correction technique [48, 104] using the standard central finite difference scheme with the five-point stencil to solve (3.7) for special choices of g. We point out that the finite element method for the Friedrichs-Keller triangulation shown in figure 3.1 results in the same linear system as a five-point stencil scheme. In other words, the finite element discretization of the Poisson equation with a uniform triangulation (cf., figure 3.1) and linear trial and test functions is equivalent to a finite difference discretization based on a five-point stencil and the linear system of equations obtained from the two discretizations are the same [33, 39, 40]. 29

Our numerical results show that the modification parameters determined using the finite element method based on the Friedrichs-Keller triangulation and the modification parameters determined using the finite difference method based on the five-point stencil are almost the same as we see from table 4.1 and table 6.1 considering the L-shape domain and table 4.2 and table 6.2 considering the slit domain.

4

Correction technique using finite element discretization

In this section, we study the correction technique that overcomes the pollution effect of the re-entrant corners considering (3.7) with g given by (3.6). We introduce the necessary notations and definitions in section 4.1, where we define the correction term a`s (·, ·), the energy defect function g`s , and the modification parameter γs` such that ` refers to the grid level and s refers to the index of the shrunk auxiliary sub-domains {Ωs : s = 1, 2, . . .} of Ω = Ω1 . We provide our main theoretical results in section 4.2, where we show that it is possible to determine accurate modification parameters even under perturbations by zooming in into the sequence of the auxiliary domains {Ωs }s . We display in section 4.3 simplified schemes of the correction technique. These techniques are used to determine finite element solutions that are as accurate as the methods discussed in [48] as shown in the numerical results in section 4.5 and section 4.6. In section 4.4, we define different defect functions other than the energy defect function defined in section 4.1 and discuss the advantages and disadvantages of each one of them. We provide our numerical results in section 4.5 for an example of a re-entrant corner domain that is the L-shaped domain, though the methods introduced and presented are valid for any re-entrant corner domain. In section 4.6, we provide the slit domain as another example for a re-entrant corner domain different from the L-shaped domain, using the same methods of correction. We implemented the correction technique using the Matlab software iFEM [38].

4.1

Modified finite element approximation

Definition 4.1.1. The correction term is defined for v, w ∈ H 1 (Ωs ) as a`s (v, w) =

Z Ω`s

∇v ∇w dΩ`s ,

30

(4.1)

where Ω`s = {T ∈ T`s , τ ∈ ∂ T } is a union of the elements of the triangulation T`s of the domain Ωs at the grid level ` in a O(h` )-neighborhood of the node of singularity τ. Figure 3.2 shows the triangulation T`s and the shaded part in figure 4.1 shows the local neighborhood Ω`s of the node of singularity. From the definition of the mapping F, we have Ω`s+1 = F(Ω`s ). S

1 2s

1 2s−1

L τ

L τ

1 2s

1 2s−1

Ω`s+1

Ω`s

Figure 4.1: The shaded triangles represent, for L-shaped domains, the construction of Ω`s and Ω`s+1 after the first refinement (` = 1). The correction is supported only in the local neighborhoods Ω`s and Ω`s+1 of the singular point, as indicated in the shaded triangles for Ωs and Ωs+1 , respectively. Definition 4.1.2. The modified finite element solution u`s,γ ∈ Vs` is defined following [94] such that the in-homogeneous boundary conditions u`s,γ (P` ) = g(P` ) are satisfied for all vertices P` of T`s being on ∂ Ωs and as (u`s,γ , v` ) − γ a`s (u`s,γ , v` ) = 0, for all v` ∈ Vs` ∩ H01 (Ω).

(4.2)

Note that u`s,0 = u`s is the standard finite element solution of the weak form (3.13). If αk = 1, αt = 0, for k,t = 1, . . . , K,t 6= k we denote the modified finite element solution by S`s,γ uk , where in this case g = uk and the analytic solution is the singular function uk .

31

The definition of the modified finite element approximation results from reducing the coefficient of the Laplacian to 1 − γ and consequently relaxing the stiffness of the problem in the one-element neighborhood of the node of singularity [48]. Definition 4.1.3. The energy defect function is defined as g`s (γ, u) = as (u, u) − as (u`s,γ , u`s,γ ) + γ a`s (u`s,γ , u`s,γ ),

(4.3)

The modified finite element solution u`s,γ depends on the exact solution u and the corresponding modification parameter γ. Therefore, the energy defect function g`s (γ, u) is a non-linear function of γ. We define the modification parameter γs` as the root of the energy defect function. Note that the value of γs` depends on the domain Ωs and the Dirichlet data imposed on the boundary. In our numerical results, if s = 1, then we drop the subscript s for simplicity and denote the modification parameter by γ ` . If g = uk as a special case of (3.6), we use the notation g`s,k (γ) := g`s (γ, uk ) and ` := γ ` . For example, if g = u (i.e. k = 1) and the domain considered is Ω γs,k 1 1 s ` (i.e. s = 1), then the modification parameter is denoted by γ1,1 . An analysis of the energy defect functions of the form (4.3) is presented in [94], ` ∈ (0, 0.5) of where it is shown in lemma 3.3 that there exists a unique root γ1,1 g`1 (γ, u1 ). According to remark 5.4 of [48] and section 4.2 of [94], we assume ` converge to an that for successively finer meshes, the modification parameters γ1,1 asymptotic value γ0 ; ` γ0 = lim γ1,1 . (4.4) `→∞

4.2

The modification parameter and the scaling relation between the energy defect functions

Lemma 4.2.1. Consider the model problem (3.7). Let γ ∈ (0, 1). Then for k, k0 = 1, . . . , K, i The finite element solution of (3.13) with g = uk satisfies kπ

u`s+1,γ (r, ϕ) = 2− θ u`s,γ (2r, ϕ).

(4.5)

ii Scaling relation of analytic energy. If g = uk in (3.7), then the bi-linear forms of analytic and approximate solutions on the different domains Ωs have the 32

scaling relations as+1 (uk , uk ) = 2− as+1 (uk , uk0 ) = 2−

2kπ θ

(k+k0 )π θ

as (uk , uk ) = 2−

2ksπ θ

as (uk , uk0 ) = 2−

as+1 (u`s+1,γ , u`s+1,γ ) = 2− =2

2kπ θ

a1 (uk , uk ),

(k+k0 )sπ θ

a1 (uk , uk0 ),

(4.6) (4.7)

as (u`s,γ , u`s,γ )

(4.8)

a1 (u`1,γ , u`1,γ ).

(4.9)

− 2ksπ θ

iii If g = uk in (3.7), then the correction term scales as a`s+1 (u`s+1,γ , u`s+1,γ ) = 2− =2

2kπ θ

a`s (u`s,γ , u`s,γ )

(4.10)

a`1 (u`1,γ , u`1,γ ).

(4.11)

− 2ksπ θ

`

Proof. i The modified finite element approximation is given by: find u1s,γ , . . . , uN s,γ such that for i = 1, . . . , N ` , N`

∑

j us,γ (as (φsj , φsi ) − γa`s (φsj , φsi )) = −

N ` +N∂`

∑

j

us,γ as (φsj , φsi ).

j=N ` +1

j=1

This results in a sequence of linear systems A`s,γ u`s,γ = fs`

(4.12)

with i, j

`

i, j

` j i ` j i A`s,γ = [as,γ ]N i j=1 , as,γ = as (φs , φs ) − γas (φs , φs ), i, j = 1, . . . , N , `

u`s,γ = [uis,γ ]N i=1 ,

(4.13) (4.14)

where A`s,γ is the modified stiffness matrix. Consider a triangular element 4 with vertices (x1 , y1 ), (x2 , y2 ), (x3 , y3 ) ∈ Ωs . Here, we denote the point (r, ϕ) as (x, y) in the Cartesian coordinates. Since F(r, ϕ) = ( 2r , ϕ), then ( 2x , 2y ) are the Cartesian coordinates of F(r, ϕ) and the

33

triangular element F(4) has the vertices ( x21 , y21 ), ( x22 , y22 ), ( x23 , y23 ) ∈ Ωs+1 . The `,F(4)

element or local stiffness matrices A`,4 and As+1 s A`,4 s

i, j = [as,4 ],

`,F(4) As+1

i, j as,4

Z

= 4

i, j = [as+1,F(4) ],

are given by [50]

∇φsi · ∇φsj d4, i, j = 1, 2, 3,

i, j as+1,F(4)

Z

(4.15) j

= F(4)

i ∇φs+1 · ∇φs+1 dF(4), i, j = 1, 2, 3.

(4.16) We define the 2 × 3 matrix x3 − x2 x1 − x3 x2 − x1 D= . y3 − y2 y1 − y3 y2 − y1 Then, the element stiffness matrix A`,4 is the matrix [61] (page 208) s A`,4 = s

D> D , 4 · Area(4)

where the area Area(4) of 4 is Area(4) =

|D| x1 (y2 − y3 ) + x2 (y3 − y1 ) + x3 (y1 − y2 ) = . 2 2

We can see that `,F(4) As+1

=

Thus,

1 > 4D D

4 · Area(F(4)) i, j

=

4·

1 > 4D D 1 4 Area(4)

= A`,4 s ,

i, j

as+1,4 = as+1,F(4) . and

(4.17)

j

i ), i, j = 1, . . . , N ` as (φsj , φsi ) = as+1 (φs+1 , φs+1

(4.18)

Similarly, j

i ), i = 1, . . . , N ` , j = N ` + 1, . . . , N ` + N∂` . (4.19) as (φsj , φsi ) = as+1 (φs+1 , φs+1

From (4.13), the modified local stiffness matrix A`,4 s,γ for an arbitrary ele`,4 `,4 ` ment 4 * Ωs is given by As,γ = As . Then, F(4) * Ω`s+1 and the modi`,F(4)

`,F(4)

`,F(4)

fied local stiffness matrix As+1,γ in Ωs+1 is As+1,γ = As+1 34

. Therefore, if

`,F(4)

4 * Ω`s , A`,4 s,γ = As+1,γ . ` If 4 ⊂ Ωs , then from (4.13), (4.15) and (4.16) A`,4 s,γ = [as,4 (γ)], as,4 (γ) = (1 − γ) as,4 (0), i, j = 1, 2, 3 i, j

`,F(4)

i, j

i, j

i, j

i, j

(4.20)

i, j

As+1,γ = [as+1,F(4) (γ)], as+1,F(4) (γ) = (1 − γ) as+1,F(4) (0), i, j = 1, 2, 3, (4.21) where as,4 (0) = as,4 and as+1,F(4) (0) = as+1,F(4) . From (4.17), A`,4 s,γ = i, j

i, j

i, j

i, j

`,F(4)

As+1,γ and A`s,γ = A`s+1,γ . In the next step, we show that a scaling relation between the right hand side that corresponds to the domains Ωs and Ωs+1 exists. So, we now compute the right hand side for each domain. From (3.18), N ` +N `

j

j

∂ fs` = [ fsi ], fsi = − ∑ j=N ` +1 us,γ as (φs , φsi ), i = 1, . . . , N ` .

Since Ωs+1 = F(Ωs ) and assuming that (r j , ϕ j ) are the grid nodes at the boundary ∂ Ωs+1 , for j = N ` + 1, . . . , N ` + N∂` , therefore from the boundary conditions g = uk , we have kπϕ j ), θ kπ kπ j kπϕ j = (2r j ) θ sin( ) = 2 θ us+1,γ . θ kπ

j

us+1,γ = r jθ sin( j

us,γ

Consequently and from (3.18) and (4.19) kπ

` fs+1 = 2− θ fs` .

(4.22)

Now, we have the two systems of equations A`s,γ u`s,γ = fs` , ` A`s+1,γ u`s+1,γ = fs+1 ,

which yield that the solutions of these two systems of equations are given by u`s,γ = (A`s,γ )−1 fs` , kπ

kπ

` u`s+1,γ = (A`s+1,γ )−1 fs+1 = (A`s,γ )−1 2− θ fs` = 2− θ u`s,γ ,

35

i respectively. We notice that φsi = φs+1 ◦ F, i = 1, . . . , N ` + N∂` , since for any j i (F(P j )) = δ = φ i (P j ), i, j = 1, . . . , N ` +N ` . Consequently, node Ps of T`s , φs+1 s ij s s ∂ kπ

from (3.14), u`s+1,γ = 2− θ u`s,γ . R

Ωs ∇uk · ∇uk

R

· ∇u r dr dϕ. Ωs ∇u # k " k kπ ∂ uk kπ θ −1 kπ sin kϕπ ∂r θ r θ Since uk = r θ sin kϕπ = , then ∇u = kπ k 1 ∂ uk kϕπ θ kπ θ −1 r cos r ∂ϕ θ θ 2 π 2 2kπ −2 kϕπ kϕπ k2 π 2 2kπ k 2 −2 2 Then ∇uk · ∇uk = θ 2 r θ (sin θ + cos θ ) = θ 2 r θ , so

ii First, as (uk , uk ) =

dx dy = "

# . that

2kπ k2 π 2 2kπ −1 k2 π 2 θ as (uk , uk ) = dr dϕ = r (2r) θ −1 2 dr dϕ 2 2 Ωs θ Ωs+1 θ Z 2kπ 2kπ k2 π 2 2kπ −1 =2 θ r θ dr dϕ = 2 θ as+1 (uk , uk ). 2 Ωs+1 θ

Z

Z

2kπ

This gives that as+1 (uk , uk ) = 2− θ as (uk , uk ) and consequently as+1 (uk , uk ) = 2ksπ 2− θ a1 (uk , uk ), by mathematical induction. Second, kϕπ kk0 π 2 (k+k0 )π −2 k0 ϕπ kϕπ k0 ϕπ θ (sin r sin + cos cos ) θ2 θ θ θ θ kk0 π 2 (k+k0 )π −2 (k − k0 )ϕπ θ r . = cos θ2 θ

∇uk · ∇uk0 =

Then kk0 π 2 (k+k0 )π −1 (k − k0 )ϕπ θ r cos dr dϕ 2 θ Ωs θ Z (k+k0 )π (k − k0 )ϕπ kk0 π 2 −1 θ = (2r) cos 2 dr dϕ 2 θ Ωs+1 θ Z (k+k0 )π kk0 π 2 (k+k0 )π −1 (k − k0 )ϕπ θ θ =2 r cos dr dϕ 2 θ Ωs+1 θ Z

as (uk , uk0 ) =

=2

(k+k0 )π θ

as+1 (uk , uk0 ).

(k+k0 )π

(k+k0 )sπ

So, as+1 (uk , uk0 ) = 2− θ as (uk , uk0 ) = 2 θ a1 (uk , uk0 ). kπ Third, from (4.5), u`s+1,γ (r, ϕ) = 2− θ u`s,γ (2r, ϕ). We find the relation between 36

the gradients " − kπ θ

∇u`s+1,γ = ∇(2

u`s,γ ) =

" =2

1− kπ θ

∂ ` ∂ (2r) us,γ 1 ∂ ` 2r ∂ ϕ us,γ

∂ − kπ θ u` ) (2 s,γ ∂r 1 ∂ − kπ θ u` ) s,γ r ∂ θ (2

#

# kπ

= 21− θ ∇u`s,γ .

Now, from the definition of the mapping F, as (u`s,γ , u`s,γ ) =

Z

∇u`s,γ · ∇u`s,γ r dr dϕ

Ωs

Z

=

2

2kπ θ −2

∇u`s+1,γ · ∇u`s+1,γ 2r 2dr dϕ

Ωs+1

=2

2kπ θ

as+1 (u`s+1,γ , u`s+1,γ ).

Consequently, 2kπ 2ksπ as+1 (u`s+1,γ , u`s+1,γ ) = 2− θ as (u`s,γ , u`s,γ ) = 2− θ a1 (u`1,γ , u`1,γ ). iii Using that the linear mapping F maps Ω`s onto Ω`s+1 and that kπ

∇u`s,γ = 2 θ −1 ∇u`s+1,γ , then the proof is very similar to part (ii). Corollary 4.2.2. Consider the model problem (3.7). Let γ ∈ (0, 1). Then for k, k0 = 1, . . . , K, i Scaling relation of the energy defect function. The energy defect function of problem (3.7) with g = uk has, on the different domains Ωs , the scaling relation g`s+1,k (γ) = 2−

2kπ θ

g`s,k (γ) = 2−

2ksπ θ

g`1,k (γ).

(4.23)

ii The energy defect function (4.3) of problem (3.7) with g = ∑K k=1 αk uk is given by g`s+1 (γ) =

K

∑ αk2 2−

2ksπ θ

k=1 K

+ ∑ αk ∑ αt 2 k=1

g`1,k (γ) −

(k+t)sπ θ

h a1 (uk , ut ) − a1 (S`1,γ uk , S`1,γ ut )

t6=k

i + γ a`1 (S`1,γ uk , S`1,γ ut ) . 37

iii Consider problem (3.7) with g = αk uk + αk0 uk0 , αk , αk0 ∈ R, then, from the linearity of the equation and the superposition principle, the discrete solution u`s,γ is given by u`s,γ = αk S`s,γ uk + αk0 S`s,γ uk0 . (4.24) The modified finite element solution S`s,γ uk of (3.7) with g = uk defined on the domain Ω = Ωs is defined in definition 4.1.2. Next, we consider the assumption in part (iii) of corollary 4.2.2 in more detail, where we compute the energy defect function and the limit at ∞ of a sequence of energy defect functions defined in the different domains Ωs , s = 1, 2, . . .. Theorem 4.2.3. For s ≥ 1, k > 1, consider the elliptic boundary value problem (3.7) with g = α1 u1 + αk uk defined on Ωs for α1 , αk ∈ R. Then, the energy defect function g`s+1 is given as g`s+1 (γ) = α12 2

−2sπ θ

g`1,1 (γ) + αk2 2

+ α1 αk 21−

(1+k)sπ θ

Consequently, lims→∞ 2

2sπ θ

−2ksπ θ

g`1,k (γ)

[a1 (u1 , uk ) − a1 (S`1,γ u1 , S`1,γ uk ) + γ a`1 (S`1,γ u1 , S`1,γ uk )]. g`s+1 (γ) = α12 g`1,1 (γ).

Proof. From corollary 4.2.2, the energy defect function is given by g`s+1 (γ) = α12 2

−2sπ θ

+ 2α1 αk 2

g`1,1 (γ) + αk2 2

−(1+k)sπ θ

−2ksπ θ

g`1,k (γ)

[a1 (u1 , uk ) − a1 (S`1,γ u1 , S`1,γ uk )

+ γ a`1 (S`1,γ u1 , S`1,γ uk )] Then, 2

2sπ θ

g`s+1 (γ) = α12 g`1,1 (γ) + αk2 2 + 2α1 αk 2

2sπ (1+k)sπ θ − θ

2sπ 2ksπ θ − θ

[a1 (u1 , uk ) − a1 (S`1,γ u1 , S`1,γ uk )

+ γ a`1 (S`1,γ u1 , S`1,γ uk )] This, lims→∞ 2

2sπ θ

g`1,k (γ)

g`s+1 (γ) = α12 g`1,1 (γ). 38

From [94], lemma 3.4 and lemma 4.1, there exists a refinement level `0 such that for ` ≥ `0 and γ ∈ [0, 21 ], (g`1,1 )0 (γ) ≥ 0 and (g`1,1 )00 (γ) ≥ 0, where the prime 0 stands for the derivative with respect to γ. Moreover, it is shown in [94] that h i (g`1,1 )00 (γ) = 2 a1 (S`1,γ u1 )0 , (S`1,γ u1 )0 − γ a`1 (S`1,γ u1 )0 , (S`1,γ u1 )0 , where S`1,γ u1 , as defined in definition 4.1.2, is the modified finite element solution of (3.7) with g = u1 (k = 1) defined on the domain Ω = Ω1 (s = 1), and (S`1,γ u1 )0 is the derivative of the modified finite element solution, S`1,γ u1 , with respect to γ. Our target is to prove the following theorem in which we add a technical assumption ` that is confirmed by our numerical results, where the modification that γs` ≥ γ1,1 ` are defined in definition 4.1.3. parameters γs` and γ1,1 Theorem 4.2.4. For a fixed refinement level `, an arbitrary s ∈ N, consider prob` ∈ (0, 1 ) such that lem (3.7). Assume that there exist γs` , γ1,1 2 ` g`s (γs` ) = 0 and g`1,1 (γ1,1 ) = 0.

(4.25)

` for all s ∈ N, then If γs` ≥ γ1,1

lim γ ` s→∞ s+1

` = γ1,1 .

(4.26)

Proof. From corollary 4.2.2, the energy defect function g`s+1 can be expressed as g`s+1 (γ) = 2− as

2sπ θ

K

Σ=

∑ αk2 2

α12 g`1,1 (γ) + Σ, where Σ := g`s+1 (γ) − 2−

−2ksπ θ

k=2 K

α12 g`1,1 (γ) is given

g`1,k (γ)

+ ∑ αk ∑ αt 2− k=1

2sπ θ

(k+t)sπ θ

i h a1 (uk , ut ) − a1 (Sk u`1,γ , St u`1,γ ) + γ a`1 (Sk u`1,γ , St u`1,γ ) .

t6=k

` ) = 2− By assumption, 0 = g`s+1 (γs+1

0 = 2−

2sπ θ

2sπ θ

` ) + Σ and g` (γ ` ) = 0, so α12 g`1,1 (γs+1 1,1 1,1

` ` α12 [g`1,1 (γs+1 ) − g`1,1 (γ1,1 )] + Σ.

(4.27)

Using Taylor expansion yields 0 = 2−

2sπ θ

` ` ` α12 [(γs+1 − γ1,1 ) (g`1,1 )0 (γ1,1 ) + R] + Σ,

39

(4.28)

where the integral form of the remainder R is given as R=

Z γ ` (g` )00 (γ) s+1 1,1 ` γ1,1

2!

` − γ) dγ. (γs+1

(4.29)

Then, 2sπ

` ` − γ1,1 γs+1

−2 θ α1−2 Σ − R = . ` ) (g`1,1 )0 (γ1,1

(4.30)

` − γ ` ≥ 0, then R ≥ 0, and From [94], (g`1,1 )00 (γ) ≥ 0. Therefore, since γs+1 1,1 ` ` 0 ≤ γs+1 − γ1,1

Σ0 ≤ ` 0 ` , (g1,1 ) (γ1,1 )

2sπ

where Σ0 := −2 3 α1−2 Σ. Next, we show that lims→∞ Σ0 = 0. The quantity Σ0 has the representation −1 K 2 − 2(k−1)sπ ` θ (∑ α 2 g1,k (γ) α12 k=2 k h i K (k+t−2)sπ + ∑ αk ∑ αt 2− θ a1 (uk , ut ) − a1 (Sk u`1,γ , St u`1,γ ) + γ a`1 (Sk u`1,γ , St u`1,γ ) ).

Σ0 =

k=1

t6=k

Since all terms in this sum converge to 0 as s → ∞, so lims→∞ Σ0 = 0. Then, ` − γ ` = 0. lims→∞ γs+1 1,1 We conclude from theorem 4.2.4 that in case of the existence of perturbations to the function defined at the boundary such that the solution is a sum of singular functions, then it is possible to determine accurate modification parameters by zooming in toward the singularity with a sequence of smaller auxiliary domains. In this case the accurate modification parameter is the limit of the sequence of the modification parameters determined at the different auxiliary domains.

4.3

Modification of the stiffness matrix

The representation of A`s u`s in (3.16) using the element stiffness matrices of (4.15) can be written as `,T (4.31) A`s u`s = ∑ A`,T s us , T ∈T`s

40

3×3 is the element stiffness matrix, 3 such that for each T ∈ T`s , A`,T s = [as,T ]i, j=1 ∈ R where ij as,T = aT (φsi , φsj ), i, j = 1, 2, 3 ij

1 2 3 ` i 3 and u`,T s = [us (PT )]i=1 , where PT , PT , PT are the vertices of the triangular element T. Let Es` = {E : E is an edge of an element T ∈ T`s and E ∈ / ∂ Ωs }. For each E ∈ ij 2 `,E ` Es , we define the local edge stiffness matrix As = [as,E ]i, j=1 ∈ R2×2 as ij

as,E = aT1 ∪T2 (φsi , φsj ), i, j = 1, 2. where T1 , T2 ∈ T`s such that E is a common edge of T1 and T2 . Then, it is possible to represent A`s u`s in an edge oriented fashion A`s u`s =

∑

`,E A`,E s us ,

(4.32)

E∈Es` ` 1 ` 2 1 2 where u`,E s = [us (PE ), us (PE )], and PE , PE are the endpoints of the edge E. Following the modification defined in [48,94], the representation of A`s,γ u`s,γ in (4.12) through the element stiffness matrices is

A`s,γ u`s,γ =

` `,T ωs,γ (T )A`,T s us ,

∑

(4.33)

T ∈T`s ` are given by where the weight functions ωs,γ ` ωs,γ (T ) =

1−γ 1

if T ∈ Ω`s , otherwise.

(4.34)

In equation (4.33) and (4.34), the variable γ refers to the modification parameter. The representation of A`s,γˆ u`s,γˆ through the local edge stiffness matrices is A`s,γˆ u`s,γˆ =

∑

`,E ωs,` γˆ (E)A`,E s us ,

(4.35)

E∈Es`

such that the weight functions ωs,` γˆ are given by   1 − cE γˆ ` ωs,γˆ (E) = 1 − γˆ  1 41

if E ∈ ∂ Ω`s \ ∂ Ωs , if E ∈ Es` , otherwise,

(4.36)

where Es` = {E : E is an edge of an element T ∈ Ω`s and E ∈ / ∂ Ω`s } and cE is the entry of the element stiffness matrix that represent the connection between the endpoints of the edge E. This is illustrated in figure 4.2(b) for the L-shaped domain and figure 4.3(b) for the slit domain, where the variable γˆ refers to the modification parameter. We see in figure 4.2(b) that if E ∈ ∂ Ω`s \ ∂ Ωs , then ˆ where cE = 12 for the triangulation of Friedrichs-Keller (see ωs,` γˆ (E) = 1 − cE γ, ij

figure 3.1) presented in this figure, and 12 is the value of as,T such that i 6= j and ˆ T ∈ T`s is any triangular element. If E ∈ Es` = {Pa τ, Pb τ}, then ωs,` γˆ (E) = 1 − γ. The modification parameter determined from the element oriented representation (4.34) and the edge oriented representation (4.36) are not the same. As an alternative to the modification technique described using the definition of the weight function from (4.36), we can use a simpler structure as it has already been proposed and used in older papers [104, 105]. Therefore, in this work, we define the ` as weight function ωs,γ ` ωs,γ (E) =

1−γ 1

if E ∈ Es` , otherwise.

(4.37)

This will turn out to be enough to compute modification parameters that result in a second-order convergence [104, 105] as we see in figure 4.9. This is illustrated in figure 4.2(a) for the L-shaped domain and figure 4.3(a) for the slit domain. To distinguish between the values of the modification parameters determined using (4.34), (4.36) and (4.37), we denote the values in (4.34) by γ, the values in (4.36) by γˆ and γ remains for the values in (4.37). The determined modification parameters γ and γˆ in figure 4.2 and figure 4.3, that lead to second-order convergence, differ in their values as in figure 4.8. Remark 4.3.1. According to the element representation of the stiffness matrix using a symmetric mesh as in figure 4.4 and figure 4.5, we provide the modified stencil representation of every grid node of the triangulation T`s . We consider the L-shaped and the slit domains in our numerical results as specific example, although the method is valid for a general re-entrant corner domain. Every grid

42

1

1

1

1

1

1

1

Pb

1

1

1−γ

1 Pc

1−γ

1 1

1

Pa

1

1

1

1

1

1

τ

1 1

Pd

1

1 1

1

1

(a)

1

1

1

1

1

1

1

Pb

1 − γˆ

1−

1 1 1

Pd

1

ˆ

1 − 2γ

1 − γˆ

1 1

1

Pa

1

1

1

τ

Pc ˆ

1 − 2γ

1 1

Pe

γˆ 2 ˆ

1 − 2γ

Pf

1 1

1

(b)

Figure 4.2: L-shape domain (θ = 3π 2 ): The representation of the stencil at triangular edges. In our work, we follow sub-figure (a). The correction technique used in [48, 94], correspond to the situation illustrated in sub-figure (b).

43

1

1

1

1

1

1

1

Pb

1

1−γ

Pd

1

Pf

1

1 Pe

1

1

1

1

1

1

1

1

τ

1

1 Pc

1−γ

1 1

1

1−γ

1 1

1

Pa

1

1

1

1

1

1

1

1

1

1

(a)

1

1

1

1

1

1

Pa

1

1

1 1

Pb

1 1

Pd

γˆ 2

1

Pc 1 − 2γ

τ

Pe

ˆ

1 − 2γ

Pf

1

1 1

1 1

1

ˆ

1 − γˆ

1 − γˆ

1

1

ˆ

1 − 2γ

1 − γˆ

1−

1

1

1 1

1 1

1 1

(b)

Figure 4.3: Slit domain (θ = 2π): The representation of the stencil at interior grid nodes. In our work, we follow the left sub-figure (a). In [48, 94], the authors present the correction technique clarified in sub-figure (b).

44

Pg

Pa

Pb Pd

τ

Pa

Pb Pd

Pe

Pf

Figure 4.4: L-shape domain (θ = Pg

Pc

3π 2 ):

A symmetric coarse mesh. Pc Pe

Pf

Ph

Figure 4.5: Slit domain (θ = 2π): A symmetric coarse mesh. node P ∈ / Ω`s will have the default 5-point stencil         0 0 −1 0  −1 1 0  +  0 1 −1  +  0 1 0  +  0 1 0  0 0 0 −1   −1  −1 4 −1  . = −1 The stencils are modified for each P ∈ ∂ Ω`s \∂ Ωs . This means that for an L-shaped domain, we modify the stencil at P ∈ {Pa , Pb , Pc , Pd , Pg } The modified stencil of the node Pa above the singularity is       0 0 −1 ` ` γˆ γˆ Pa : (1 − )  −1 1 0  + (1 − )  0 1 −1  +  0 1 0  2 2 0 0 0     −1 0 ` ` ` ˆ γ + (1 − γˆ )  0 1 0  =  2 − 1 4 − 2γˆ` γˆ2 − 1  . −1 γ` − 1 45

For the grid node beside Pb the node of singularity, this is the modified stencil       −1 0 0 ` γˆ Pb :  −1 1 0  + (1 − γˆ` )  0 1 −1  + (1 − )  0 1 0  2 0 0 0     ` γˆ 0 2 −1 γˆ`     + (1 − ) 0 1 0 =  −1 4 − 2γˆ` γˆ` − 1  . 2 γˆ` −1 −1 2

The modified stencil for the grid node Pc is       0 0 −1 ` γˆ Pc : (1 − )  −1 1 0  +  0 1 −1  +  0 1 0  2 0 0 0     −1 0 ` γˆ  ˆ`  + (1 − )  0 1 0  =  γ2 − 1 4 − γˆ` −1  . 2 γˆ` −1 2 −1 For the grid node Pd , the modified stencil is       0 0 −1 ` ` γˆ γˆ Pd :  −1 1 0  + (1 − )  0 1 −1  + (1 − )  0 1 0  2 2 0 0 0     ` γˆ 0 2 −1     + 0 1 0 =  −1 4 − γˆ` γˆ` − 1  . 2 −1 −1 The modified stencil for the grid node Pg is    0 ` γˆ Pg :  −1 1 0  + (1 − )  0 2 0    0 γˆ`  + (1 − )  0 1 0  =  2 −1 46

   0 −1 1 −1  +  0 1 0  0 0  −1  ˆ` −1 4 − γˆ` γ2 − 1  . γˆ` 2

−1

For the slit domain, we do the same modification of the stencils at the grid nodes {Pa , Pb , Pc , Pd , Pg } and add stencil modification to the two grid nodes {Pf , Ph }. We modify the stencil of the node Pf below the node of singularity as       0 0 −1 ` ` γˆ γˆ Pf : (1 − )  −1 1 0  + (1 − )  0 1 −1  + (1 − γˆ` )  0 1 0  2 2 0 0 0     ` γˆ − 1 0 ` ` ˆ +  0 1 0  =  γ − 1 4 − 2γˆ` γˆ − 1  . 2 2 −1 −1 The grid node Ph has the modified stencil       0 −1 0 ˆγ ` ˆγ ` Ph : (1 − )  −1 1 0  +  0 1 −1  + (1 − )  0 1 0  2 2 0 0 0     ` γˆ 0 2 −1   +  0 1 0  =  γˆ` − 1 4 − γˆ` −1  . 2 −1 −1 For the Friedrichs-Keller triangulation as in figure 4.2(b) and figure 4.3(b), the modified stencils at the grid nodes of ∂ Ω`s \ ∂ Ωs have different representation. First, we consider the L-shaped domain. For the grid node Pa , the stencil is modified as       0 0 −1 ` γˆ Pa :  −1 1 0  + (1 − )  0 1 −1  +  0 1 0  2 0 0 0     −1 0 ` ` + (1 − γˆ` )  0 1 0  =  −1 4 − 32γˆ γˆ2 − 1  . −1 γˆ` − 1

47

The modified stencil at Pb is    0 Pb :  −1 1 0  + (1 − γˆ` )  0 0    0 γˆ`  + (1 − )  0 1 0  =  2 −1

   0 −1 1 −1  +  0 1 0  0 0  −1  ˆ` −1 4 − 32γ γˆ` − 1  . γˆ` 2

−1

For the grid node Pc , the modified stencil is       0 0 −1 ˆγ ` Pc : (1 − )  −1 1 0  +  0 1 −1  +  0 1 0  2 0 0 0     −1 0 γˆ`   ˆ` + (1 − )  0 1 0  =  γ2 − 1 4 − γˆ` −1  . 2 γˆ` −1 2 −1 The modified stencil for the grid node Pd is       0 0 −1 ` ` γˆ γˆ Pd :  −1 1 0  + (1 − )  0 1 −1  + (1 − )  0 1 0  2 2 0 0 0     ` γˆ 0 2 −1     + 0 1 0 =  −1 4 − γˆ` γˆ` − 1  . 2 −1 −1 For the slit domain, we use the same stencil modification as in the L-shaped domain except that we add the stencil modification at the grid node Pf as       0 0 −1 ` γˆ Pf : (1 − )  −1 1 0  +  0 1 −1  + (1 − γˆ` )  0 1 0  2 0 0 0     γˆ` − 1 0 ` ` +  0 1 0  =  γˆ − 1 4 − 3γˆ −1  . 2 2 −1 −1 48

For each s ∈ N, assume that the sequence {T`s }` of triangulations of the domain Ωs are considered in the sense of Friedrichs-Keller. If 4 ∈ Ω`s , then from (4.20), the modified local stiffness matrix A`,4 s,γˆ is given by `,4 ˆ `,4 A`,4 s,γˆ = As − γ As ,

where  A`,4 = s

1 −1 2 −1 2

−1 2 1 2

0

 − 12 0 . 1 2

`,4 If 4 ∈ / Ω`s , then A`,4 s,γˆ = As . For an L-shaped domain Ω, the contributions of the edges of the triangular elements to the global stiffness matrix are illustrated in figure 4.2(b), where all the edges that do not belong to Ω`s have a contribution with weight 1. If an edge of a triangular element is a common edge between an element in Ω`s that gives a ˆ contribution with weight 12 − 2γ and an element outside Ω`s , that gives a contribution ˆ with weight 21 , then the sum 1− 2γ is the corresponding entry’s weight of the global stiffness matrix. If an edge of a triangular element is a common edge between two triangles in Ω`s , then its contribution in the global stiffness matrix has weight ˆ The grid nodes that does not belong to Ω`s have an entry 4 in the global 1 − γ. ˆ stiffness matrix. The contributions of the grid nodes Pa and Pb are 4 − 32γ , while ˆ the contributions of Pc and Pd are 4 − γ. However, in our computations in this thesis (if not stated otherwise), we modify only the contributions of the edges Pa τ and Pb τ and the grid nodes Pa and Pb in the global stiffness matrix as shown in figure 4.2(a). The contributions of the nodes Pa and Pb will be modified from 4 to 4 − γ. The contributions of the edges Pa τ and Pb τ have weights 1 − γ instead of 1. This turned out to be enough to compute modification parameters that result in a second order convergence [104,105]. The determined modification parameters γ in figure 4.2(a) and γˆ in figure 4.2(b) are not the same. If Ω is a slit domain, the contributions of the edges of the triangular elements to the global stiffness matrix is illustrated in figure 4.3, where sub-figure 4.3(b) follows the modification discussed in [48, 94] and sub-figure 4.3(a) illustrates the modification discussed in this thesis. ` The modified local stiffness matrix A`,4 s,γ , for 4 ∈ Ts in our computations is

49

given by `,4 `,4 A`,4 s,γ = As − γBs ,

where B`,4 is given if 4 = Pa τPc by s  B`,4 = s

−1 2

−1 2 1 2

0

0

1

 0 0 . 1 2

If 4 = τPa Pb , then B`,4 = A`,4 and if 4 = Pb Pd τ, then s s   1 0 −1 2 B`,4 =  0 12 0  . s −1 0 12 2 If Ω is an L-shaped domain, then for any other element 4 in Ω`s , B`,4 = 0. If s Ω is a slit domain, then we modify the other two elements in Ω`s as follows. If 4 = Pf τPd or 4 = τPf Pe , then  B`,4 = s

−1 2

−1 2 1 2

0

0

1

 0 0 . 1 2

Then for any other element 4 in Ω`s , B`,4 = 0. s

4.4

Defect functions `, j

Definition 4.4.1. The modification parameter γs is a level-dependent value that `, j is determined alternatively as the root of the defect functions gs (γ), j = 1, 2, 3, 4 that are defined as ` g`,1 s (γ) = u(P) − us,γ (P), ` g`,2 s (γ) = gs (γ), `−1 `−1 `−1 `−1 `−1 ` ` ` ` ` , u ) , , u ) − a (u , u ) − γa (u , u ) − γa (u g`,3 (γ) = a (u s s,γ s s,γ s,γ s,γ s,γ s s,γ s s,γ s,γ s ` `−1 g`,4 s (γ) = us,γ (P) − us,γ (P),

(4.38) 50

where u is the analytic solution of (3.7), u`s,γ is the modified finite element solution at the level `, P is a grid node at a fixed location in the domain. The grid node P is chosen to be far enough from the node of singularity. For simplicity, we omit the second superscript in the notation of the modification `, j parameter γs , j = 1, 2, 3, 4 and we denote the modification parameter instead by γs` ; knowing that the modification parameter γs` depends on the respective defect function. ` The defect function g`,1 s is the discretization error u(P) − us,γ (P) measured `,3 at the node P and g`,2 s is the energy defect function. The defect function gs is the difference between the modified discrete energies as (u`s,γ , u`s,γ ) − γa`s (u`s,γ , u`s,γ ) `,4 `−1 `−1 `−1 `−1 and as (u`−1 s,γ , us,γ ) − γas (us,γ , us,γ ) at the grid levels ` and ` − 1 while gs is the difference between the modified finite element solutions u`s,γ (P) and u`−1 s,γ (P) measured at the grid node P at the grid levels ` and ` − 1. The defect function g`,3 is provided in [94] as an alternative characterization of the modification parameter γs` such that the energy defect function g`,2 s is approximately required to be the same at two consecutive grid levels ` and ` − 1. For the energy defect function g`,2 s , the analytic solution has to be known and the analytic energy has to be computed. Also, g`,2 s may require complicated analytic calculations that may be difficult or impossible. The defect function g`,1 s does not need any analytic calculations, however, the analytic solution has to be known. So, using the defect function g`,1 s is an advantage if we know the analytic solution and do not want to go through the analytic calculations of the energy. But if the analytic solution is not available, we can not use this defect function in our computations. Using the defect function g`,3 s does not require the complexity of the analytic calculation of the energy that might appear, but we have to go through the computation of the discrete energy that is more expensive than only computing the finite element `,4 solution as in g`,4 s . The defect function gs is computationally the cheapest one because it uses only the modified finite element solution measured point-wise. The four defect functions do not necessarily result in the same root. Different modification parameters are determined at each grid level using the bisection method (see algorithm 3.2.1) applied to each one of the four defect functions. Therefore, the modification parameter is determined in four different methods; called method M1, method M2, method M3 and method M4. Method M1 `,2 uses g`,1 s in determining the modification parameter, method M2 uses gs , method `,3 `,4 M3 uses gs and method M4 uses gs . To measure whether the value of the modification parameter is appropriate

51

for the correction technique, we compute the so called convergence rate. The convergence rate is defined to be the logarithm to base 2, log2 , of the ratio of a chosen function between two grid levels ` − 1 and `. In the optimal case, if an appropriate value of the modification parameter is determined, the convergence rate is of second order [48, 94, 95] - as in the regular case, where the solution is smooth. In our numerical results, we compute the convergence rates of the error norms, ||u − u`s,γ ||L2 (Ω\S) and ||u − u`s,γ ||L∞ (Ω\S) , and the convergence rates of different defect functions that are defined in definition 4.4.1. The defect functions are also defined later in definition 6.1.2 using the central finite difference method. The sub-domain S is chosen in our numerical results to be a fixed rectangular neighborhood of the singular point that does not depend on the step-size [70]. Definition 4.4.2. The convergence rate of a sequence of functions F ` , ` = 2, 3, . . . is defined as `−1 F . (4.39) Convergence rate = log2 F` When we compute the convergence rates considering F ` to be one of the defect functions, we use the value γs` of the modification parameter determined at the finest grid level as an approximation of the asymptotic value. An extrapolated value of the modification parameter γs`∗ is determined using the Richardson extrapolation as γs`∗ = β γs` − (β − 1)γs`−1 .

(4.40)

p

We set the extrapolation parameter β = 2 p2−1 . The value p represents the convergence rate of the modification parameter and it is predicted in [48] to be p = 2− 2π θ for the energy defect function g`,2 and we will use this value as a fact for the other defect functions. So since θ = 3π 2 for the L-shaped domain, we assume in our 2 experiments that p = 3 and for the slit domain (θ = 2π), we assume that p = 1. Alternatively, the value of p can be approximated numerically from `−1 γs − γs` p = − log2 `−2 . (4.41) γs − γs`−1

4.5

L-shaped domain

The target of this section is to first determine appropriate modification parameters and then use these values to solve the considered problem. An appropriate value of 52

the modification parameter recovers a second-order convergence of the different defect functions as well as the L2 - norm, ||u − u`1,γ ||L2 (Ω\S) , and the L∞ -norm ||u − u`1,γ ||L∞ (Ω\S) || of the discretization error measured in Ω \ S [48, 70, 94]. Let us consider the model problem (3.7) defined on an L-shaped domain Ω Ω = ((−1, 1) × (−1, 1))/([0, 1] × [−1, 0]).

(4.42)

` by We study (3.7) with g = u1 and determine the modification parameter γ1,1 finding the root of each of the defect functions (4.38). As already shown previously, e.g. in [48, 94], this correction technique can be used on each level ` to compute modified finite element solutions that do not suffer from the pollution effect and that exhibit second order convergence away from the singularity. We want to see whether a perturbation of the function defined on the boundary affects the value of the modification parameter. Based on this, we study the case when g is perturbed and in particular, we use g = u1 + αk uk for k = 1, . . . , K such that αk ∈ R is large. This perturbation of g results in a perturbation of the modification parameter, that means that the perturbed values of the modification parameters are not capable to recover a second order of convergence of the values of the defect functions and the error norms ||u − u`1,γ ||L2 (Ω\S) and ||u − u`1,γ ||L∞ (Ω\S) ||. Increasing the value of αk leads to a higher perturbation in the modification parameter that will not be appropriate for the correction; in the sense that the convergence rates of the defect functions and the error norms ||u − u`1,γ ||L2 (Ω\S) and ||u − u`1,γ ||L∞ (Ω\S) || are not of second order. We then consider (3.7) defined in the auxiliary shrunk domains Ωs , s ∈ N, of Ω. The determined modification parameter γs` are closer in ` when the domain is shrunk, i.e. when s is large enough. We proved value to γ1,1 ` as s theoretically in theorem 4.2.4 that the modification parameter γs` tends to γ1,1 tends to ∞. This is confirmed numerically in figure 4.12, for s = 1, . . . , 14. Let us first assume that g = u1 .

The analytic solution is u = u1 . ` and the extrapolated values γ `∗ of the modiTable 4.1 displays the values γ1,1 1,1 fication parameter at the different grid levels ` using the value p = 23 as predicted from the literature [48]. The values of the modification parameter are determined using the four methods of correction. For method M1, the first defect function g`,1 is measured at the grid node P = ( 34 , 34 ) that is far enough from the singular point (0, 0). However, we compute the convergence rates of g`,1 measured at arbitrary chosen grid nodes P` as shown in figure 4.6. 53

` . Table 4.1: L-shape domain (FE): Extrapolated modification parameters γ1,1 Method M1 Method M2 ` `∗ ` `∗ ` γ1,1 γ1,1 γ1,1 γ1,1 1 1.811790e-01 1.939932e-01 2 1.869812e-01 1.950618e-01 3 1.906433e-01 1.968777e-01 1.957355e-01 1.968823e-01 4 1.929626e-01 1.969111e-01 1.961594e-01 1.968810e-01 5 1.944580e-01 1.970037e-01 1.964264e-01 1.968810e-01 6 1.953125e-01 1.967672e-01 1.965942e-01 1.968800e-01 Method M3 Method M4 ` `∗ ` `∗ ` γ1,1 γ1,1 γ1,1 γ1,1 1 1.911913e-01 1.661072e-01 2 1.932895e-01 1.773300e-01 3 1.946185e-01 1.968810e-01 1.845551e-01 1.968551e-01 4 1.954560e-01 1.968819e-01 1.891174e-01 1.968845e-01 5 1.959839e-01 1.968825e-01 1.920166e-01 1.969522e-01 6 1.963158e-01 1.968808e-01 1.938477e-01 1.969649e-01

` and the extrapolated values After determining the modification parameter γ1,1 `∗ , we solve problem (3.7) applying the correction technique and using the exγ1,1 `∗ determined at the finest grid trapolated values of the modification parameter γ1,1 level. The modification parameters determined at each grid level are appropriate only for the correction at the corresponding grid level or the coarser ones but not for the finer grid levels [48]. Therefore, for implementing the correction technique at each grid level, we use the fixed extrapolated value determined at the finest grid level. We can see from figure 4.6 the convergence rates of the first defect function g`,1 that is pointwise the discretization error measured at certain grid nodes P` . The convergence rates are almost of second order as postulated in [48, 94]. We see in figure 4.6 that from the symmetry property, the values of the convergence rates at the grid nodes ( 43 , 34 ) and (− 34 , 34 ) coincide. Figure 4.7 displays the values of the convergence rates of the three defect `∗ defunctions g`,2 , g`,3 and g`,4 using the extrapolated modification parameter γ1,1 termined at the finest grid level. These values are almost of second order as confirmed in [48, 94]. They are compared to the values of the convergence rates determined using the standard finite element method without the correction by

54

2.2

2.15

2.1

2.05

2

1.95

1.9

1.85 0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

Figure 4.6: L-shaped domain (FE): Convergence rates of g`,1 at grid nodes P` . setting γ ` = 0, that are of order ≈ 34 as confirmed in the literature [48]. Figure 4.8 displays the values and the extrapolated values of the modification ` and γˆ` determined as described in figure 4.2(a) and figure 4.2(b) parameters γ1,1 1,1 using method M2; that uses the energy defect function g`,2 , and we observe that ` is the value of the modification paramthe values are not the same. The value γˆ1,1 ` is the value of the modification eter determined using equation (4.36), while γ1,1 parameter determined using equation (4.37). So, different techniques or strategies of the correction lead to different values of the modification parameter. We also ` using a symmetric mesh determine the values and the extrapolated values of γˆ1,1 for the L-shaped domain as illustrated in figure 4.4. We computed the extrapo`∗ assuming that p = 2 , as in [48] and this value of p is confirmed lated values γ1,1 3 by our numerical results. Note that [48] uses slightly different triangulation, as illustrated here in figure 4.4. For this triangulation, we computed different modi55

2.002

1.335

2 1.33 1.998 1.325 1.996 1.994

1.32

1.992

1.315

1.99 1.31 1.988 1.305 1.986 1.3

1.984 0

0.05

0.1

0

0.05

0.1

Figure 4.7: L-shaped domain (FE): The convergence rates of the defect functions g`,2 , g`,3 , g`,4 . `∗ = 0.1376405 at ` = 6 (see figure 4.8), that are in a good fication parameters, γˆ1,1 agreement with the asymptotic modification parameter, γ0 = 0.13802, published in [48]. One approach to measure the pollution effect away from the singularity is to compute the L2 -norm or the L∞ -norm of the error u − u`s,γ on the domain Ω \ S, where S = [− 14 , 41 ] × [− 14 , 41 ]. We see in figure 4.9 the convergence rates of the L2 -norm and the L∞ -norm of the discretization error u − u`1,γ on the domain Ω \ S, and the convergence rates of

the energy defect function g`,2 1 at different grid levels. The convergence rates are `∗ and γˆ`∗ computed with the values of the extrapolated modification parameters γ1,1 1,1 determined at the finest grid level ` = 6 using the Friedrichs-Keller triangulation and the symmetric triangulation, respectively. We can observe from figure 4.9 that 56

0.2

0.19

0.18

0.17

0.16

0.15

0.14

0.13 0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

Figure 4.8: L-shaped domain (FE): Values of γ ` and γˆ` using the Friedrichs-Keller (FK) triangulation and values of γˆ` using the symmetric triangulation compared to the extrapolated values of all. the convergence rates are almost of order 2 at the fine grid levels as expected from the literature [48]. We change the function defined on the boundary and make a perturbation to 2 the non-smooth singular function u1 = r 3 sin( 2πϕ 3 ). So we consider problem (3.7) with g = u1 + uk , where k = 2, 3, 4. We use for the correction at each grid level, the extrapolated modification parameter γ `∗ determined at the finest grid level ` = 6. Figure 4.10 displays the convergence rates of the discretization error g`,1 determined at the grid node P` = ( 34 , 34 ) for the different options of k = 2, 3, 4. We observe in figure 4.10 that 57

2

1.95

1.9

1.85 0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.02

0.025

0.03

0.035

(a) Friedrichs-Keller triangulation. 2.01

2

1.99

1.98

1.97

1.96

1.95 0

0.005

0.01

0.015

(b) Symmetric triangulation.

Figure 4.9: L-shaped domain (FE): Convergence rates. the convergence rates are perturbed for k = 2. This is because the values of the modification parameter are perturbed due to the perturbation in the function de58

7 6 5 4 3 2 1 0 -1 0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

Figure 4.10: L-shaped domain (FE): Convergence rates of the first defect function g`,1 ; k = 2, 3, 4. 3.5

3

2.5

2

1.5

1

0.5 0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

Figure 4.11: L-shaped domain (FE): Convergence rates of g`,2 , g`,3 , g`,4 .

59

` fined in the boundary. If we use the values of the modification parameter γ1,1 determined when g = u1 is defined on the boundary without any perturbation, then we get a second-order convergence rates of all of the four defect functions g`,1 , g`,2 , g`,3 , g`,4 and the error norms ||u − u`1,γ ||L2 (Ω\S) , ||u − u`1,γ ||L∞ (Ω\S) ||. We will show numerically at the end of this section in figure 4.12 that we can compute non-perturbed values of the modification parameter by shrinking the domain Ω to the auxiliary domains Ωs , s > 1.

0.22 α2 = 0 α =1 2

0.215

α2 = 2

Modification parameter

α2 = 5 α2 = 10 α2 = 100

0.21

0.205

0.2

0.195 0

2

4

6

8

10

12

14

s

Figure 4.12: L-shaped domain (FE): The modification parameters γs` for different L-shaped domains Ωs at the same grid level ` = 6 where g = u1 + α2 u2 . In figure 4.11, we present the convergence rates of the three defect functions for k = 2. We can observe in figure 4.11 that the values of the convergence rates are not of second order - except at the finest grid level- because the values of the convergence rates are affected by the perturbed modification parameters that in turn are affected by the perturbation g = u1 + u2 done at the g`,2 , g`,3 , g`,4

60

` determined when g = u , we boundary. If we use the modification parameter γ1,1 1 get a second-order convergence rate. We then consider problem (3.7) with

g = u1 + α2 u2 , where α2 = 2, 5, 10, 100. We added α2 u2 to the singular function u1 and considered high values of α2 to increase the perturbation in g. We redefine problem (3.7) so that it is defined on the auxiliary shrunk domains Ω = Ωs , for s ≥ 1 and we determine the corresponding modification parameter γs` to each problem at a fixed grid level ` = 6. We can see from figure 4.12 that the perturbation in the values of the modification parameter γs` increase when α2 increases. With shrinking the domain; meaning that when s increases, the perturbation in the values of the modification parameter γs` decreases. The value of the modification parameter ` (if s = 1). For example, we observe from figure 4.12 that for at α2 = 0 is γ1,1 α2 = 100, when the shrunk domain considered is Ω = Ω14 (s = 14), we find that ` = γ ` ; confirming theorem 4.2.4. For α = 5 and s = 8, then γ ` = γ ` as we γ14 2 1,1 8 1,1 see from figure 4.12.

4.6

Slit domain with a fixed tip at the origin

In this section, we consider the slit domain as a second geometry such that Ω = ((−1, 1) × (−1, 1))/([0, 1] × {0}),

(4.43)

where θ = 2π. The problem to solve is (3.7) with g = u1 . The analytic solution 1 of this problem is u = u1 = r 2 sin( ϕ2 ). We see in figure 4.13 the values of the modification parameter γˆ` and the extrapolated values γˆ`∗ such that the correction is done as in (4.36), where we used both the Friedrichs-Keller triangulation as in figure 4.3(b) and a regular triangulation of the symmetric mesh as in figure 4.5 to compare the results with [48]. The extrapolated modification parameter at the finest grid level ` = 6 of the symmetric mesh has the value γˆ`∗ = 0.2803057 that is very close to the value 0.28187 as published in [48]. In our numerical results, we follow the correction realized as in (4.37). We present in table 4.2 the values γ ` and the extrapolated values γ `∗ of the modification parameter determined at the different grid levels `, using the four methods of correction. Our numerical values of the convergence rate p of the modification parameter are almost 1, therefore we use p = 1 as predicted in [48] to compute the 61

0.315

0.31

0.305

0.3

0.295

0.29

0.285

0.28

0.275 0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

Figure 4.13: Slit domain (FE): Values γˆ` and extrapolated values γˆ`∗ using both the Friedrichs-Keller (FK) and the symmetric triangulation. Table 4.2: Slit domain (FE): Extrapolated modification parameters γ `∗ . Method M1 Method M2 ` γ` γ `∗ γ` γ `∗ 1 3.391361e-01 3.464447e-01 2 3.432522e-01 3.469127e-01 3 3.453178e-01 3.473835e-01 3.471471e-01 3.475461e-01 4 3.463516e-01 3.473854e-01 3.472643e-01 3.474638e-01 5 3.468628e-01 3.473740e-01 3.473229e-01 3.474228e-01 6 3.471375e-01 3.474121e-01 3.473520e-01 3.474015e-01 Method M3 Method M4 ` `∗ ` ` γ γ γ γ `∗ 1 3.445837e-01 3.227882e-01 2 3.459764e-01 3.349838e-01 3 3.466783e-01 3.478734e-01 3.411789e-01 3.517255e-01 4 3.470298e-01 3.476282e-01 3.442841e-01 3.495703e-01 5 3.472056e-01 3.475050e-01 3.458252e-01 3.484489e-01 6 3.472934e-01 3.474427e-01 3.466187e-01 3.479694e-01

62

2.1 2.08 2.06 2.04 2.02 2 1.98 1.96 1.94 1.92 0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

Figure 4.14: Slit domain (FE): Convergence rates of the first defect function g`,1 , considering different grid nodes P` . 1.12 1.1 1.08 1.06 1.04 1.02 1 0.98 0.96 0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

Figure 4.15: Slit domain (FE): Convergence rates of g`,1 at grid nodes P` , γ ` = 0.

63

extrapolated modification parameter γ `∗ from (4.40). In determining the values of the modification parameters with method M1, we assume as in section 4.5 that the first defect function g`,1 is the pointwise discretization error measured at the grid node P = ( 43 , 43 ) that is far enough from the singular point (0, 0). 2 1.98 1.96 1.94 1.92 1.9 1.88 1.86 1.84 1.82 1.8 0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

Figure 4.16: Slit domain (FE): Convergence rates of the error norms ||u − u`1,γ ||L2 (Ω\S) and ||u − u`1,γ ||L∞ (Ω\S) . Using the extrapolated modification parameter γ `∗ at the finest grid level ` = 6, we present in figure 4.14, the values of the convergence rates of the pointwise discretization error g`,1 measured at the different grid nodes P` . We can see that the convergence rates are of order 2 as in [48]. In figure 4.15, we see that the convergence rates are of order 1, if γ ` = 0, since the slit domain has a stronger singularity than the L-shaped domain. We get second order convergence rates as well when we use for the correction the other three defect functions g`,2 , g`,3 , g`,4 . Figure 4.16 shows that the convergence rates of the L2 -norm ||u − u`1,γ ||L2 (Ω\S) and the L∞ -norm ||u − u`1,γ ||L∞ (Ω\S) of the discretization error are of almost second order as postulated and shown in [48, 94].

64

4.7

Re-entrant corner domains in 3D with the third co-ordinate being constant in the solution

We consider in this section problem (3.7) defined on a three dimensional domain Ω3D = Ω2D × (−1, 1), where Ω2D = Ω is a two dimensional re-entrant corner domain. As a direct extension, we consider the cylindrical coordinates (r, ϕ, ρ) p r = x2 + y2 , y ϕ = tan−1 , x ρ = z. The correction is considered to each two-dimensional grid, parallel to the xyplane, in the re-entrant corner domain. We consider problem (3.7) with g = u1 as a direct extension of the two-dimensional case, where the analytic solution u = u1 is constant in the third coordinate, z. Our test geometries are the three-dimensional L-shaped domain Ω3D such that Ω2D = Ω is the two-dimensional L-shaped domain defined in (4.42) in section 4.5 and the three-dimensional slit domain Ω3D such that Ω2D = Ω is the twodimensional slit domain defined in (4.43) in section 4.6 Figure 4.17 shows the finite element solutions considering both the L-shaped and the slit domains. The modification parameters γ ` and the extrapolated values γ `∗ are determined in figure 4.18.

4.8

Analytic energy

In this section, we provide the analytic computation of the energy a(u, u) of the different solutions u given in section 4.5, section 4.6, and section 4.7. The analytic energy is used in the computation of the energy defect function g`,2 (see definition 4.4.1). The analytic energy computed using polar coordinates in two dimensions is given as Z a(u, u) =

∇u.∇u r dr dϕ,

(4.44)

Ω

while the analytic energy a(u, u) in three dimensions is given by Z

a(u, u) =

∇u.∇u r dr dϕ dz, Ω3D

We evaluate the analytic energy of the different cases 65

(4.45)

1

1.2

0.8 0.6

1

z

0.4 0.2 0.8

0 -0.2 0.6

-0.4 -0.6 -0.8

0.4

-1 -1 -0.5

0.2

0

x

0.5 1

-1

0

-0.5

0.5

1 0

y

(a) L-shaped domain. 1.2

1 0.8 0.6

1

z

0.4 0.2 0.8

0 -0.2 -0.4

0.6

-0.6 -0.8 0.4

-1 -1 -0.5

0.2

0

x

0.5 1

-1

0

-0.5

0.5

1

y

(b) Slit domain.

Figure 4.17: The finite element solution of (3.7) with g = u1 on three dimensional L-shaped and slit domains.

66

0.32 0.31 0.3 0.29 0.28 0.27 0.26 0.25 0.24 1

2

3

4

5

(a) L-shaped domain. 1.2

1.15

1.1

1.05

1

0.95

0.9

0.85 1

2

3

4

5

(b) Slit domain.

Figure 4.18: Values and extrapolated values of γ ` of (3.7) with g = u1 on threedimensional L-shaped and slit domains.

67

1. u(r, ϕ) = r2/3 sin( 2ϕ 3 ), defined in the L-shaped domain Ω such that Ω = ((−1, " 1) × (−1, 1) × (−1, 0)). # 1))/((0, " # ∂u 2ϕ 2 − 31 r ) sin( 3 ∇u = 1∂∂ru = 23 − 1 . 2ϕ 3 r cos( r ∂ϕ 3 3 ) 2

∇u.∇u = 94 r− 3 . Z

a(u, u) =

∇u.∇u r dr dϕ Ω

1 4 r 3 dr dϕ 9 Ω Z πZ Z π Z sec( π −ϕ) Z 3π Z sec(ϕ− π ) 2 2 4 2 4 4 sec ϕ + π = [ + π 9 0 0 0 0 4 2

Z

=

Z π Z sec(π−ϕ)

+

+

3π 4

0

1 = [ 3

π 4

Z

Z π

+

3π 4

Z

5π 4

Z sec(ϕ−π)

+ 0

π

Z

4 3

sec (ϕ) dϕ +

0 4 3

sec (π − ϕ) +

Z

Z

π 4 5π 4

π 2

3π 2 5π 4

Z sec( 3π −ϕ) 2 0

π sec ( − ϕ) + 2 4 3

4 3

sec (ϕ − π) +

π

1

]r 3 dr dϕ

Z

Z π 2 3π 2

5π 4

3π 4

4 π sec 3 (ϕ − ) 2 4

sec 3 (

3π − ϕ)] dϕ 2

= 1.8362, using Matlab. The functions to be integrated are induced from considering the triangles 1

•ϕ −1

0

r

1

First integral: r = sec(ϕ), 0 ≤ ϕ ≤ π4 .

68

1

• −1

ϕ

0

1

Second integral: r = sec( π2 − ϕ), π4 ≤ ϕ ≤ π2 . 1

•ϕ −1

0

1

Third integral: r = sec(ϕ − π2 ), π2 ≤ ϕ ≤

3π 4 .

1

• −1

ϕ 0

1

Fourth integral: r = sec(π − ϕ), 3π 4 ≤ ϕ ≤ π. −1

ϕ •

1 0

−1 Fifth integral: r = sec(ϕ − π), π ≤ ϕ ≤

69

5π 4 .

−1

1

ϕ •

0

−1 5π Sixth integral: r = sec( 3π 2 − ϕ), 4 ≤ ϕ ≤ 2

2k

3π 2 .

2kϕ 3 2. u(r, ϕ) = r 3 sin( 2ϕ 3 ) + αk r sin( 3 ), defined in the L-shaped domain Ω such that " Ω =#((−1, # " 1) × 1(−1, 1))/((0, 1) ×2k(−1, 0)). 2ϕ 2kϕ du 2k 3 −1 2 −3 r sin( ) + α r sin( ) k 3 3 3 ∇u = 1drdu = 23 − 1 . 2ϕ 2kϕ 2k 2k −1 3 3 r cos( ) + α r ) cos( r dϕ k 3 3 3 3

70

∇u.∇u = 49 r

−2 3

2

4k

4

2k

2(k−1)ϕ 3 − 3 cos( + αk2 4k9 r 3 −2 + αk 8k ). 3r 3

Z

a(u, u) = ∇u.∇u r dr dϕ Ω Z 4k2 4k −1 8k 2k − 1 2(k − 1)ϕ 4 1 2 r 3 + αk r 3 + αk r 3 3 cos( ) dr dϕ = 9 3 3 Ω 9 π 4

Z

=[

Z sec ϕ

π 2

Z

+

0

π 4

0

Z π Z sec(π−ϕ)

+

3π 4

Z sec( π −ϕ) 2

+ 0

+ π 2

0

Z π

5π 4

3π 4

Z

Z sec(ϕ−π)

0

Z

+ 0

Z sec(ϕ− π ) 2 3π 2 5π 4

Z sec( 3π −ϕ) 2

]·

0

4k2 4k −1 8k 2k − 1 2(k − 1)ϕ 4 1 2 r 3 + αk r 3 + αk r 3 3 cos( ) dr dϕ 9 9 3 3 Z π 4 4k 2k 2 4 1 k 4k 2(k − 1)ϕ sec 3 + 3 cos( )] =[ [ sec 3 (ϕ) + αk2 sec 3 (ϕ) + αk 3 k+1 3 0 3 Z π 4 π 4k π 2k 2 2 1 k 4k 2(k − 1)ϕ + π [ sec 3 ( − ϕ) + αk2 sec 3 ( − ϕ) + αk sec 3 + 3 cos( )] 3 2 3 2 k+1 3 4

3π 4

4 4k 2k 2 1 π π 4k 2(k − 1)ϕ 2k 3 (ϕ − 3 (ϕ − 3 + 3 cos( [ sec ) + α sec ) + α sec )] k k π 3 2 3 2 k+1 3 2 Z π 4 4k 2k 2 1 k 4k 2(k − 1)ϕ + 3π [ sec 3 (π − ϕ) + αk2 sec 3 (π − ϕ) + αk sec 3 + 3 cos( )] 3 3 k+1 3 4

Z

+

5π 4

4 4k 2k 2 k 4k 2(k − 1)ϕ 1 sec 3 + 3 cos( )] [ sec 3 (ϕ − π) + αk2 sec 3 (ϕ − π) + αk 3 3 k+1 3 π Z 3π 4 3π 4k 3π 2 1 k + 5π [ sec 3 ( − ϕ) + αk2 sec 3 ( − ϕ) 3 2 3 2 4 2k 2 4k 2(k − 1)ϕ + αk sec 3 + 3 cos( )] ] dϕ. k+1 3

Z

+

Specifying k and αk , the analytic energy can be computed easily using Matlab. 1

3. u(r, ϕ) = r 2 sin( ϕ2 ), defined in the slit domain Ω such that Ω = ((−1, 1) × (−1, 1))/((0, 1) {0}). " # ×" # ϕ du 1 − 21 r sin( 2 ) ∇u = 1drdu = 12 − 1 . ϕ 2 cos( ) r r dϕ 2 2 71

∇u.∇u = 14 r−1 . Z

a(u, u) =


1 dr dϕ 4 Ω Z πZ Z π Z sec( π −ϕ) Z 3π Z sec(ϕ− π ) 2 2 4 2 1 4 sec ϕ = [ + π + π 4 0 0 0 0 4 2 Z

=

Z π Z sec(π−ϕ)

+ +

+

3π 4 Z 7π 4

0

π 4

Z

+ +

3π 2

+

Z sec( 3π −ϕ) 2

5π 4

0

Z 2π Z sec(2π−ϕ)

+

] dr dϕ

7π 4

0 π 2

Z

sec(ϕ) dϕ +

0

sec(π − ϕ) +

3π 4 Z 7π 4

3π 2

Z

0

0

1 = [ 4

Z sec(ϕ−π)

π

Z sec(ϕ− 3π ) 2

3π 2

Z π

5π 4

Z

Z

π 4 5π 4

π sec( − ϕ) + 2

sec(ϕ − π) +

π

3π sec(ϕ − ) + 2

Z 2π 7π 4

Z

Z π 2 3π 2

5π 4

The last two terms come from considering the triangles ϕ 0

1

•

−1 3π Seventh integral: r = sec(ϕ − 3π 2 ), 2 ≤ ϕ ≤

72

7π 4 .

π sec(ϕ − ) 2

sec(

sec(2π − ϕ)] dϕ

= 1.7627, using Matlab.

−1

3π 4

3π − ϕ) 2

ϕ

−1 0

•

1

−1 Last integral: r = sec(2π − ϕ), 7π 4 ≤ ϕ ≤ 2π. 1

4. u(r, ϕ) = r 2 sin( ϕ2 ) + r2 sin(2ϕ), defined in the slit domain Ω such that Ω = ((−1, " 1) × (−1, 1) × {0}). # 1))/((0, " # ϕ du 1 − 21 r ) + 2r sin(2ϕ) sin( 2 ∇u = 1drdu = 12 − 1 . ϕ 2 r cos( r dϕ 2 2 ) + 2r cos(2ϕ)

73

1

∇u.∇u = 14 r−1 + 4r2 + 2r 2 cos( 3ϕ 2 ). Z

a(u, u) =


3 1 3ϕ [ + 4r3 + 2r 2 cos( )] dr dϕ 2 Ω 4

Z

=

π 4

Z

=[

Z sec ϕ

π 2

Z

+

0

0

π 4

Z π Z sec(π−ϕ)

+

3π 4

Z sec(ϕ− π ) 2

Z

3π 2

+ π 2

0

5π 4

3π 4

Z

Z π Z sec(ϕ−π)

+ 0

Z sec( π −ϕ) 2

+ 0

0

5π 4

Z sec( 3π −ϕ) 2

]·

0

3 1 3ϕ ( + 4r3 + 2r 2 cos( )) dr dϕ 4 2 Z π 5 4 1 4 3ϕ =[ [ sec(ϕ) + sec4 (ϕ) + sec 2 (ϕ) cos( )] 5 2 0 4 Z π 5 2 1 π π 4 π 3ϕ + π [ sec( − ϕ) + sec4 ( − ϕ) + sec 2 ( − ϕ) cos( )] 4 2 2 5 2 2 4 3π 4

5 π 1 π π 4 3ϕ 4 2 (ϕ − [ sec(ϕ − ) + sec (ϕ − ) + sec ) cos( )] π 4 2 2 5 2 2 Z 2π 5 1 4 3ϕ + 3π [ sec(π − ϕ) + sec4 (π − ϕ) + sec 2 (π − ϕ) cos( )] 4 5 2 4

Z

+

5π 4

5 1 4 3ϕ [ sec(ϕ − π) + sec4 (ϕ − π) + sec 2 (ϕ − π) cos( )] 4 5 2 π Z 3π 5 3π 2 1 3π 3π 4 3ϕ + 5π [ sec( − ϕ) + sec4 ( − ϕ) + sec 2 ( − ϕ) cos( )] 4 2 2 5 2 2 4

Z

+

7π 4

5 1 3π 3π 4 3π 3ϕ 4 2 (ϕ − [ sec(ϕ − ) + sec (ϕ − ) + sec ) cos( )] 3π 4 2 2 5 2 2 2 Z 2π 5 1 4 3φ + 7π [ sec(2π − ϕ) + sec4 (2π − ϕ) + sec 2 (2π − ϕ) cos( )] ] dϕ 4 5 2 4

Z

+

= 12.4294, using Matlab. 2

5. u(r, ϕ) = r 3 sin( 2ϕ 3 ) defined in the three-dimensional L-shaped domain Ω3D = Ω2D ×(−1, 1) such that Ω2D = ((−1, 1) × (−1, 1)) / ((0, 1) × (−1, 0)). Then,

74

  ∇u = 

∂u ∂r 1 ∂u r ∂ϕ ∂u ∂z





  =



2 − 13 sin( 2ϕ 3r 3 ) 2ϕ 2 − 31 cos( 3 ) 3r

 .

0

Therefore, ∇u.∇u = 94 r

− 23

. So, the analytic energy is given by Z

aΩ (u, u) =

∇u.∇u r dr dϕ dz Ω3D

1 4 = r 3 dr dϕ dz 9 Ω3D Z Z 1 4 1 = −1 r 3 dr dϕ dz 9 Ω2D Z 1Z 1 4 =2 r 3 dr dϕ dz 9 0 Ω2D Z 1 4 =2 r 3 [z]10 dr dϕ 9 Ω2D Z 1 4 r− 3 dr dϕ =2 9 Ω2D

Z

Z

=2

∇u.∇u r dr dϕ, Ω2D

= 2 aΩ2D (u, u), where the last term represents the analytic energy of the solution of the twodimensional Poisson equation (3.7) defined in the two-dimensional L-shape domain Ω2D . 1

6. u(r, ϕ) = r 2 sin( ϕ2 ) defined in the three-dimensional slit domain Ω3D = Ω2D ×(−1, 1)suchthat Ω2D = ((−1,  1) × (−1, 1)) / ((0, 1) × {0}). Then, ∂u ϕ 1 − 12 r sin( 2 )  1∂∂ru   12 − 1  ∇u =  r ∂ ϕ  =  r 2 cos( ϕ )  . 2 2 ∂u 0 ∂z

75

Therefore, ∇u.∇u = 14 r−1 and so the analytic energy is given by Z

a(u, u) =

∇u.∇u r dr dϕ dz Ω3D

1 = dr dϕ dz 4 Ω3D Z Z 1 1 = dr dϕ dz 4 −1 Ω2D Z Z 1 1 dr dϕ dz =2 4 0 Ω2D Z 1 =2 [z]1 dr dϕ 4 Ω2D 0 Z 1 =2 dr dϕ 4 Ω2D Z

Z

=2

∇u.∇u r dr dϕ, Ω2D

= 2 aΩ2D (u, u), where the last term represents the analytic energy of the solution of the two-dimensional Poisson equation (3.7) defined in the two-dimensional slit domain Ω2D .

5

Periodicity of the modification parameter

In this section, we apply the correction technique to the Poisson equation with singular solution. The domain considered is a slit domain with a slit tip being not necessarily accurately represented as a grid node. As a beginning step, we first assumed that the tip of the slit domain is not a grid node and this tip is a fixed point in the slit domain that does not depend on the refinement level. We computed the convergence rates of the L2 - and L∞ -norms of the discretization error, excluding from the domain a fixed neighborhood of the singular point that does not depend on the step-size. The convergence rates were not of second order as in the case where the tip of the slit domain is the origin or is a grid node. Therefore, in the next step, we assumed that the tip τ of the slit domain depends on the refinement level such that τ = (ch` , 0), for example, such that c is a fixed constant that does not depend on the refinement level `. This means that the tip of the slit domain changes its position as the refinement ` level changes. In this case the values of 76

the convergence rates are of second order. As a consequence of this choice of the tip τ of the slit domain, both the analytic solution u and the finite element solution u` , or the modified finite element solution u`γ , change when the refinement level changes. As a motivation to this problem, we refer to the crack growth problems [68, 83, 88, 100] that attract a special interest in fracture mechanics, thermoelasticity, and material sciences. These fields are concerned with crack growth problems to study the propagation of cracks in materials, since material failure can appear in crack domains where the tip of the crack moves. A crack domain is a slit domain and it means a domain with cracks (or slits). The crack growth problems are considered as a motivation to the problem considered in this section because in a crack growth problem, a sequence of elliptic equations in a crack domain are solved such that the crack changes. Each problem in this sequence is the problem that we consider in this section. Although the equations that describe the crack growth problems in fracture mechanics are different from the equation considered in this section, the Poisson equation considered in the current section serves as a model. For the correction technique used for solving a system of equations, we refer to [65, 77]. The crack domain has a singularity at the tip of the crack since the crack is considered to be a re-entrant corner with an angle θ greater than π (θ = 2π). Solving the crack growth problems using the traditional finite element method has a disadvantage. In solving crack growth problems using the traditional finite element method, the crack tip singularity has to be accurately represented at each step [101, 103]. This means that the crack tip has to be a grid node. Therefore, in the traditional finite element method, a sequence of finite element problems are considered. Therefore, using the traditional finite element framework becomes expensive because the mesh has to be regenerated and the stiffness matrix has to be reset at each step. An alternative approach to the traditional finite element method in dealing with the crack growth problems is the extended finite element method (XFEM) [14, 36, 56, 103]. The discretization of the XFEM method does not need to conform to the cracks and this is an advantage. This means that the tip of the crack domain does not need to be a grid node and the crack does not have to lay on a grid line. A disadvantage of the standard extended finite element method from the viewpoint of performance is the dependence of the approximation space on the growth of the crack. To represent the discontinuity and the singularity of cracks, the degrees of freedom at nodes around the surfaces and tips of the cracks are enriched by additional degrees of freedom. These additional degrees of freedom 77

make the structure of the system matrix more complicated. This can be seen because the function u1 is one of the enrichment functions that is added to represent the discontinuity on the element containing the tip of the crack domain [36]. Other enrichment functions are added to get accurate results with relatively coarse meshes. We consider in this section a slit domain, where the slit represents the crack. The tip of the slit is assumed to be a non-grid node and it is approximately represented to be a grid node (see figure 5.1 and figure 5.2). This means that the tip of the slit domain is not accurately represented. We consider the boundary value problem (3.7) with g = u1 and assume that Ω is a slit domain with the slit line being parallel to the x-axis. For solving this problem, there is no need for a remeshing and the degrees of freedom do not change. We use the correction technique discussed earlier in section 3 and section 4, where we do not set up a new linear system at each step. We assume that the tip τ = (τ1 , τ2 ) of the slit domain is a general point that is not necessarily the origin (0, 0). The polar coordinates (r, ϕ) considered for the computation of the analytic energy and the solution are defined as q (5.1) r = (x − τ1 )2 + (y − τ2 )2 , y − τ2 ϕ = tan−1 ( ), (5.2) x − τ1 where (x, y) are the Cartesian coordinates. In our numerical results, we show that the relation between the tip of the slit domain and the modification parameter is periodic with its period being the stepsize h` . This means that the values of the modification parameters at all the grid nodes are the same and these values repeat themselves after each period.

5.1

Numerical results

We assume that the slit line is a grid line and in our discretization, we choose the tip of the slit domain to be the closest grid node to the tip τ in the positive x-direction. We denote this tip by τd and call it the discrete tip of the slit domain. We apply the correction technique to the neighborhood of the discrete tip τd as we see in figure 5.2 and we determine the modification parameters using equation (4.37) with Ω` being the one-element neighborhood of the discrete tip τd . For determining the modification parameter, we use method M2 that means we use the energy defect function g`,2 (see definition 4.4.1). 78

τ

Figure 5.1: The slit domain with the tip being not the origin (0, 0) at the center of the domain. This partitioning of the domains is used in the computation of the analytic energy in section 5.2.

1

1

1

1

1

1

1

1

1 1 1

Pb τ

1−γ

1

1

1 1

1

1

1

1

τ τd

1 1

1

Pa

1−γ

1 1

1

1−γ

1

1

Pf

1

1 1

1 1

1

1 1

Figure 5.2: The representation of the entries of the stiffness matrix that correspond to the different grid nodes around the discrete tip τd of the slit domain. The tip τ lays on the line connecting Pb and τd .

79

0.6

0.4

0.2

0

-0.2

-0.4

-0.6

-0.8

-1 -1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Figure 5.3: The modification parameter γ ` for different τ ∈ (−1, 1) × {0}. The relation is periodic in the interval (−1 + h` , 1 − h` ) with the period being h` . y yP

(xP − h` , yP )

τ = (xP + ch` , yP )

(xP , yP )

xP − h`

xP + ch`

xP

x

Figure 5.4: The tip τ = (xP + ch` , y) such that c ∈ (−1, 0] and the two neighboring grid nodes (xP , yP ) and (xP − h` , yP ). The tip τ is not a grid node. We found numerically that if the tip τ of the slit domain is a grid node, then the values of the modification parameters are almost the same as the values in 80

0.21

0.205

0.2

0.195

0.19

0.185

0.18

0.175

0.17 0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

Figure 5.5: The values of γ ` and γ `∗ for τ = (ch` , 0), c = −0.2. table 4.2 determined when the tip τ of the slit domain is (0, 0), and these values have the same quality of correction, meaning that the convergence rates of the L2 - and L∞ -norms of the discretization error measured in Ω \ S and that of the different defect functions are of order 2 as in the regular case. Therefore, we plot the values of the modification parameters γ ` determined considering different tips τ ∈ (−1, 1) × {0} to see the relation between γ ` and τ. We refer to that the rectangular neighborhood S is considered to be a neighborhood of the tip τ := (τ1 , τ2 ). Therefore, we consider S to be S := [τ1 − 14 , τ1 + 14 ] × [τ2 − 14 , τ2 + 14 ]. Figure 5.3 shows the plot of the modification parameter γ ` for different tips τ ∈ (−1, 1) × {0}, where h` = 14 is considered. We can observe from figure 5.3 that the modification parameter is periodic in the interval [−1 + h` , 1 − h` ] where the peaks of the periods are at the tips that are grid nodes. This explains the results that the modification parameter determined when the tip of the slit domain is the origin (0, 0) can be used when the tip of the slit domain is any other grid 81

`

h node at the x-axis. Therefore, we can use the interval (− h` 2 , 2 ] as a representative interval. This means that we consider the tip of the slit domain to be any point in (−1 + h` , 1 − h` ) × {0} and we determine the value of the modification parameter ` ` assuming that the tip of the slit domain is instead in (− h2 , h2 ] × {0}. Then, we use this modification parameter for the correction of the problem defined on the slit ` ` domain whose tip is in (xP − h2 , xP + h2 ] × {0}, where P = (xP , 0) is any grid node in (−1 + h` , 1 − h` ) × {0}.

2.02 2 1.98 1.96 1.94 1.92 1.9 1.88 1.86 1.84 1.82 0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

Figure 5.6: The convergence rates of ||u − u`1,γ ||L2 (Ω\S) , ||u − u`1,γ ||L∞ (Ω\S) ||, g`,2 , and g`,3 . The tip of the slit domain is τ = (ch` , 0), c = −0.2. The next step is to consider the tip τ of the slit domain to be not a fixed point in the slit domain. As mentioned before, this is useful in problems in fracture mechanics, where the crack grows. We assume that τ = (xP + ch` , yP ), such that P = (xP , yP ) is an inner grid node and c ∈ (−1, 0] is a fixed constant that does not depend on `. We can see in figure 5.4, the grid node P = (xP , yP ) and the tip (xP + ch` , yP ). From the periodicity, the values of the modification parameter 82

γ ` are almost the same as the values determined considering τ = (ch` , 0) with P = (xP , yP ) = (0, 0). Figure 5.5 shows the values γ ` and the extrapolated values γ `∗ of the modification parameters for c = −0.2. We compute the convergence rates of the L2 -norm ||u − u`1,γ ||L2 (Ω\S) and the L∞ -norm ||u − u`1,γ ||L∞ (Ω\S) || of the discretization error on Ω \ S and the convergence rates of the defect functions g`,2 and g`,3 . We see from figure 5.6 that the convergence rates are almost of second order as for the case where the tip of the slit domain is a fixed grid node in [48]. So far, we considered a slit domain with a slit line laying on a horizontal grid line in a Friedrichs-Keller triangulation that is a regular grid. Our result can be generalized if the slit line lays on a vertical grid line in a regular grid.

5.2

Analytic energy

For the computation of the analytic energy and in order to be able to compute the integrals, we divide the slit domain into four rectangles according to the position of the tip τ and divide each rectangle into two triangles, so that the value of the coordinate r in each triangle can be determined easily (see figure 5.1). We can compute the gradient of u = r1/2 sin( ϕ2 ) and find that ∇u.∇u = 14 r−1 . Then the analytic energy can be computed as

83

Z

a(u, u) =


1 dr dϕ 4 Ω Z Z Z π Z (1−τ2 ) sec( π −ϕ) Z θ2 Z (1−τ2 ) sec(ϕ− π ) 2 2 2 1 θ1 (1−τ1 ) sec(ϕ) = [ + + π 4 0 0 θ1 0 0 2 Z

=

Z π Z (1+τ1 ) sec(π−ϕ)

+ 0

θ2

+

Z θ3 Z (1+τ1 ) sec(ϕ−π)

+

3π 2

1 = [ 4

Z θ2

+ π 2

Z θ3

+

0

+

3π 2

θ3

+

Z (1+τ2 ) sec( 3π −ϕ) 2 0

] dr dϕ 0

θ4

(1 − τ1 ) sec(ϕ) dϕ +

Z

π 2

θ1

π (1 − τ2 ) sec(ϕ − ) + 2 (1 + τ1 ) sec(ϕ − π) +

Z π

π (1 − τ2 ) sec( − ϕ) 2

(1 + τ1 ) sec(π − ϕ)

θ2

Z

π

Z θ4

3π 2

Z 2π Z (1−τ1 ) sec(2π−ϕ)

0

Z θ1

0

π

Z θ4 Z (1+τ2 ) sec(ϕ− 3π ) 2

Z

+

3π 2

(1 + τ2 ) sec(

θ3

3π (1 + τ2 ) sec(ϕ − ) + 2

Z 2π

3π − ϕ) 2

(1 − τ1 ) sec(2π − ϕ)] dϕ

θ4

The computation of these integrals come from regarding these eight triangles that divide the slit domain

1

1 1 − τ2

−1

θ1 •ϕ τ 1 − τ1

1 − τ2 θ1 −1

1

τ 1 − τ1

1

2 First integral: r = (1 − τ1 ) sec(ϕ), 0 ≤ ϕ ≤ θ1 , where θ1 = tan−1 ( 1−τ 1−τ1 ).

84

1 1 − τ1 1 − τ2 • ϕ −1

τ

1

Second integral: r = (1 − τ2 ) sec( π2 − ϕ), θ1 ≤ ϕ ≤ π2 . 1 + τ1 1 θ2 ϕ

1 − τ2 • −1

τ

Third integral: r = (1 − τ2 ) sec(ϕ −

1 π π 2 ), 2

≤ ϕ ≤ θ2 , where θ2 =

π 2

1 + tan−1 ( 1+τ 1−τ2 ).

1 1 − τ2

ϕ •

−1

1 + τ1

τ

1

Fourth integral: r = (1 + τ1 ) sec(π − ϕ), θ2 ≤ ϕ ≤ π. θ3 ϕ −1 1 + τ1 1 τ 1 + τ2

−1 2 Fifth integral: r = (1 + τ1 ) sec(ϕ − π), π ≤ ϕ ≤ θ3 , where θ3 = π + tan−1 ( 1+τ 1+τ1 ).

85

ϕ −1

1 •

τ 1 + τ2

1 + τ1

−1

Sixth integral: r = (1 + τ2 ) sec( 3π 2 − ϕ), θ3 ≤ ϕ ≤ θ4 ϕ −1

3π 2 .

1 τ

•

1 + τ2

−1 1 − τ1 3π 3π −1 1−τ1 Seventh integral: r = (1+τ2 ) sec(ϕ − 3π 2 ), 2 ≤ ϕ ≤ θ4 , where θ4 = 2 +tan ( 1+τ2 ). ϕ 1 − τ1 1 −1 • τ

1 + τ2

−1 Eighth integral: r = (1 − τ1 ) sec(2π − ϕ), θ4 ≤ ϕ ≤ 2π. Knowing the values of τ1 and τ2 , the values of θi , i = 1, 2, 3, 4 can be determined and the analytic energy a(u, u) can be computed.

86

6

Correction technique using finite difference discretization

Different from section 4 where we used the finite element discretization, we follow in this section the correction technique as in [48,94] using the finite difference discretization as originally introduced in [104]. We implement a finite difference scheme based on the five-point stencil that is equivalent to the standard finite element method based on the Friedrichs-Keller triangulation [33, 39, 40] as stated in section 3.4. We consider the same problem as in section 4, however at the end of the current section, we define Dirichlet boundary conditions different from u1 but as monotonic as u1 . In this case the solution is unknown and we get the same results, i.e., the same modification parameters that recover a second-order convergence. We also use different Dirichlet boundary conditions that are not as monotonic as u1 and we get the same modification parameters as well. We implemented a code in C to solve equation (3.7) using the Multigrid method with the Red-Black Gauss-Seidel as a smoother such that we present in section 6.1 the modification of the 5-point stencil in an O(h) neighborhood of the singular point. Section 6.2 shows the numerical results considering the L-shaped domain as a specific example of a re-entrant corner domain Ω. We consider the slit domain as a second geometry in section 6.3. Although we do not provide new results in this section, we want to show that, using the slit domain as a type of a re-entrant corner domain other than the L-shaped domain, it is possible to determine appropriate values of the modification parameter such that these determined modification parameters recover second order convergence rates of the defect functions and the error norms.

6.1

Modification of the 5-point stencil `

Definition 6.1.1. If u`γ = [uiγ ]ni=1 is the modified solution where n` is the number of the grid nodes, then the discrete energy of the modified solution u`γ is given by 1 a(u`γ , u`γ ) = ∑ ωi, j 2 (i, j) =

j

uiγ − uγ h

!2

1 j ωi, j (uiγ − uγ )2 , ∑ 2 (i, j)

87

h2 ,

(6.1) (6.2)

such that the summation goes over the indices i, j of the grid nodes connecting one edge of the triangulation and ωi, j is defined as   1 − γ if i ∈ I and j = jτ ωi, j = 1 − γ if j ∈ I and i = jτ , (6.3)   1 otherwise where jτ is index of the node of singularity τ and I is the set of indices of the grid nodes that are connected to the singular point. In our numerical computations, we consider I = {Pa , Pb }, if Ω is a L-shaped domain and if Ω is a slit domain, then I = {Pa , Pb , Pf } (see figure 4.2(a) and figure 4.3(a)). In an analogy to the correction technique applied to the standard finite element method in section 4, we introduce the definition of the defect functions and the modification parameter for the finite difference scheme. Definition 6.1.2. The modification parameter γ `, j is a level-dependent value that is determined as the root of one of the defect functions g`, j (γ), j = 1, 2, 3, 4, g`,1 (γ) = u(P) − u`γ (P), g`,2 (γ) = a(u, u) − a(u`γ , u`γ ), `−1 g`,3 (γ) = a(u`γ , u`γ ) − a(u`−1 γ , uγ ),

(6.4)

g`,4 (γ) = u`γ (P) − u`−1 γ (P), where u is the analytic solution, u`γ is the modified solution computed after the correction at level ` and P is a grid node that will be chosen to be far enough from the singular point. We notice that the modification parameter γ `, j , j = 1, 2, 3, 4 depends on the respective defect function g`, j . However, we omit the second superscript for simplicity and denote the modification parameter by γ ` . As in section 4, the defect function g`,1 refers to the pointwise discretization error measured at the node P and g`,2 is the energy defect function. The defect function g`,3 is the difference of the discrete energy between the levels ` and ` − 1 as introduced in [94] while g`,4 is the difference of the modified solution at the grid node P between the levels ` and ` − 1. The four methods of correction; M1, M2, M3 and M4 are used to refer to the use of the four defect functions; g`,1 , g`,2 , g`,3 and g`,4 , respectively in determining the modification parameter γ ` . 88

We now introduce how the 5-point stencils are modified in an O(h) neighborhood of the singular point (see figure 4.1). We present the modification in a neighborhood of the singular point of the L-shaped domain and the slit domain that are considered as examples in our numerical results in section 6.2 and section 6.3. The 5-point stencil notation without modification can be expressed as the sum of 4 stencils; each one represents the connection with a neighboring grid node, as         0 0 −1 0  −1 1 0  +  0 1 −1  +  0 1 0  +  0 1 0  0 0 0 −1   −1 =  −1 4 −1  . −1 If Ω is an L-shaped domain, then the set of the grid nodes at which the stencil will be modified is PM = {Pa , Pb }, and if Ω is a slit domain, then PM = {Pa , Pb , Pf }. First, we consider the L-shaped domain and begin with the grid node Pa . The stencil at Pa is modified such that the term expressing the connection of Pa to the singularity is multiplied with 1 − γ ` , as         0 0 −1 0 Pa :  −1 1 0  +  0 1 −1  +  0 1 0  + (1 − γ ` )  0 1 0  0 0 0 −1   −1  = −1 4 − γ ` −1  . γ` − 1 The stencil of the grid node Pb is modified such that the term expressing the connection of Pb to the singularity is multiplied with 1 − γ ` , as         0 0 −1 0 Pb :  −1 1 0  + (1 − γ ` )  0 1 −1  +  0 1 0  +  0 1 0  0 0 0 −1   −1 =  −1 4 − γ ` γ ` − 1  . −1

89

Second, we consider the slit domain and we add the modification of the stencil of the grid node Pf as         0 0 −1 0 Pf :  −1 1 0  +  0 1 −1  + (1 − γ ` )  0 1 0  +  0 1 0  0 0 0 −1   ` γ −1 =  −1 4 − γ ` −1  . −1 This stencil representation is represented in figure 4.2(a) and figure 4.3(a).

6.2

L-shaped domain

Let us define the L-shaped domain that corresponds to an interior angle θ = 1 1 1 1 1 1 Ω = (− , ) × (− , ) / (0, ) × (− , 0) 2 2 2 2 2 2

3π 2

and consider the Dirichlet boundary value problem (3.7) with g = u1 . The modified solution is given in figure 6.1. We aim to determine appropriate values of the modification parameter with our numerical results. Next, we use these modification parameters to solve the problem. Then, we compute the convergence rates of the different defect functions and the error norm ||u − u`γ ||L∞ (Ω\S) , that is of second order according to [48, 94, 104]. We assume that S = (− 18 , 18 ) × (− 18 , 18 ). All the four methods of correction that are used for determining the modification parameter γ ` can reach the correct value. This correct value exhibits a second order of convergence rates, after some iterations of the bisection methods. By the correct value, we mean a value of the modification parameter at which the estimated convergence rate, after solving, is of order 2. In figure 6.2 and figure 6.3, that show the asymptotic behavior of the modification parameter, we can see a comparison between the four methods of correction and which one is faster in determining the correct value of γ ` . Figure 6.2 and figure 6.3 show the behavior of the modification parameter when the mesh is refined; that is when the step-size gets smaller. It is clear that method M2 is faster in reaching the correct value, that is given at the finest level, than the other methods. The graph that represents method M2 gives values of γ ` closer to the correct value than the other graphs. Method M3 comes at the second stage, i.e., it is less fast than method M2 but faster than the other methods in reaching the correct value. 90

0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0.6 0.4 0.2 -0.6

-0.4

0 -0.2 x

0

-0.2 0.2

0.4

y

-0.4 -0.6 0.6

Figure 6.1: L-shaped domain (FD): The modified solution of the model problem (3.7) with g = u1 . We provide in table 6.1 the values γ ` and the extrapolated values γ `∗ of the modification parameter using p = 32 as predicted from the literature [48] and using the four methods of correction; methods M1, M2, M3 and M4. We see from table 6.1 that the four methods of correction yield values of the modification parameter γ ` that are slightly different. However, they lead to the same quality of correction. For method M1, we use the first defect function g`,1 measured at the grid node ( 83 , 38 ), but the convergence rates of γ ` are measured at randomly chosen grid nodes P` as we will see in figure 6.4. In comparison to the values determined in table 4.1, we conclude that the standard finite element method based on the Friedrichs-Keller triangulation and the finite difference method based on the five-point stencil determine almost the same value of the modification parameter and this is because the two methods are equivalent and produce the same linear system of equations [33, 39, 40]. The convergence rates of the first defect function or the point-wise discretiza-

91

0.2 0.195

Modification Parameter

0.19 0.185 0.18 0.175 0.17 0.165 0.16 0.155 0.015625

0.03125

0.0625

0.125

0.015625

0.03125

0.0625

0.125

0.0078125

0.00390625

0.00195312

0.000976562

0.000488281

step-size

(a) Method M1. 0.2 0.195


0.19 0.185 0.18 0.175 0.17 0.165 0.16 0.155 0.0078125

0.00390625

0.00195312

0.000976562

0.000488281

step-size

(b) Method M2.

Figure 6.2: Asymptotic behavior of the modification parameter.

92

0.2 0.195


0.19 0.185 0.18 0.175 0.17 0.165 0.16 0.155 0.015625

0.03125

0.0625

0.125

0.015625

0.03125

0.0625

0.125

0.0078125

0.00390625

0.00195312

0.000976562

0.000488281

step-size

(a) Method M3. 0.2 0.195


0.19 0.185 0.18 0.175 0.17 0.165 0.16 0.155 0.0078125

0.00390625

0.00195312

0.000976562

0.000488281

step-size

(b) Method M4.

Figure 6.3: Asymptotic behavior of the modification parameter.

93

Table 6.1: L-shaped domain (FD): Modification parameters γ ` and extrapolated values γ `∗ at different grid levels `. Method M1 Method M2 ` `∗ ` γ γ γ` γ `∗ 2 0.158954 0.189677 3 0.172171 0.192304 4 0.181178 0.196512 0.193993 0.196869 5 0.186987 0.196877 0.195062 0.196881 6 0.19064 0.196859 0.195735 0.196881 7 0.192975 0.196949 0.196159 0.196882 8 0.194458 0.196983 0.196426 0.196881 9 0.195337 0.196833 0.196594 0.19688 Method M3 Method M4 ` `∗ ` γ γ γ` γ `∗ 3 0.187948 0.149628 4 0.191191 0.166107 5 0.19329 0.196862 0.177332 0.19644 6 0.194618 0.196862 0.184558 0.196861 7 0.195456 0.196881 0.189111 0.196863 8 0.195984 0.196883 0.191992 0.196897 9 0.19632 0.196891 0.19375 0.196743 tion error g`,1 measured at randomly chosen grid nodes P` can be observed in figure 6.4, where we use method M1 of the correction such that the discretization error g`,1 is computed at the grid node P = ( 38 , 38 ). The extrapolated modification parameter γ `∗ determined at the finest grid level ` = 9 is used for the correction at each grid level `. We computed the discretization error g`,1 at different grid nodes P` ; P` = (− 41 , 14 ), (− 41 , 0), ( 14 , 14 ), (0, 38 ) and we see from figure 6.4 that the convergence rates computed considering the grid nodes P` = (− 14 , − 14 ), (− 14 , 14 ) coincide, because of the symmetry. We observe that the convergence rates are almost of order 2, that coincides with the conclusion in [48, 104]. In order to compare the convergence rates of g`,1 presented in figure 6.4 with the numerical values computed without using the correction technique or with setting γ ` = 0, we present in figure 6.5 the convergence rates of the discretization error g`,1 . The discretization error g`,1 is computed at the same grid nodes P` considered in figure 6.4. We observe from figure 6.5 the sup-optimality of the values of the convergence rates; that they are of order 34 as predicted in the 94

2.15

2.1

2.05

2

1.95

1.9 3

4

5

6

7

8

9

Figure 6.4: L-shaped domain (FD): Convergence rates of g`,1 at grid nodes P` . 1.5

1.45

1.4

1.35

1.3

1.25

1.2 3

4

5

6

7

8

9

Figure 6.5: L-shaped domain (FD): Convergence rates of g`,1 using γ ` = 0.

95

literature [48, 94, 104]. 1.35 2 1.3 1.95 1.25

1.9

1.2

1.15 1.85 1.1 1.8 1.05

1.75

1 0

5

10

0

5

10

Figure 6.6: L-shaped domain (FD): Convergence rates of the 3 defect functions g`,2 , g`,3 , g`,4 and the error norm ||u − u`γ ||L∞ (Ω\S) using extrapolated modification parameter γ `∗ at the finest level. We can use the other three defect functions to compute the modification parameter as in table 6.1 and we then solve the problem and compute the convergence rates of each of the three defect function g`,2 , g`,3 and g`,4 as in figure 6.6. We observe from figure 6.6 that the values of the convergence rates of each of the three considered defect function are of order almost 2. We can see also from figure 6.6 that the convergence rates of the error norm ||u − u`γ ||L∞ (Ω\S) are of order almost 2 at the fine grid levels. In the same figure 6.6, we compare the numerical results of the convergence rates with the values determined using γ ` = 0 that are found to be sub-optimal of order ≈ 1.3, as predicted in [48]. In the previously discussed boundary value problem (3.7) with g = u1 , the 96

analytic solution is known. Next, we consider a boundary value problem where the analytic solution is unknown and we use the correction technique for determining the modification parameters. We consider the same boundary value problem (3.7) but change the boundary condition g=G where G is a function that is defined so that it is as monotonic as g = u1 . So, we define G as follows (see figure 6.8): on the right edge of the L-shaped domain, where x = 21 and 0 ≤ y ≤ 21 , G is an increasing linear function that increases from 0 to 12 as y increases. On the upper edge, where − 12 ≤ x ≤ 21 and y = 12 , g2 is a decreasing linear function from 1 to 12 as x increases. On the left edge, where x = − 12 and − 21 ≤ y ≤ 12 , G is an increasing linear function that decreases from 1 to 12 as y decreases. On the lower edge, where − 12 ≤ x ≤ 0 and y = − 21 , G is a decreasing linear function from 12 to 0 as x increases. On the two edges connected to the re-entrant corner, where 0 ≤ x ≤ 12 , y = 0 and x = 0, − 12 ≤ y ≤ 0, G is zero. The modified finite difference solution u`γ is associated with the vector u`γ (n` i + j), i, j = 0, 1, . . . , n` − 1, where 2n` is the total number of grid nodes (Figure 6.7 represent the numbering of the grid nodes). We define the vector G(n` i + j) associated with the function G as  j if i = 0 and j = 0, 1, . . . , n` − 1, 1 + n` −1    `  i   if i = 1, . . . , n2 and j = 0, 1 − ( n` )   2  ` `  0 if i = n2 and j = 0, 1, . . . , n2 − 1, ` ` (6.5) 2G = 0 if i = n2 + 1, . . . , n` − 1 and j = n2 ,    `   ` i− n2   if i = n` − 1 and j = n2 , . . . , n` − 1, `  n  −1   2 i if i = 0, 1, . . . , n` − 2 and j = n` − 1, 2 − n` −1 The analytic solution of the boundary value problem (3.7) with g = G is unknown and consequently, we can not compute the analytic energy. Therefore, among the four methods of correction, the methods that can be used in determining the modification parameters are method M3 and method M4 since the analytic knowledge of the solution is not required there. We observe in figure 6.9 the values of the extrapolated modification parameters γ `∗ determined using method M3, that uses the defect function g`,3 and 97

n` − 1

n` + (n` − 1)

n` (n` − 1) + (n` − 1)

.. . `

n`

n` 2 +

0

n`

...

n` (n` − 1) + n2

n` 2

`

n` n2

Figure 6.7: L-shaped domain (θ =

3π 2 ):

The numbering of the grid nodes. 1 2

1

0

1 2

0

0

Figure 6.8: L-shaped domain (θ =

3π 2 ):

The function G.

method M4, that uses the defect function g`,4 . The values are close to the values in table 6.1 considering g = u1 . This means that the modification parameters determined considering g = u1 can be used for the correction of problem (3.7) considering g = G.

98

0.1995

0.199

0.1985

0.198

0.1975

0.197

0.1965 5

6

7

8

9

Figure 6.9: L-shaped domain (FD): Values of the extrapolated modification parameter γ `∗ for problem (3.7) with g = G.

6.3

Slit domain

As a second example of a re-entrant corner domain, we consider the slit domain (θ = 2π) 1 1 1 1 1 Ω = (− , ) × (− , ) / [0, ] × {0} , 2 2 2 2 2 where the problem to be solved is (3.7) with g = u1 . Figure 6.10 shows the modified solution of this problem. As we mentioned in section 4, the slit domain has a stronger singularity, compared to the L-shaped domain, as the reduced convergence rates due to the pollution effect are of order 2π θ as confirmed in [48, 104]. In section 6.2, a standard 5-point stencil, with no correction applied, results in a convergence of the different defect functions and the error norm ||u − u`γ ||L∞ (Ω\S) , of order 43 for a L-shaped domain with an interior angle θ = 3π 2 . However we expect a reduction of the convergence rates for the slit domain to be of order one as we observe in figure 6.12. Table 6.2 displays the values γ ` and the extrapolated values γ `∗ of the modifi99

0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

-0.6 -0.4 -0.2 x

0 0.2 0.4 0.6 -0.6

-0.4

-0.2

0.2

0

0.4

0.6

y

Figure 6.10: Slit domain (FD): The numerical solutions of the model problem (3.7) with g = u1 . cation parameter using p = 1 as predicted from the literature [48, 94] at different grid levels `. For the determined modification parameters, we used the four methods of correction; method M1-M4. We observe that the values of the modification parameter in table 6.2 using the finite difference method based on the five-point stencil and the values in table 4.2 using the finite element method based on the Friedrichs-Keller triangulation are almost the same since the two methods are equivalent [33, 39, 40]. We can observe from figure 6.11 that the convergence rates of the discretization error g`,1 determined at some grid nodes P` are of second order as postulated from the literature [48, 94]. The value of the modification parameter used at each grid level ` is the extrapolated value γ `∗ determined in table 6.2 at the finest grid level ` using method M1. We see in figure 6.12 the values of the convergence rates of the first defect function g`,1 using γ ` = 0, and we observe that the values are sub-optimal; of order 1 as predicted in [48, 104]. We compute the convergence rates of the other three defect function g`,2 , g`,3

100

Table 6.2: Slit domain (FD): Values γ ` and extrapolated values of the modification parameter at different grid levels `. Method M1 Method M2 ` `∗ ` γ γ γ` γ `∗ 2 0.31595 0.343706 3 0.331058 0.345515 4 0.339136 0.347214 0.346445 0.347375 5 0.343253 0.347369 0.346913 0.347381 6 0.345319 0.347385 0.347147 0.347381 7 0.34635 0.347382 0.347264 0.347382 8 0.346875 0.3474 0.347321 0.347377 9 0.347119 0.347363 0.347321 0.347382 Method M3 Method M4 ` γ` γ `∗ γ` γ `∗ 3 0.341897 0.299731 4 0.344584 0.322788 5 0.345976 0.347369 0.334985 0.347183 6 0.346678 0.34738 0.34118 0.347376 7 0.346678 0.347383 0.344287 0.347394 8 0.347205 0.347378 0.345825 0.347412 9 0.34729 0.347376 0.346582 0.347412 and g`,4 and the convergence rates of the error norm ||u − u`γ ||L∞ (Ω\S) in figure 6.13 and we can observe the second order convergence rates as for the convergence rates of the first defect function g`,1 . We can see also from the same figure 6.13 that the values of the convergence rates of the three defect functions and the error norm ||u − u`γ ||L∞ (Ω\S) are of first order if γ ` = 0; that is without the correction. An advantage of the defect functions g`,3 and g`,4 is that the knowledge of the analytic solution is not required and therefore, we assume a boundary value problem with an unknown solution such that the solution is as monotonic as u1 . We then compute the modification parameters and compare the values with table 6.2. Our numerical results show that the values are almost the same as the values in table 6.2. We consider problem (3.7) with g = G, where G is a function that is defined to be almost as monotonic as g = u1 . We define G as follows (see figure 6.14): on the right edge of the boundary of the slit domain, where x = 12 and − 12 ≤ y ≤ 12 , G is an increasing linear function that increases from − 12 to 12 as y increases, and 101

2.2

2.15

2.1

2.05

2

1.95

1.9 3

4

5

6

7

8

9

Figure 6.11: Slit domain (FD): Convergence rates of the first defect function g`,1 using extrapolated modification parameter γ `∗ determined at the finest level ` = 9. 1.15

1.1

1.05

1

0.95

0.9 3

4

5

6

7

8

9

Figure 6.12: Slit domain (FD): Convergence rates of the first defect function g`,1 using γ ` = 0.

102

2.05

1.05

1

2

0.95 1.95 0.9 1.9 0.85 1.85 0.8 1.8

0.75

1.75

0.7 0

5

10

0

5

10

Figure 6.13: Slit domain (FD): Convergence rates of the 3 defect functions g`,2 , g`,3 , g`,4 and the error norm ||u − u`γ ||L∞ (Ω\S) . on the slit line, G = 0. On the upper edge, where − 12 ≤ x ≤ 12 and y = 12 , G is a decreasing linear function that decreases from 1 to 12 as x increases. On the left edge, where x = − 21 and − 12 ≤ y ≤ 12 , G is an increasing linear function from 21 to 1 as y increases. On the lower edge, where − 12 ≤ x ≤ 21 and y = − 12 , G is a decreasing function from 12 to − 12 as x increases. The analytic solution of (3.7) with g = G is unknown. Therefore, we can use only method M3 and method M4 in determining the modification parameters. The modified solution after the correction can be seen in figure 6.16. We see in figure 6.17 that the values of the modification parameter for problem (3.7) with g = G are almost the same as the values of the modification parameter of problem (3.7) with g = u1 in table 6.2. This means that the values of the modification parameter for problem (3.7) with g = u1 can be used for the correction of 103

1 2

1

0

0

− 12

1 2

Figure 6.14: Slit domain (θ = 2π): The function G. 1 2

1

1 2

0

0

1 2

1

Figure 6.15: Slit domain (θ = 2π): The function G. the same problem changing the boundary conditions to a function like G, that is almost as monotonic as u1 . Even if the function defined on the boundary of the slit domain is not as monotonic as u1 for problem (3.7), the modification parameters can be the same. For example, we consider problem (3.7) with g = G such that G is defined as (see figure 6.15): on the right edge of the boundary of the slit domain, where x = 12 and − 12 ≤ y ≤ 12 , G is a linear function that decreases from 21 to 0 as y decreases from 12 to 0 and then increases from 0 to 12 as y decreases from 0 to − 12 . On the upper edge, where − 12 ≤ x ≤ 12 and y = 12 , G is a decreasing linear function from 1 to 12 as x increases. On the left edge, where x = − 12 and − 12 ≤ y ≤ 21 , G is a decreasing linear function that decreases from 1 to 12 as y increases from − 12 to 0 and then G increases from 21 to 1 as y increases from 0 to 12 . On the lower 104

1 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 0.6 0.4 0.2 -0.6

-0.4

0 -0.2

0

x

y

-0.2 0.2

0.4

-0.4 -0.6 0.6

(a) g = G

1 0.8 0.6 0.4 0.2 0 0.6 0.4 0.2 -0.6

-0.4

0 -0.2 x

0

-0.2 0.2

0.4

y

-0.4 -0.6 0.6

(b) g = G

Figure 6.16: Slit domain (FD): Numerical solutions of problem (3.7) with analytic solutions being unknown.

105

0.35 0.34 0.33 0.32 0.31 0.3 0.29 0.28 0.27 0.26 0.25 5

6

7

8

9

Figure 6.17: Slit domain (FD): Modification parameters for problem (3.7) with g = G or g = G. edge, where − 21 ≤ x ≤ 21 and y = − 12 , G is a decreasing linear function from 1 to 21 as x increases. On the slit edge connecting the origin (0, 0) to ( 12 , 0), where 0 ≤ x ≤ 12 , y = 0, the function is zero. We present in figure 6.17 the values of the modification parameter γ ` for problem (3.7) with u = G defined in the boundary. We see in figure 6.17 that the values of the modification parameter are very close to the determined values if g = u1 defined in the boundary as in table 6.2. This means that the values of the modification parameter γ ` of problem (3.7) with g = u1 can be used for the correction of the same problem but with a function like G defined in the boundary, even if G is not as monotonic as u1 .

106

7

Conclusion

In this thesis, we consider the Poisson equation defined on a polygonal re-entrant corner domain in R2 with the angle π < θ < 2π (see figure 2.1). Local corrections are studied to eliminate the pollution effect of the re-entrant corners if the solution is a sum of singular functions with large coefficients. To determine accurate modification parameters, even under perturbations, we zoom in with a sequence of smaller auxiliary domains enclosing the singularity. Also, simplified schemes are studied to apply the correction technique and we show that equally accurate finite element solutions as previous methods that are discussed in the literature can be determined using these schemes. Although the correction is performed in a small neighborhood of the singularity where we modify the contribution of every inner node and edge in this small neighborhood to the stiffness matrix, we can perform the correction by modifying the contribution of some of the nodes and the edges of that neighborhood of the singularity. At the end, we propose a set of new defect functions that permit to determine the modification parameters with less numerical cost and without a necessity of knowing the analytic solution. Beside the energy defect function, we use especially the point-wise discretization error. The different defect functions do not necessary lead to the same value of the modification parameter. If the domain has a general re-entrant corner, the discussed correction technique is still valid although we consider in our work the L-shaped domain as a specific example. We assumed that the function defined on π the boundary is the non-smooth singular function r θ sin( πϕ θ ). Then, the sequence of the modification parameters, determined for this problem but defined on a sequence of domains zoomed in toward the fixed singularity has a limit. This limit is the modification parameter determined for this special problem being defined on the original domain. If the slit line of the slit domain is parallel to the x-axis, then the relation between the tip τ of the slit domain and the modification parameter is periodic with its period being the step-size. If the tip of the slit domain is not a fixed grid node, but a point that lays at each grid level at a fixed relative position to a grid node, then it is possible to determine appropriate modification parameters that recover a second order convergence of the defect functions and the L2 - and the L∞ -norms of the discretization error. These norms are measured excluding a neighborhood of the singular point not depending on the step-size. The correction technique applied using the finite element discretization of the Poisson equation with a uniform triangulation and linear test functions, and the correction technique applied using the central finite difference discretization with 107

the five-point stencil, yields almost the same modification parameters since the two methods are equivalent.

108

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