Approximation methods for boundary integral ...

UNIVERSITI BRUNEI DARUSSALAM DOCTORAL THESIS

Approximation methods for boundary integral equations and applications

Author:

Supervisor:

Anh My VU

Prof. Dr. Victor. D. DIDENKO

A thesis submitted in fulfilment of the requirements for the degree of Doctor of Philosophy in the Department of Mathematics Faculty of Science

March 2016

Abstract Approximation methods for boundary integral equations and applications by Anh My VU

In this work, the stability problem of Nyström method and spline Galerkin methods for the double layer potential equation and Sherman–Lauricella equation on contours with corners is considered. We establish necessary and sufficient conditions for the stability of the methods under consideration. It is shown that the approximation method is stable if and only if the corresponding operator and additional operators associated with the corner points are invertible. The invertibility of such operators depends on certain parameters related to the methods and on the opening angles of corner points. However, at present there is no efficient analytical tool to verify the invertibility of the operator involved. Therefore, a numerical approach to this problem is proposed. Thus the corresponding method is applied to equations on model curves which allows us to detect angles where the operators in question are not invertible. It is shown that for the Galerkin methods based on the splines of degree 0, 1, 2, the auxiliary operators associated with the corners having opening angles in the interval [0.1π, 1.9π] are always invertible. On the other hand, the Nyström method for the double layer potential equation has four instability angles in the interval [0.1π, 1.9π]. Moreover, approximate solutions for the equations considered on various curves are found and convergence of the corresponding methods is studied.

Acknowledgements This work was carried out when the author was a PhD student in the Faculty of Science, Universiti Brunei Darussalam under the supervision of Prof. Victor. D. Didenko. The research was partially supported by Grant UBD/GSR/S&T/19. This financial support is highly appreciated. I would like to thank my supervisor, Prof. Victor Didenko for his support, patience, knowledge and enthusiasm. I also want to thank all the lecturers, staffs and students of the Faculty of Science and University for creating a friendly and stimulating environment in which I regain all my energy to do the research. Especially, I would like to thank Dr. Arosha Senayake, Dr. Chong Kim Onn for unrestricted access to the Alienware computers where some of my computations have been done. Finally, I wish to express my deep gratitude to my family: my father Son, my deceased mother Tuan, my stepmother Suu, my wife Linh and my little daughter Minh for their support. I definitely can not go this far without your precious support. Thank you.

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Contents Abstract

i

Acknowledgements

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Contents

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List of Figures

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List of Tables

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1 Introduction 1.1 Boundary integral equations and boundary element methods 1.2 Double layer potential equation . . . . . . . . . . . . . . . . . . 1.3 Biharmonic and Sherman–Lauricella equations . . . . . . . . . 1.4 Organization of the thesis . . . . . . . . . . . . . . . . . . . . . .

1 1 2 6 9

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List of Publications

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2 Auxiliary results 2.1 Algebraic tools . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Banach and C ∗ − algebras . . . . . . . . . . . 2.1.2 Local principle . . . . . . . . . . . . . . . . . . 2.2 Sequence of approximation operators and stability 2.3 Mellin operators . . . . . . . . . . . . . . . . . . . . . . 2.4 Curves and special curves . . . . . . . . . . . . . . . . 2.5 Spline spaces . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Quadrature formulae . . . . . . . . . . . . . . . . . . .

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3 Stability of approximation methods for double layer potential equation 3.1 Fredholm property of the operator of double layer potential equation 3.2 Spline Galerkin methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Spline Galerkin methods: Description and examples . . . . . . 3.2.2 Stability conditions for spline Galerkin method . . . . . . . . . iii

12 12 12 16 17 19 22 25 26 28 28 30 31 34

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Contents 3.2.3 Numerical experiments . . . . . . . . . . . . . . 3.3 Nyström method for double layer potential equation 3.3.1 Nyström method: Description and examples 3.3.2 Stability of the Nyström method . . . . . . . . 3.3.3 Numerical experiments . . . . . . . . . . . . . .

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40 41 42 44 48

4 Spline Galerkin method for Sherman-Lauricella equation 4.1 Spline Galerkin method: Description and examples . . . . . . . . . . . 4.2 Stability of Galerkin method . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Numerical experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

51 51 56 62

5 Application to the approximate solutions of boundary value problems for biharmonic equation 5.1 Boundary value problems with homogeneous and nonhomogeneous boundary data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Boundary integral equation method for approximate solutions of boundary value problems for biharmonic equation . . . . . . . . . . . . . . . . 5.2.1 Approximation method for biharmonic problem with homogeneous boundary data . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 Approximation method for a class of general boundary value problems for biharmonic equation . . . . . . . . . . . . . . . . .

67 67 73 74 76

6 Conclusions 6.1 Approximation methods for double layer potential equation . . . . . . 6.2 Spline Galerkin method for Sherman–Lauricella equation . . . . . . . 6.3 Application to approximate solutions of boundary value problems for biharmonic equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

78 78 79

Bibliography

81

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List of Figures 2.1 The curves L1 and L2 with all corner points angle θ = 0.3π. . . . . . . . . . . . . . . . . . . . 2.2 Curves L4 (θ ) for various θ . . . . . . . . . . . . . 2.3 The curve Γ and the model curve Γτ . . . . . . .

of the same . . . . . . . . . . . . . . . . . . . . . . . .

opening . . . . . . . . . . . . . . . . . .

3.1 ’Pacman’ curve and ’battleax’ curve . . . . . . . . . . . . . . . . . . . . . 3.2 Spline Galerkin solution of the double layer potential equation in the case n = 1024 on ’pacman’ curve L and ’battleax’ curve M with various right hand sides. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Spline Galerkin method for double layer potential equation. Logarithm log10 of condition numbers versus opening angles for various contours and spline spaces. . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Nyström solutions of double layer potential equation on various contours with different right hand sides . . . . . . . . . . . . . . . . . . . . 3.5 Nyström method: condition numbers vs opening angles . . . . . . . . 3.6 Nyström method: condition numbers for some opening angles with varying number of discretization points . . . . . . . . . . . . . . . . . . 4.1 Galerkin solution of Sherman–Lauricella equation on unit square. . . 4.2 Galerkin solution of Sherman–Lauricella equation on rhombus with opening angle π/3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Galerkin solution of Sherman–Lauricella equation on rhombus with opening angle π/4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Galerkin solution of Sherman–Lauricella equation on rhombus with opening angle π/5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Condition numbers for Galerkin method with n = 128. . . . . . . . . . 4.6 Condition numbers for Galerkin method with n = 256. . . . . . . . . . 4.7 Condition numbers for Galerkin method with n = 256 with a verification by n = 512. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8 Condition numbers for Galerkin method with n = 256. The onecorner curve is rotated. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Approximate solution of biharmonic problem with nonhomogeneous right hand side and boundary conditions . . . . . . . . . . . . . . . . . .

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23 24 24 32

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41 44 49 50 54 55 55 56 64 65 66 66 77

List of Tables 3.1 Convergence of Galerkin solution for double layer potential equation 33 3.2 Convergence of Nyström solutions for double layer potential equation 44 4.1 Convergence of Galerkin solution for Sherman–Lauricella equation. . 5.1 Convergence of Galerkin solutions of BVP for biharmonic equation with homogeneous boundary conditions . . . . . . . . . . . . . . . . . . 5.2 Convergence of Galerkin solutions of general BVP for biharmonic equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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53 76 76

Dedicated to my beloved mother. . .

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Chapter 1 Introduction The aim of this work is to study the stability of approximation methods for some classes of integral equations arising in mathematical physics. In this work, the stability problem is studied by means of Banach algebra technique. Using this tool, necessary and sufficient conditions for the stability of some approximation methods for boundary integral equations are obtained.

1.1

Boundary integral equations and boundary element methods

Boundary integral equations are a classical tool for the analysis of boundary value problems for partial differential equations. The term boundary elements method (BEM) means any numerical methods for the solution of boundary integral equation. After obtaining an approximate solution on the boundary, one can restore the solution of the boundary value problem by the solution of the boundary integral equation. The BEM has some advantages in comparison with finite element methods (FEM) or methods of finite difference. In BEM, only the value of the solutions on the boundary needs to be discretized, which allows simple data input and storage. Although the resulting matrices of linear systems are usually dense, their sizes are smaller than ones in FEM. Using BEM, the exterior problems on unbounded domains with bounded boundary and interior problems can be treated almost by the same manner. Furthermore, in some applications, the physically relevant data are represented not by the solution in the interior of the domain but by the boundary 1

Chapter 1. Introduction

2

values of the solution or its derivatives. After an approximate solution of the corresponding integral equation is found, the solution of the boundary value problem and its derivatives in the interior domain can be calculated with a high accuracy. However, BEM also has a number of disadvantages. For example, the integral equations arising are, in general, not ordinary Fredholm integral equations of second kind but equations with singular kernels. Therefore, the standard theory for Fredholm equations of second kind is not applicable and there are difficulties in the study of the approximation methods used. In particular, this happens if the boundary has corner points. The existence of corner points also gives rise to singularities of solutions of boundary value problem at the boundary. In any treatment for BEM, we have to deal with these singularities more directly than for FEM.

1.2

Double layer potential equation

Let D be a simply connected domain in R2 with boundary Γ , and let nτ denote the outer normal to Γ at the point τ ∈ Γ . It is well known that many boundary value problems for Laplace equation can be reduced to boundary integral equations. For example, let us consider the interior Dirichlet problem for Laplace’s equation: ∆u(x) = 0,

x∈D

u(x) = f (x),

x ∈ Γ.

(1.1)

If we represent the unknown function u(x) as double layer potential u(x) =

Z Γ

∂ log(|x − y|)ω(y)dΓy , ∂ ny

where ny denotes the outer normal at the point y, then the density ω(x) is the solution of the following integral equation [57]

ω(x) +

Z Γ

∂ log(|x − y|)ω(y)dΓy = 2 f (x), x ∈ Γ. ∂ ny

(1.2)

In this work, we consider a more general equation which contains an additional compact operator. Moreover, the corresponding equation (1.2) is considered on spaces of complex valued functions. More precisely, let us identify the points (x, y) ∈


3

R2 and t = x + i y ∈ C and consider the equation 1 (AΓ ω)(t) = ω(t) + π

Z

ω(τ)

Γ

d log |t − τ| dsτ + (T ω)(t) = f (t), dnτ

t ∈Γ

(1.3)

where T is a compact operator and dsτ refers to the arc length differential. In the following, we denote, by VΓ , the double layer potential operator, 1 (VΓ ω)(t) = π

Z

ω(τ)

Γ

d log |t − τ| dsτ , dnτ

(1.4)

and rewrite (1.3) in operator form as AΓ ω = (I + VΓ + T )ω = f .

(1.5)

It is well known [72] that the operator VΓ can be represented as VΓ =

1 (SΓ + M SΓ M ) , 2

where SΓ is the Cauchy singular integral operator 1 SΓ u(t) = πi

Z Γ

u(τ) dτ, τ− t

(1.6)

and M is the operator of the complex conjugation. It is also known that the operator VΓ : L 2 (Γ ) → L 2 (Γ ) is continuous and AΓ := I + VΓ invertible in L 2 (Γ ) [83] and in Sobolev spaces H s (Γ ) with s in a neighborhood of 1/2 [17]. The convergence of different numerical methods for (1.3) in various function spaces has been investigated by many authors (see [4, 5, 6, 8, 9, 10, 12, 18, 40, 41, 42, 43, 55, 59, 60, 46]). In a special case where Γ is a polygon, it is well-known that equation (1.3) is closely connected to the following integral equation of Mellin-type,

ω(t) +

Z1 kλ

t s

ω(s)

ds = f (t), t ∈ (0, 1]. s

(1.7)

0

The kernel kλ (x) has the following form kλ (x) = −

sin λπ x . 2 π 1 + x − 2x cos λπ

(1.8)

where (1 − λ)π, −1 < λ < 1 is the interior angle at some corner point of Γ . More


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precisely, the kernel of the double layer potential operator (1.4) is considered on each edge of the polygon. Let t be on an edge. If τ is located on the same edge then the kernel becomes zero. If τ belongs to two neighbouring edges, the kernel admits the form (1.8). For the remaining edges, the kernel is smooth. In this fashion, equation (1.3) can be reduced to a system of compact perturbations of (1.7) (see [8]). Since it is easier to approximate compact operator numerically, the resulting system presents no new difficulty comparing to the original one (1.3). In fact, many authors treated equation (1.3) on polygon by investigating (1.7). Let us rewrite equation (1.7) in the operator form,i.e, (I + K )ω = f ,

(1.9)

where K denotes the integral operator of (1.7). The norm of K on L 2 ([0, 1]) and C([0, 1]) is less than 1 [8, Lemmata 1,2]. This fact simplifies the stability analysis for many approximation method. Equations of the form (1.7) have its own attractiveness. Different kernels k are considered in certain crack problems of elasticity, e.g, cruciform crack in [40] and crack open by internal pressure in [11, 12]. In these cases, the kernels always have properties which secures convergence of the approximation methods at hand. Equation (1.7) has a fixed singularity at the endpoint of integration interval. This equation has been studied in many works [4, 10, 11, 12, 41, 42, 43, 44, 59, 60]. Equation (1.7) appears in [3, 4] where equation (1.3) on an open wedge is studied. By a change of variable, this equation is reduced to an Wiener-Hopf equation. Moreover, collocation method using piecewise linear and quadratic polynomial on uniform mesh has been studied. Under some mild restriction on the collocation points, it is shown that the sequence of approximate inverse operators is uniformly bounded . Following this work, [8] deals with the Galerkin method based on piecewise polynomial on graded meshes for equation (1.3) on polygon. It is shown that, on L 2 space, Galerkin and the iterated Galerkin solutions converge to the exact solution provided that the mesh is graded intensively enough near the corner points. The convergent rate depends on the grading level. Convergence results in uniform norm are obtained in [9] where, for the iterated Galerkin method, a nearly super convergence rate in uniform norm was obtained. A similar result was established for equation (1.7) in [10]. Convergence of collocation methods for equation (1.3) on polygons in Sobolev norms has been studied in [18].


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A modified collocation method is considered in [12] where the discontinuous piecewise polynomials are used as basis functions. The authors replace high-order piecewise polynomials by piecewise constant ones in some neighborhood of the corner points and show the stability of collocation method in L ∞ . Using this idea, Elschner investigates a variety of approximation methods for the equation of the form (1.7) in various spaces [41, 42, 43, 44]. In particular, it is shown that if the function space is appropriately modified near the corner points, then the methods under consideration are stable provided that the original operator is invertible. Continuous piecewise polynomials are used as basis functions in [41] for both Galerkin method and collocation method applying on an extended class of Mellin type singular integral equation. Stability results in a weighted Sobolev norm are obtained. Thus, the kernel k of (1.7) is considered as an inverse Mellin transform of a function a and operator K of (1.9) is considered as a pseudodifferential operator of Mellin type with symbol a. The corresponding operators contains, e.g, single layer potential operators arising from exterior Neumann problems. These results are extended to the case of Galerkin method using smoothest splines as trial functions in [42]. The ideas of [41] are further developed in [42, 43] and a comprehensive survey of this approach is presented in [45]. In order to improve the convergence rate of the methods, Richardson’s extrapolation scheme is used for iterated collocation method ([66]), Nyström method ([65]). The extrapolation procedures remedies for the singularities near corner points and discontinuities of the functional spaces. The scheme improves the accuracy of the approximate solutions remarkably. In certain physical applications, linear functionals of the solution have meaningful interpretation, for example, stress intensity factor and crack energy are respectively represented by linear functionals of inner product and point values type (see [11]). In [11], a cut-off technique for Nyström approximation method for equations (1.3) and (1.7) has been proposed. Combining the cut-off technique with mesh grading near corner points, one can establish uniform convergence results for Nyström solutions. This allows us to calculate the linear functionals with a satisfactory accuracy. The authors of [59] generalize this idea by introducing a cut-off function and applying to Nyström discretization based on trapezoidal rule on a graded mesh. In combination with graded mesh it allows to obtain uniform convergence of Nyström solutions. Later on, efforts have been made to improve the quadrature rules for Nyström method, see, e.g, [5, 6, 7, 55, 60]. However, the papers [5, 6, 7] do not contain any stability analysis.


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Further, approximation methods for singular integral equation a(t)x(t) + b(t)SΓ x(t) + T x(t) = f (t), where a(t), b(t) are discontinuous coefficients, Γ is a piecewise smooth contour and T is compact operator, have been studied. There are developed methods using results from Banach algebra theory (see the papers [75, 76, 77] and the monographs [50, 51, 79, 78]. In these works, Banach algebra techniques have been applied to various approximation methods, e.g, spline collocation [75], spline Galerkin [76], quadrature and collocation [77]. An extension of these works to the real Banach algebra settings is proposed in [25, 26] and the monograph [29]. In these works, the analysis has been adapted to the equations with complex conjugation which often arise in applications, e.g, double layer potential equation, Sherman–Lauricella and Muskhelishvili equations. The analysis of approximation methods is different from those mentioned above since the corresponding operators are not of the form I + K + T where I is the identity operator, K is a noncompact operator with small norm and T is compact. The typical results obtained state that the approximation method under consideration is stable if and only if the original operator and certain operators associated with the corner points of the curve Γ are invertible. This approach has been applied to double layer potential equation with continuous coefficients in [27]. Moreover, quadrature methods using adaptive meshes have been studied. However, necessary and sufficient conditions of stability are not easy to verify. In this thesis, we follow this line of research by using Banach algebra technique to study the Nyström method and spline Galerkin methods. To verify the stability conditions, we propose a numerical procedure.

1.3

Biharmonic and Sherman–Lauricella equations

Many problems of applied mathematics can be reduced to boundary value problems for biharmonic equation. For example, the problem of finding biharmonic Airy’s stress function U on D ⊂ R2 whose first derivatives U x and U y have prescribed values on the boundary ∂ D = Γ is well-known in the theory of linear elasticity [48, 68, 69, 71, 74]. In theory of incompressible fluid dynamics, creeping flows are governed by a biharmonic stream function U with velocity (U y , −U x ) given on the boundary, see, e.g, [61, 73]. These problems can be formulated as boundary


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value problems for biharmonic equation of the form ∆2 U (x, y) = 0,

(x, y) ∈ D

U x = f1 , U y = f2 ;

(x, y) ∈ Γ .

(1.10)

Another boundary value problem for biharmonic equation encountered in the literature is the clamped plate equation [15, 16], ∆2 U (x, y) = 0, ∂U U Γ = g1 , = g2 ; ∂n Γ

(x, y) ∈ D, (1.11)

(x, y) ∈ Γ .

Let α = α(x, y) denote the angle between real axis and the outer normal n at (x, y) ∈ Γ and l the unit vector such that the angle between l and the real axis is α − π/2. It is known that boundary value problems (1.10) and (1.11) can be reduced to the following boundary problem for holomorphic function of complex variable: Find two holomorphic functions ψ and φ such that φ(t) + tφ 0 (t) + ψ(t) = h(t), t = x + i y, (x, y) ∈ Γ , ∂ g1 where h(t) = f1 (t) + i f2 (t) or h(t) = e iα g2 (t) − i (t) for the problems (1.10) ∂l and (1.11), respectively. The functions φ(t) and ψ(t) are sought in the form 1 φ(t) = 2πi

Z

1 ψ(t) = 2πi

Z

Γ

Γ

ω(τ) dτ, τ− t ω(τ) 1 dτ + τ− t 2πi

Z Γ

ω(τ) 1 dτ − τ− t 2πi

Z Γ

τω(τ) dτ, t ∈ D. (τ − t)2

(1.12)

This choice of the functions φ and ψ is closely related to the familiar single and double layer potentials which are used to solve Laplace’s equations. Letting t tend to a point on the boundary Γ and using classical formulae for the limits of Cauchy integrals, one obtains the following equation 1 ω(t) + 2πi

Z

τ− t

1 ω(τ) d ln − 2πi τ− t Γ

Z Γ

ω(τ) d

τ− t τ− t

= h(t),

t = x +iy ∈ Γ, (1.13)

where the bar denotes the complex conjugation and ω is an unknown function. This equation is called the Sherman–Lauricella equation. If we have an exact or approximate solution ω(t) of (1.13), then the solution U (x, y) of (1.10) or (1.11)


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in the domain D can be obtained by using representations (1.12) and the famous Goursat’s formula U (x, y) = ℜ(tφ(t) + χ(t)), χ 0 (t) = ψ(t). Equation (1.13) originated in works of G. Lauricella [62]. He was the first who used the method of integral equations in elasticity. Later D.I. Sherman rewrites Lauricella equation in a complex form and proposes a new simple way to derive it [81]. The equation (1.13) is uniquely solvable in appropriate functional spaces, provided h satisfies certain smoothness conditions and Z Re

h(t) d t = 0,

(1.14)

Γ

see, e.g, [37, 68, 74]. It is well known that the operator of the left-hand side of (1.13) is not invertible in main functional spaces so approximation methods can not be applied to this equation directly. However, this operator can be corrected by a compact operator in such a way that the solution of the corrected equation exists uniquely and is simultaneously the solution of the original equation. Sherman-Lauricella equation corresponding to equation (1.10) arising from different physical scenarios has been considered in [48, 49, 61]. The authors of these works solve it by using a Nyström method based on trapezoidal rule. The boundaries of the corresponding domains are supposed to be smooth curves. Numerical results obtained in these papers show a good convergence but a rigorous stability analysis is missing. Recently, Nyström method applied to equation (1.13) in the case of piecewise smooth contours is studied in [21, 22]. Necessary and sufficient conditions for stability of the method have been established. In addition, a numerical approach to verify these conditions has been proposed. It is shown that for equations on piecewise smooth curves, the Nyström method is not always stable. The stability depends on opening angles of the corners. More precisely, there are 8 critical angles which cause instability and these points are distributed symmetrically with respect to π. In [23], an approximation method for the velocity and vorticity fields of a Stokes flow in non-smooth domains has been proposed. This approximation method shows a good accuracy even in the case of discontinuous boundary data. Recently, a modification of Galerkin method called "fast Fourier–Galerkin method" has been proposed for the equation (1.13) in smooth domain [56]. The accuracy of numerical results presented is comparable with ours.


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In the present work, we study stability problem of spline Galerkin method for equation (1.13) by using Banach algebra technique. A numerical verification for stability conditions obtained is proposed.

1.4

Organization of the thesis

This thesis consists of six chapters and a bibliography. Chapter I contains a short survey of literature on approximation methods for double layer potential equation and for Sherman–Lauricella equation. In Chapter II, definitions and auxiliary results are presented. Section 1 recalls results of general theory of Banach and C ∗ − algebras and local principle which are needed to study the stability of numerical methods. Section 2 deals with sequences of approximation operators and their stability. In Section 3, theory of Mellin operators which arises in the localization process is discussed. Section 4 is devoted to main curves and special curves used in this work. Section 5 considers splines spaces constructed on curves introduced in Section 4. Quadrature formulae frequently used in this work are the topic of Section 6. Chapter III deals with approximation methods for double layer potential equation. Fredholmness of the operator of double layer potential equation is discussed in Section 1. Section 2 considers spline Galerkin method. Three subsections of this Section are devoted to the description and illustration of the method, stability analysis and numerical experiments, respectively. Nyström method is addressed in Section 3. Description together with illustration of the method, stability conditions and numerical verifications are presented in its three subsections, respectively. In Chapter IV, we discuss the Galerkin method for the Sherman–Lauricella equation. Section 1 contains description and illustration of the method. Stability conditions are established in Section 2. Section 3 is devoted to numerical verifications of the stability conditions obtained in Section 2. Chapter V considers applications of boundary integral equation method. As an auxiliary result, we first discuss a transition between boundary value problems for biharmonic equation with homogeneous right-hand side and homogeneous boundary conditions in Section 1. Section 2 is devoted to the approximate solution of boundary value problem with homogeneous boundary conditions. In Section 3, an approximate solution for a full boundary value problem for biharmonic equation is proposed. Numerical experiments are included in each Section.


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In Chapter VI, we discuss the work done in the thesis and a possibility for further developments.

List of Publications Paper 1 Didenko, V. D. and Vu, A. M. Critical angles of approximation methods of boundary integral equations, 4th European Seminar on Computing, Plzen, Czech Republic, 15–20 June, 2014, Book of Abstracts, p. 44.

Paper 2 Didenko, V. D. and Vu, A. M. Approximation methods for boundary integral equations on curves with corners, International Congress of Mathematicians, Seoul, Korea, August, 13–23, 2014, Abstracts. Short Communications. p. 525.

Paper 3 Didenko, V. D. and Vu, A. M. The Nyström method for the double layer potential equations on contours with corners, submitted to Applied and computational mathematics, ArXiv:1410.3044, 2014.

Paper 4 Didenko, V. D., Tao, T. and Vu, A. M., Spline Galerkin methods for the ShermanLauricella equation on contours with corners, SIAM J. Numer. Anal., 53(6), 2015.

Paper 5 Didenko, V. D. and Vu, A. M., Spline Galerkin methods for the double layer potential equations on contours with corners, accepted in Recent Trends in Operator Theory and Partial Differential Equations - The Roland Duduchava Anniversary Volume, Operator Theory. Advances and Applications, Eds.:V. Maz’ya, D. Natroshvili, E. Shargorodsky, and W. L. Wendland, Springer/Birkhäuser, 2015.

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Chapter 2 Auxiliary results In this Chapter, we present relevant definitions and related results used in this work.

2.1 2.1.1

Algebraic tools Banach and C ∗ − algebras

Let C and R denote the field of complex and real numbers, respectively. A complex algebra is an associative ring A which is also a complex vector space. Assume that vector space addition and ring addition coincide and that the operations of multiplication and multiplication by scalars satisfy the relation λ(x y) = (λx) y = x(λ y)

(2.1)

for all x, y ∈ A and for all λ ∈ C. A subset of an algebra which is, with respect to the operations inherited, an algebra again, is called a subalgebra of the given algebra. Similarly, a real algebra is an associative ring A which is a real vector space with the ring and vector space additions coinciding, and the relation (2.1) holds for all x, y ∈ A and for all λ ∈ R. Obviously, every complex algebra is a real algebra but the reverse claim is not always correct. One of the reasons is the absence of the operation of multiplication with complex scalars in A. However, many algebras emerging from applications have this operation but fail to satisfy relation (2.1). Therefore, standard tools used in complex situation either are not available or have to be adapted properly. 12

Chapter 2. Auxiliary results

13

Definition 2.1. A complex (real) Banach algebra is a complex (real) algebra A equipped with a Banach space norm k · k such that kx yk ≤ kxkk yk for all x, y ∈ A. Definition 2.2. A complex algebra A is called an algebra with involution or a ∗ algebra if there exists a map ∗ : A → A called involution, such that (a + b)∗ = a∗ + b∗ , (ab)∗ = b∗ a∗ , (λa∗ ) = λa∗ , (a∗ )∗ = a for all a, b ∈ A and for all λ ∈ C. Let A be a real or complex Banach algebra. A subset J of A is called an ideal of A if

J is a linear subspace of A and ja, a j ∈ J for all j ∈ J and a ∈ A. An ideal J ⊂ A is called proper if J 6= 0 and J 6= A. A proper ideal J is said to be maximal if it is not strictly contained in any other proper ideal of A. Note that any maximal ideal J is closed, viz., the subspace J is closed.

A Banach algebra A is called unital if it has an identity element e. An element a of an unital Banach algebra A is called invertible if there exists b ∈ A such that ba = ab = e.

To each Banach algebra A and each closed ideal J of A one can assign another Banach algebra A/J which is called quotient algebra. The elements of A/J are the cosets a0 := a + J, a ∈ A. It is easily see that the product (a + J)(b + J) := ab + J is well defined. The norm on A/J is given by ka + Jk := inf ka + jk. j∈J

It is easily seen that e0 := e + J is the identity element of the quotient algebra A/J if e is the identity element of A. A complex algebra homomorphism is a mapping between unital complex algebras which preserves addition, multiplication, scalar multiplication and the identity element. A complex ∗ −algebra homomorphism has to preserve the operation of involution as well. Corresponding real algebra homomorphisms are defined similarly.


14

Note that there is a close connection between ideals and homomorphisms. Thus the kernel Ker W = {a ∈ A : W a = 0} of every homomorphism W is an ideal of A and conversely, given an ideal J of an algebra A there exists an algebra B as well as a homomorphism W : A → B such that J is the kernel of W . Such a homomorphism is given by A → A/J,

a 7→ a + J and called canonical homomorphism.

Definition 2.3. A complex C ∗ −algebra is a complex Banach ∗ −algebra such that ka∗ ak = kak2 . Definition 2.4. A real algebra A is called a ∗ −algebra or algebra with involution if there exists a map ∗ : A → A such that (a + b)∗ = a∗ + b∗ , (ab)∗ = b∗ a∗ , (λa∗ ) = λa∗ and (a∗ )∗ = a for all a, b ∈ A and for all λ ∈ R. The definition of a real Banach algebra repeats the corresponding definition for the complex case. Definition 2.5. A real C ∗ −algebra is a real Banach ∗ −algebra such that ka∗ ak = kak2 for all a ∈ A, and such that the element e + a∗ a is invertible in A for any a ∈ A. Some basic properties of complex C ∗ −algebra are as follows [50]: a) If A is a C ∗ −algebra and J is an ideal of A then J is self-adjoint (J∗ = J), and

A/J provided with the involution (a + J)∗ = a∗ + J is a C ∗ −algebra.

b) If A is a C ∗ −algebra, B is a C ∗ −subalgebra of A, and J is an ideal of A then

B + J is a C ∗ −subalgebra of A, and the C ∗ −algebra (B + J)/J and B/(B ∩ J) are isometrically isomorphic. c) If A is a unital C ∗ −algebra and B is a C ∗ −subalgebra of A containing the identity element then B is inverse closed in A. Recall that subalgebra B is inverse closed in A if an element b ∈ B is invertible in B if and only if it is invertible in A. This fact allow us to switch to an appropriate C ∗ −subalgebra of a given C ∗ −algebra when dealing with invertibility problem.


15

The inverse closedness property mentioned in c) is crucial in many applications of complex C ∗ −algebra. However, for real C ∗ −algebra, we need some appropriate modifications. Thus, let us consider the so-called real extension of complex ∗

−algebras [29]. Let R be a real algebra. Assume that R contains a complex uni-

tal C ∗ −algebra A and an element m which does not belong to A and satisfy the following assumptions: (A1 ) For each a ∈ A the element mam belongs to A as well. (A2 ) For each a ∈ A and for each λ ∈ C the relation m(λa) = λma holds. (A3 ) m2 = e and me = m. (A4 ) The null element 0 of A is also that of R. (A5 ) For each a ∈ A, (mam)∗ = ma∗ m, where ∗ denotes the given involution on A. ˜ ⊂ R, which consists of all elements a˜ ∈ R having the form a˜ = The subset A ˜ is a real b + cm, b, c ∈ A is called the extension of A by the element m. Note that A algebra. Moreover, it can be made into a real ∗ −algebra by introducing an involution ? ˜ ˜ by the rule a˜? := b∗ + mc ∗ if a˜ = b + cm. :A→A

For real C ∗ −subalgebra of complex C ∗ −algebra, the inverse closedness property remains unchanged. Proposition 2.6 ([29], Corollary 1.4.5). Let B be a real C ∗ −subalgebra of the complex C ∗ −algebra A. Then B is inverse closed in the algebra A. Let A be a complex C ∗ −algebra as before and C be complex C ∗ −subalgebra of A and ˜ be the real extension of C by m an element of R which does not belong to A. Let C ˜ of A. m in the real extension A Definition 2.7. A complex C ∗ −subalgebra C of a real C ∗ −algebra A is called m−closed if mCm ⊆ C, and if the axioms (A2 ) − (A5 ) with respect to A and C are satisfied. Proposition 2.8 ([29], Corollary 1.4.10). If C is an m−closed C ∗ −subalgebra of an ˜ is inverse closed in m−closed complex C ∗ −algebra A, then the real C ∗ −subalgebra C ˜. the real algebra A


2.1.2

16

Local principle

Definition 2.9. Let A be a real or complex Banach algebra with the identity e. The center Cen A of A is the set of all elements z ∈ A with the property that za = az for all a ∈ A. Clearly, Cen A is a closed commutative subalgebra of A. Let B be a closed subalgebra of Cen A containing e. Thus B is also commutative. If N ⊂ B is a maximal ideal of

B, then let JN denote the smallest closed ideal of A containing N, i.e, ¨

JN := clos

m X

« x k a k : x k ∈ N , a k ∈ A, m ∈ N .

k=1

Suppose that B is a complex C ∗ −algebra if A is a complex Banach algebra, and that B is a strictly real C ∗ −algebra if A is a real algebra. The latter means that the involution in B is the identity operator. This makes sense because B is commutative. Equivalently, B is a commutative real Banach algebra such that kb2 k = kbk2 and e + b2 is invertible for all b ∈ B. Note that in both cases there is a Hausdorff compact M such that B is isometrically isomorphic to the algebra of all continuous complex-or real-valued functions on M. For the complex case, this is the well-known Gelfand-Naimark theorem, and for the real case the assertion follows from [47, Chapter 11]. In what follows we identify algebra B with the related function algebra. Recall that the maximal ideals of B are precisely the sets N = Nτ , τ ∈ M where

Nτ := { f ∈ B : f (τ) = 0}, τ ∈ M. We are now in position to state a variant [29, Theorem 1.9.5] of the Allan’s local principle [1]. Theorem 2.10. Let A be a real or complex Banach algebra with identity e, and let

B ⊂ Cen A be a subalgebra of the specified type containing e. If a ∈ A then a is invertible in A if and only if for each N ∈ M the coset aN := a + N is invertible in the quotient algebra AN := A/JN .


2.2

17

Sequence of approximation operators and stability

Let X be a real or complex Banach space. Given a linear bounded operator A on X , consider the operator equation Ax = y, y ∈ X .

(2.2)

For the approximate solution of this equation, we choose a sequence of appropriate closed finite dimensional subspaces X n of X . Approximate solutions of (2.2) are sought in the subspaces X n . Assume that X n ’s are ranges of certain projection operators L n : X → X n and that s − lim L n = I where s − lim refers to the strong limit and I is the identity operator. Having fixed the sequence of subspaces X n , we choose linear operators An : X n → X n and consider the equations An x n = L n y,

n = 1, 2, . . . .

(2.3)

Definition 2.11. A sequence (An ) of operators An ∈ L (Im L n ) is called an approximation method for A ∈ L (X ) if An L n converges strongly to A as n → ∞. Definition 2.12. The approximation method (An ) for the operator A is said to be applicable if there exists a number n0 such that equations (2.3) are uniquely solvable in X n for every n ≥ n0 and every right hand side y ∈ X and if the sequence of approximate solutions converges to a solution of (2.2) in the norm of X . Definition 2.13. A sequence (An ) of operators An ∈ L (Im L n ) is said to be stable if there exists a number n0 such that the operators An are invertible for every n ≥ n0 and if the norms of their inverses are uniformly bounded: L n k < ∞. sup kA−1 n

n≥n0

Now let us turn to algebras of sequences of approximation operators. Let A refer to the set of all sequences (An )∞ of operators An ∈ L (Im L n ) which are uniformly n=0 bounded: supn≥0 kAn L n k < ∞. Provided with the operations (An ) + (Bn ) := (An + Bn ), (An )(Bn ) := (An Bn ), λ(An ) := (λAn ),


18

the set A becomes an algebra. This algebra is unital with the identity (L n ). It is well known that A is a Banach algebra with the norm k(An )k := supn≥0 kAn L n k. Let us introduce the set G of all sequence (Gn ) ∈ A with limn→∞ kGn L n k = 0. Note that G is a closed ideal of A . The following observation is due to Kozak [58]. Theorem 2.14. A sequence (An ) ∈ A is stable if and only if its coset (An ) + G is invertible in the quotient algebra A /G . The following corollary of Theorem 2.14 shows that a small perturbation of a stable sequence is still stable. Corollary 2.15. Let (An ) ∈ A be a stable sequence and let (Sn ) be a sequence such that lim sup kSn L n k < lim inf kA−1 L n k−1 . n Then the sequence (An ) + (Sn ) is stable. Usually, the quotient algebra A /G is too large in order to study invertibility problem efficiently. The problem becomes more manageable if one find a smaller algebra B ⊂ A and an ideal J ⊃ G such that for elements from B the invertibility problems in A /G and B/J are equivalent. Assume now that the adjoint projections L n∗ also converge to the identity operator I ∗ of the dual space X ∗ . Consider now the set B of all the sequence (An ) ∈ A for which both sequences (An L n ) and (A∗n L n∗ ) are strongly convergent in X and X ∗ , respectively. Let us denote the strong limit of a sequence (An ) by W (An ). We have the following result. Theorem 2.16. The set B is a closed subalgebra of A which contains the identity element of A and the mapping W : B → L (X ), (An ) 7→ W (An ) is a continuous unital homomorphism with norm 1. It is clear that the ideal G of A is a closed ideal of B but this algebra possesses a larger ideal which consists of all possible compact perturbations of approximation sequences [29, Proposition 1.6.4]. Thus, let K (X ) denote the collection of all compact operators on X . Theorem 2.17. The set J of all sequences (L n K L n ) + (Gn ) where (Gn ) ∈ G and K ∈ K (X ) is a closed ideal of B. Furthermore, a sequence (An ) ∈ B is stable if and only if W (An ) is invertible and if its coset (An ) + J is invertible in the quotient algebra B/J .


19

This theorem implies that all compact perturbations of a stable sequence are stable again, provided only that the strong limits of them are invertible.

2.3

Mellin operators

The invertibility problem for the cosets (An ) + J can be tackled efficiently by using local principle. Localization process in this work usually leads us to a position where "model problems" on "special curves" get involved. It turns out that the integral operators under consideration on such "special curves" are isometrically isomorphic to certain block Mellin operators, therefore it is reasonable to mention Mellin operator here. Let p and α be fixed real numbers satisfying p > 1, 0 < 1/p + α < 1 and consider the weighted Lebesgue space L p (R, α) endowed with the norm kf k =

Z

αp

| f (t) |t| d t p

1/p .

R

The Lebesgue space L p (R+ , α) is defined analogously. The singular integral operator SR (SR+ ) is defined by 1 (SR f )(t) := πi

Z f (s) f (s) 1 ds (SR+ f )(t) := ds , s−t πi R+ s − t R

Z

where the singular integral should be understood in the sense of Cauchy principle value. The singular integral operator SR (SR+ ) is bounded on the space L p (R, α) (L p (R+ , α)). It is well known that SR is a Fourier convolution operator. To be more specific, let F denote the Fourier transform acting on Schwarz space S(R) of rapidly decreasing functions via (F f )(z) =

Z

e−2πi xz f (x)d x, z ∈ R,

R

and write F −1 for the inverse Fourier transform (F −1 f )(x) =

Z

e2πi xz f (z)dz, x ∈ R.

R

One can show that the restriction of the singular integral operator SR onto L p (R, α)∩ L2 (R, 0) coincides with the restriction of the Fourier convolution operator F −1 a0 F


20

with the generating function a0 (z) = sign z on the same space, viz., SR = F −1 a0 F . Moreover, if b is function with bounded total variation, then the operator F −1 bF is well defined on L p (R, α) ∩ L2 (R, 0). If this operator extends boundedly onto all of L p (R, α) then this extension is called the Fourier convolution operator and denoted by W 0 (b), and the function b is referred to as an L p (R, α)-Fourier multiplier. Each L p (R, α)-multiplier is a bounded function. Conversely, if b is a bounded function with finite total variation then b is an L p (R, α)-multiplier. The Mellin transform M is defined (depending on p and α) by

(M f )(z) =

Z∞

x 1/p+α−zi−1 f (x)d x, z ∈ R,

0

and the inverse Mellin transform M −1 by +∞ Z

1 (M−1 f )(x) = 2π

x zi−1/p−α f (z)dz, x ∈ R+ .

−∞

Further, we introduce operators E p,α by 1 (E p,α f )(x) = f 2π

1 ln x x −1/p−α , x ∈ R+ . 2π

One can easily check that kE p,α f k L p (R+ ,α) = (2π)1/p−1 k f k L p (R,0) , and that ME p,α = F . If b is an L p (R, 0)-multiplier then the operator −1 M 0 (b) := E p,α W 0 (b)E p,α

is bounded on L p (R+ , α), and we call it the Mellin convolution by b or the Mellin convolution operator, or simply by Mellin operator. The function b is called the symbol of the corresponding Mellin operator M 0 (b). Another representation of Mellin operators is available for some special class of symbol b, viz., b together with its Mellin image k = M−1 b are continuously differentiable and there exist constants m1 , m2 and polynomials P1 , P2 such that sup |b(z)(1 + |z|)| ≤ m1 , sup |b0 (z)(1 + |z|)2 | ≤ m2 z∈R

z∈R


21

and |k(x)(1 + x)| ≤ P1 (| ln x|), |k0 (x)(1 + x)2 | ≤ P2 (| ln x|). In this case, one has the following integral representation Z∞ x ds (M 0 (b) f )(x) = k f (s) , x ∈ R+ . s s

(2.4)

0

Some examples of Mellin operators are as follows: For each complex number β with 0 < ℜβ < 2π, the generalized Hankel operator Z

1 (Nβ f )(t) = πi

R+

f (s) ds, t ∈ R+ s − e iβ t

is the Mellin convolution operator M 0 (nβ ) with the symbol e(z+i(1/p+α))(π−β) nβ (z) = . sinh π(z + i(1/p + α)) Further, the weighted Hankel operator 1 (Nβ,γ f )(t) = πi

t γ

Z R+

s

f (s) ds, t ∈ R+ s − e iβ t

for parameters γ satisfying 0 < ℜ(1/p + α + γ) < 1, is again the Mellin convolution operator M 0 (nβ,γ ) with the symbol nβ,γ (z) =

e(z+i(1/p+α+γ))(π−β) . sinh π(z + i(1/p + α + γ))

The weighted singular integral operator 1 (SR+ ,γ f )(t) = πi

t γ f (s) ds, t ∈ R+ s s−t R+

Z

is also a Mellin operator with the symbol sγ (z) = coth π(z + i(1/p + α + γ)).


2.4

22

Curves and special curves

Let Γ be a simple piecewise smooth positively oriented contour in the complex plane. Let L 2 (Γ ) denote the set of all Lebesgue measurable functions f such that Z || f || L 2 :=

1/2

| f (t)| |d t| 2

< ∞.

Γ

For the study of approximation methods and numerical computations we need an appropriate parametrization for the curve. Thus, let γ : R 7→ C be a 1-periodic parametrization of Γ . By MΓ we denote the set of all corner points τ0 , τ1 , . . . , τq−1 of Γ . We suppose further that j τj = γ , j = 0, 1, . . . , q − 1. q

(2.5)

In addition, we also assume that the function γ is two times continuously differentiable on each subinterval ( j/q, ( j + 1)/q) and γ0 j + 0 = γ0 j − 0 , j = 0, 1, . . . , q − 1. q q

(2.6)

Let L p := L p (Γ , w) denote the collection of all Lebesgue measurable functions f satisfying the condition Z

1/p

| f (t)| w(t) |d t| p

|| f || L p :=

< ∞,

1 < p < ∞,

Γ

where w(t) :=

Qq−1 j=0

|t − τ j |α j , 0 < α j + 1/p < 1.

For numerical experiments in the upcoming Chapters, we essentially use the following curves: Curve L1 with the parametrization γ1 (s) = sin(πs) exp(iθ (s − 0.5)),

s ∈ [0, 1].


23

0.2

0.5 0.4

0.15

0.3

0.1 0.2

0.05

0.1

0

0 −0.1

−0.05

−0.2

−0.1 −0.3

−0.15

−0.4

−0.2 0

0.2

0.4

0.6

0.8

1

−0.5

−0.1

−0.05

0

0.05

0.1

0.15

FIGURE 2.1: The curves L1 , left, and L2 , right. All corner points have the same opening angle θ = 0.3π.

Curve L2 with the parametrization  1 1  exp(iθ (2s − 0.5)) − cot(θ /2) + 2 2 sin(θ /2) γ2 (s) = 1 1   cot(θ /2) − exp(iθ (2s − 1.5)) 2 2 sin(θ /2)

if 0 ≤ s ≤ 1/2; if 1/2 < s ≤ 1.

The curves L1 and L2 have, respectively, one and two corner points of the magnitude θ ∈ (0, 2π). The graphs of them are presented in Figure 2.1. Moreover, we use an additional 4-corner curve L4 constructed as follows: First, we connect the two points A = (1 − i) and B = (1 + i) by an arc representing a Hermit interpolation polynomial such that Û Û −→ − −→ − → → OA, t A = π − θ /2, OB, t B = θ /2, − → − → where O denotes the origin, t A and t B are tangential vectors at the points A and B,

_

respectively (see Figure 2.2). After obtaining the arc AB, we rotate the arc around the origin by angles 0.5π, π and 1.5π. Some curves from this family are presented in Figure 2.2. Note that the Hermit interpolation polynomial used is P3 (s) = 1 − i + as + (2i − a)s2 + (a + b − 4i)s2 (s − 1),

0 ≤ s ≤ 1,

where 3π θ 3π θ π θ π θ a = 3 sin + + 3i cos + , b = 3 sin − + 3i cos − . 4 2 4 2 4 2 4 2

With each point τ ∈ Γ we associate a special curve Γτ as follows. Let θτ be the angle between the right and the left semi-tangents to Γ at the point τ, and let βτ refer to


24

2 θ=0.2π

1.5

θ=0.5π

B(1,1)

θ=π θ=1.5π

1 0.5 0 −0.5 −1 A(1,−1)

−1.5 −2 −2

−1.5

−1

−0.5

0

0.5

1

1.5

2

FIGURE 2.2: Curves L4 (θ ) for various θ .

FIGURE 2.3: The curve Γ and the model curve Γτ .

the angle between the right semi-tangent to Γ and the real line R. Consider now the curve Γτ := e i(βτ +θτ ) R+− ∪ e iβτ R++

(2.7)

where R+− and R++ denote the positive semi-axis R+ correspondingly directed to and away from the origin.


2.5

25

Spline spaces

Let f and g be functions defined on the real line R, and let f ∗ g denote the convolution ( f ∗ g)(s) :=

Z

f (s − x)g(x)d x

R

of f and g. If χ is the characteristic function of the interval [0, 1), ¨ χ(s) :=

1

if s ∈ [0, 1),

0

otherwise,

Ò= φ Ò(d) (s) refers to the function defined by then φ ¨ Ò(d) (s) := φ

χ(s)

if d = 0,

Ò(d−1) )(s) (χ ∗ φ

if d = 1, 2 . . . .

Ò generates spline Recall that for any given non-negative integer d, the function φ spaces on R. Thus if an n ∈ N is fixed, then closure in the L 2 -norm of the set of all Òn j (s) := φ(ns Ò − j), j ∈ Z constitutes a finite linear combinations of the functions φ spline space on R. Using the above defined spline functions, one can introduce spline spaces on the contour Γ . These spline spaces will play the role of the target subspaces of our projections. More precisely, for a fixed non-negative integer d and an n ∈ N, n ≥ d + 1, we denote by Snd = Snd (Γ ) the set of all linear combinations of the functions Òn j (t) := φ Ò(d) (ns − j), φ

t = γ(s) ∈ Γ ,

j = 0, 1, . . . , n − (d + 1),

s ∈ R.

This spline space will be used as the subspace where the approximate solution of double layer potential equation is sought by Nyström method. For the approximate solution sought by Galerkin method, the space is slightly modified. Thus, by I (n, d) we denote the set of all integers j ∈ {0, 1, . . . , n − (d + 1)} such that the interval d [ j/n, ( j + d + 1)/n] does not contain any point sk = k/q, k = 0, 1, . . . , q. Let Sn,∗ = d Sn,∗ (Γ ) be the set of all linear combinations of the functions

Òn j (t) = φ Ò(d) (ns − j), t = γ(s) ∈ Γ , φ

s ∈ R,

j ∈ I (n, d).

Ò of the function φ Ò is contained This definition is correct since the support supp φ Òn j in the interval [0, d + 1] [80] and γ is a 1-periodic function. For each spline φ


26

d belonging to Snd or Sn,∗ let

where

p Òn j , φn j := νd nφ −1/2  d+1 Z 2 Ò(d) . (s) ds νd =  φ

(2.8)

0

It is easily seen that φn j are normalized functions, i.e. ||φn j || = 1. These normalized splines will be used in what follows. Similarly, on model curve Γτ associated with a en j , point τ ∈ Γ , we consider the following functions φ   φ Ò(d) (ns − j) if t = e iβτ s    j ≥ 0,   0 otherwise en j (t) =  φ  φ Ò(d) (ns − j + d) if t = e i(βτ +θτ ) s      0 otherwise

. j 1.

k=0

where the nodes x k are zeros of the r th Legendre polynomial Pr (x) defined by the formula Pr (x) =

1 dr 2 r (x − 1) , 2n n! d x r


27

and the weights wk are determined accordingly by wk =

2 2 = . r Pr−1 (x k )Pr0 (x k ) (1 − x 2 ) P 0 (x ) 2 k k r

For integrals over an arbitrary interval [a, b], the nodes and weights are scaled and shifted linearly, viz., Zb

f (x)d x ≈

r−1 X

w k f ("k ),

(2.9)

k=0

a

where w k = wk (b − a)/2 and "k = x k (b − a)/2 + (b + a)/2. In our particular case, R1 the integration is over [0, 1]. The integral 0 f (x)d x is approximated by Z

1

f (x)d x ≈

0

r−1 X

w p f (" p ),

(2.10)

p=0

where w p = w p /2 and " p = 1/2x k + 1/2. Consider the error E r ( f ) :=

Z 0

1

f (x)d x −

r−1 X

w p f (" p )

p=0

of the approximation (2.10). If f is continuously differentiable up to order 2r in [0, 1] then the error can be evaluated as follows [19, pp 98]: Ed ( f ) =

(r!)4 f 2r (ξ), 0 < ξ < 1. (2r + 1)[(2r)!]3

In order to approximate integral operators and inner products, we use composite Gauss–Legendre quadrature formula describes as follows: The interval [0, 1] is divided uniformly into m subintervals, on each subinterval, a Gauss–Legendre quadrature rule of the form (2.9) is performed. The resulting quadrature formula is Z 0

1

m−1 X r−1 X 1 f (x)d x ≈ w p f (x l p ), m l=0 p=0

(2.11)

where x l p = (l + " p )/m and " p , w p are the Gauss–Legendre nodes and weights from (2.10), respectively.

Chapter 3 Stability of approximation methods for double layer potential equation Let Γ be the piecewise smooth curve defined in Section 2.4 and L 2 (Γ ) be the corresponding L 2 space on Γ . In this Chapter, we consider the double layer potential equation: 1 (AΓ ω)(t) = ω(t) + π

Z Γ

ω(τ)

d log |t − τ| dsτ + (T ω)(t) = f (t), dnτ

t ∈Γ

(3.1)

where T is a compact operator. We establish the Fredholmness of the operator of (3.1) in L 2 (Γ ) and then study stability problems in L 2 (Γ ) of Nyström method and Galerkin method based on splines.

3.1

Fredholm property of the operator of double layer potential equation

It is well-known that the invertibility of the operator AΓ is a necessary condition for the applicability of many numerical methods. The invertibility of AΓ depends on the curve Γ , on the compact operator T and on the space where the operator AΓ acts. In this section we include certain conditions of the Fredholmness of the operator AΓ considered in the space L p (Γ , w), 1 < p < ∞. For each corner point τ j , j = 0, 1, . . . , q − 1, let Γ j denote the corresponding model curve Γτ j defined in Section 2.4 and L p (Γ j , α j ) denote the weighted L p -space on Γ j with the weight w j (t) = |t|α j . 28

Chapter 3. Double layer potential equation

29

With each corner point τ j we associate the operator AΓ j : L p (Γ j , α j ) → L p (Γ j , α j ) defined by AΓ j := I + VΓ j ,

(3.2)

where VΓ j is the double layer potential operator on Γ j . Application of Theorem 2.10 to the operator AΓ of (1.5) with T = 0 leads to the following result, cf. [29, Theorem 5.2.1] Proposition 3.1. Let Γ be a simple closed piecewise smooth curve in the complex plane C. Then the operator AΓ is Fredholm if and only if all the operators AΓ j : L p (Γ j , α j ) → L p (Γ j , α j ) are invertible for all j = 0, 1, . . . , q − 1. Consider now the operator Nθ : L p (R+ , t α j ) → L p (R+ , t α j ) defined by 1 (Nθ (φ))(σ) = πi

Z

+∞

0

φ(s)ds s − σe iθ

It is easily seen that AΓ j is isometrically isomorphic to the matrix operator Aθ j : L p (R+ , t α j )2 → L p (R+ , t α j )2 ,  Aθ j = 

(1/2) (Nθ − N2π−θ )

I (1/2) (Nθ − N2π−θ )

I

 ,

(3.3)

where L p (R+ , t α j )2 := L p (R+ , t α j ) × L p (R+ , t α j ), and the corresponding isomorphism is given by the relation A 7→ ηAη− 1 with the mapping η : L p (Γ j , t α j ) → L p (R+ , t α j )2 defined by

η( f )(s) = ( f (se i(β j +θ j ) ), f (se iβ j )) T , s ∈ R+ .

(3.4)

As stated in Section 2.3, Nθ is the Mellin convolution operator M 0 (nθ ) with the symbol

e(π−θ j ) y , nθ j ( y) = sinh π y

1 y =z+ + α j i, z ∈ R. p

This immediately leads to the formula sinh(π − θ j ) y smb (1/2)(Nθ j − N2π−θ j ) = , sinh π y

(3.5)


30

where y as above. Thus 

1  smb Aθ j ( y) =  sinh(π − θ ) y j

sinh(π − θ j ) y  sinh π y 1

sinh π y

 .

(3.6)

Note that the Mellin operator (1/2)(Nθ j − N2π−θ j ) can be also represented in the integral form (2.4) with the kernel k = kθ j having the form k(z) = kθ j (z) =

iz sin θ j 1 πi (1 − ze iθ j )(1 − ze−iθ j )

(3.7)

Corollary 3.2. Let Γ be a simple closed piecewise smooth contour satisfying the conditions of Section 2.4. Then the operator AΓ of (3.1) is Fredholm in the space L 2 (Γ ). Proof. The matrix Mellin operator Aθ j is invertible in L 2 (Γ ) if and only if its symbol (3.6) is invertible. The determinant of smb Aθ j is 1 − sinh2 (π − θ j ) y/ sinh2 π y, and it vanishes if and only if sinh(π − θ j ) y = sinh π y or sinh(π − θ j ) y = − sinh π y. Consider, for example, the first of these equations in the case p = 2 and α j = 0. Separating the real and imaginary parts, one obtains the following system of equations cosh((π − θ j )z) sin sinh((π − θ j )z) cos

π − θj 2 π − θj 2

= cosh(πz) = 0,

where z ∈ R. Since cos((π − θ j )/2) 6= 0 for any θ j ∈ (0, 2π), the second equation of the system is satisfied if z = 0 or π − θ j = 0. If z = 0, the first equation of the system becomes sin((π − θ j )/2) = 1 which has no solution for θ j ∈ (0, 2π). On the other hand, if π − θ j = 0, the first equation becomes cosh(πz) = 0 which obviously has no solution. Thus, the symbol of Aθ j does not vanish on the line R + i/2. Therefore, the operator Aθ j is invertible for any j = 0, 1, . . . , q − 1 and so are the operators AΓ j . Now one can apply Proposition 3.1 and obtain Fredholmness of the operator AΓ .

3.2

Spline Galerkin methods

In this Section, we study spline Galerkin methods for equation (3.1) on close piecewise smooth curves.


3.2.1

31

Spline Galerkin methods: Description and examples

Let us consider equation (3.1) in L 2 (Γ ) and provide the space with the inner product

( f , g) =

Z1

f (γ(s))g(γ(s))ds.

0 d Given two positive integers n, d, let Sn,∗ (Γ ) be the corresponding spline space with

the index set I (n, d) defined in Section 2.5. An approximate solution ωn of the equation (3.1) is sought in the form ωn (t) =

X

a j φn j (t)

(3.8)

j∈I (n,d)

with the coefficients a j obtained from the system of linear algebraic equations (AΓ ωn , φn j ) = ( f , φn j ),

j ∈ I (n, d).

(3.9)

The stability of this Galerkin method will be studied in Section 3.2.2. However, here we would like to illustrate the efficiency of the method by a few examples. For simplicity, we only consider equations with the operator T = 0. Although special, this case is of the utmost importance. It occurs when reducing boundary value problems (1.1) for Laplace equation to boundary integral equations. In particular, we determine Galerkin solutions of the double layer potential equation with various right-hand sides on two curves L, M with two and four corners, respectively. Let us describe the curves used in the examples below. The curves L and M are obtained from the ellipse γe (s) = a cos(2πs) + i b sin(2πs),

s ∈ R,

by cutting parts of it and connecting the cutting points by arcs representing cubic Hermit interpolation polynomials in such a way that each common point of the curve obtained becomes a corner point satisfying the conditions (2.5), (2.6). In Figure 3.1, the semi-axes of the ellipse are a = 3, b = 4. The curve L has two corner points obtained by cutting off the part of the ellipse corresponding to the parameter s ∈ [3/8, 5/8]. On the other hand, two parts of the ellipse corresponding to the parameter s ∈ [3/8, 5/8] ∪ [7/8, 9/8] are cut off to create the curve M. The parametrization of the remaining parts of the curves L and M is scaled and shifted


32

so that the conditions (2.5) and (2.6) are satisfied.

Let f1 , f2 and f3 be the following functions defined on the curves L and M, f1 (z) = −z|z|,

f2 (z) = and f3 (z) =

 −1 + iz

if Im z < 0,

 1 + iz

if Im z ≥ 0,

 −2 + iz

if Im z < Im z0 ,

 2 + iz

if Im z ≥ Im z0 ,

where z0 = γe (3/8). Note that f1 is continuous on both curves and f2 has two discontinuity points neither of which coincides with any corner of L or M. On the other hand, one of the corner points of L is a discontinuity point for the function f3 , and two discontinuity points of f3 are located at the corner points of M. Let ωn = ωn ( f j , Γ ) be the Galerkin solution (3.8), (3.9) of the double layer potential equation with right-hand side f j f ,Γ

considered on a curve Γ , and let Enj be the quantity kω2n ( f j , Γ ) − ωn ( f j , Γ )k2

f ,Γ

En j =

kω2n ( f j , Γ )k2

,

which shows the rate of convergence of the approximation method under consideration. Table 3.1 illustrates how the spline Galerkin method with d = 0 performs for the curves L and M and for the right-hand sides f1 , f2 and f3 . Note that the 4

4

3

3

2

2

1

1

0

0

−1

−1

−2

−2

−3

−3

−4 −3

−2

−1

0

1

2

3

−4 −2.5

−2

−1.5

−1

−0.5

0

0.5

1

FIGURE 3.1: Left: ’pacman’ curve L; Right: ’battleax’ curve M.

1.5

2

2.5

Chapter 3. Double layer potential equation En( f1 ,L) 0.0257 0.0129 0.0054

n 128 256 512

En( f1 ,M) 0.0279 0.0147 0.0073

En( f2 ,L) 0.0248 0.0125 0.0052

33 En( f2 ,M) 0.0261 0.0141 0.0070

En( f3 ,L) 0.0445 0.0286 0.0186

En( f3 ,M) 0.0383 0.0230 0.0153

TABLE 3.1: Convergence of the approximate solution for double layer potential equation by spline Galerkin method with d = 0.

integrals in the scalar products (AΓ ωn , φn j ), j ∈ I (n, d) are approximated by the Gauss-Legendre quadrature formula (2.9) with quadrature points coinciding with the zeros of the Legendre polynomial of degree 24 on the canonical interval [−1, 1] scaled and shifted to the intervals [ j/n, ( j + d + 1)/n]. More precisely, we employ the formula

(AΓ ωn , φn j ) =

Z1

AΓ ωn (γ(s))φn j (γ(s))ds ≈

24 X

w k AΓ ωn (γ(sk ))φn j (γ(sk )),

(3.10)

k=1

0

where w k , sk are weights and Gauss-Legendre points on the interval [ j/n, ( j + d + 1)/n]. Composite Gauss-Legendre quadrature (2.11) is also used in approximation of the integral operators of AΓ ωn (γ(sk )), cf. [22]. Thus Z

k(t, τ)x(τ)dτ =

Γ

≈

Z1

k(γ(σ), γ(s))x(γ(s))γ0 (s)ds

0 m−1 r−1 XX

w p k(γ(σ), γ(sl p ))x(γ(sl p ))τ0l p /m

l=0 p=0

where τ0l p = γ0 (sl p ) and m = 40, r = 24. For the discrete norms, let h0 = 1/128, h = 1e − 3, we choose the meshes (0 + h0 : h : 0.5 − h0 )

[ (0.5 + h0 : h : 1 − h0 )

and (0+h0 :h:0.25−h0 )

[ [ (0.25+h0 :h:0.5−h0 ) (0.5+h0 :h:0.75−h0 )

[ (0.75+h0 :h:1−h0 )

due to the fact that L, M have two and four corner points, respectively, cf. Condition 2.5. In the graphs of Figure 3.2, jumps appear when the corner points of M and the


34

discontinuity points of the right-hand side f3 coincide. At the same time, it is quite remarkable that the condition numbers of the methods are relatively small. For the interval considered, they do not exceed 10 and 5 for the curve L and M, respectively. Let us also point out that the results presented in Table 3.1 are comparable with the convergence rates of the spline Galerkin methods for the Sherman-Lauricella [56] and Muskhelishvili [31] equations on smooth curves. These estimates can still be improved if one uses a more accurate approximation of the integrals arising in the Galerkin method, see, e.g, [52, 54]. Nevertheless, the approximate solutions presented in Figure 3.2 demonstrate a good accuracy. We also computed Galerkin solutions of the double layer potential equation with the right-hand sides f1 , f2 and curves L1 , L2 as in Section 3.3.1. Although these results are not reported here, there is a good correlation with approximate solutions obtained by the Nyström method in that Section.

3.2.2

Stability conditions for spline Galerkin method

Let us consider the Galerkin method in the space L 2 (Γ ). Let Pnd denote the orthogd onal projection from L 2 (Γ ) onto Sn,∗ (Γ ). The spline Galerkin method (3.9) can be

rewritten as Pnd AΓ Pnd ωn = Pnd f , n ∈ N.

(3.11)

For the convenience of the reader, let us recall the following definition, see Section 2.2. Definition 3.3. The approximation sequence (Pnd AΓ Pnd ) is said to be stable if there exists n0 ∈ N and a constant C > 0 such that for all n ≥ n0 the operators Pnd AΓ Pnd : d d Sn,∗ (Γ ) → Sn,∗ (Γ ) are invertible and k(Pnd AΓ Pnd )−1 Pnd k ≤ C.

Let A Γ denote the set of all bounded sequences of bounded linear operator An : Im Pnd → Im Pnd such that there exist strong limits s − lim An Pnd = A,

s − lim(An Pnd )∗ Pnd = A∗ .

Moreover, let K (L 2 (Γ )) denote the ideal of all compact operators in L (L 2 (Γ )), and let G ⊂ A Γ be the set of sequences which converge uniformly to zero. Recall that the sequence of orthogonal projection (Pnd ) in L 2 (Γ ) converges strongly to identity


35

15

15

10

10

5

5

0

0

−5

−5

−10

−10

−15

−10

−5

0

5

−15 −10

10

4

4

3

3

2

2

1

1

0

0

−1

−1

−2

−2

−3

−3

−4 −3

−2

−1

0

1

2

3

−4 −3

4

4

3

3

2

2

1

1

0

0

−1

−1

−2

−2

−3

−3

−4 −6

−5

−4

−3

−2

−1

0

1

2

3

−4 −5

−8

−6

−4

−2

−4

−2

−1

−3

0

2

0

−2

4

1

−1

6

8

2

0

3

1

FIGURE 3.2: Spline Galerkin solution of the double layer potential equation in the case n = 1024. Left: ’pacman’ curve L; Right: ’battleax’ curve M. First row: with r.-h. s. f1 (z); Second row: with r.-h. s. f2 (z); Third row: with r.-h. s. f3 (z).

operator and (Pnd )∗ = Pnd . It follows that s − lim Pnd AΓ Pnd = AΓ , n→∞

s − lim (Pnd AΓ Pnd )∗ Pnd = A∗Γ . n→∞

Therefore, the stability analysis for elements of A Γ can be applied to (Pnd AΓ Pnd ). Proposition 3.4 (cf. Theorem 2.17). The set of sequences J Γ = {(An ) ∈ A Γ : An = Pnd K Pnd + Gn , K ∈ K (L 2 (Γ )), (Gn ) ∈ G }

10

2


36

forms a close two-sided ideal of A Γ . The sequence (Pnd AΓ Pnd ) is stable if and only if the operator AΓ ∈ L (L 2 (Γ )) and the coset (Pnd AΓ Pnd ) + J Γ ∈ L (A Γ /J Γ ) are invertible. Recall that both Fredholm properties and invertibility of the operator AΓ in various spaces have been studied in literature, see [2, 14, 72, 83] or the introduction in Chapter 1. Therefore, our main task here is to investigate the behavior of the coset (Pnd AΓ Pnd ) + J Γ . For, it is more convenient to consider this coset as an element of a smaller algebra. Thus, let CR (Γ ) denote the set of all continuous real-valued functions defining on Γ and B Γ denote the smallest closed C ∗ -subalgebra of A Γ which contains the sequences (Pnd M SΓ M Pnd ), (Pnd SΓ Pnd ), all sequences (Gn ) ∈ G , and all sequences (Pnd f Pnd ) with f ∈ CR (Γ ). It follows from [78, 79] that J Γ ⊂ B Γ and (Pnd AΓ Pnd ) ∈ B Γ . Therefore, B Γ /J Γ is a C ∗ -subalgebra of A Γ /J Γ , hence the coset (Pnd AΓ Pnd )+J Γ is invertible in A Γ /J Γ if and only if it is invertible in B Γ /J Γ (inverse closedness). However, the invertibility of the coset (Pnd AΓ Pnd ) + J Γ in the quotient algebra B Γ /J Γ can be showed by a local principle. Thus, consider the smallest closed C ∗ −subalgebra U Γ of A Γ which contains all sequences (Fn ) of the form Fn = Pnd f Pnd + Gn where f ∈ CR (Γ ) and (Gn ) ∈ J Γ . Since for any f , f1 , f2 ∈ CR (Γ ) one have Pnd f1 Pnd Pnd f2 Pnd = Pnd f1 f2 Pnd = Pnd f2 Pnd Pnd f1 Pnd and f SΓ − SΓ f ∈ K (L 2 (Γ )) [70], the algebra U Γ /J Γ is a subalgebra of the center of B Γ /J Γ . Following the discussion in Section 2.1.2, U Γ /J Γ is identified with the function algebra CR (Γ ). The maximal ideal space of U Γ /J Γ is homeomorphic to Γ and the maximal ideal associated with a point τ ∈ Γ is {(Pnd fτ Pnd ) + J Γ : fτ (τ) = 0, fτ ∈ CR (Γ )}.

(3.12)

Let Iτ be the smallest closed ideal of B Γ /J Γ generated by the maximal ideal (3.12). By Allan’s local principle, the coset (Pnd AΓ Pnd )+J Γ is invertible in B Γ /J Γ if and only if the cosets Aτn := ((Pnd AΓ Pnd ) + J Γ ) + Iτ are invertible in BτJ := (B Γ /J Γ )/Iτ . Let us consider two cases concerning the nature of the point τ. If τ is not a corner point of Γ , let Uτ be a sufficiently small neighborhood of τ such that Uτ ∩ MΓ = ; and let χUτ be the characteristic function of Uτ . It follows from [79, Corollary 4.6.3]


37

that χUτ VΓ χUτ is compact hence Aτn = (Pnd + J Γ ) + Iτ which is obviously invertible. Therefore, if τ 6∈ MΓ then the coset Aτn is invertible in the quotient algebra BτJ . The case where τ ∈ MΓ is much more involved. Thus, let us consider the model curve Γτ defined by (2.7). On the curve Γτ consider the corresponding double layer potential operator AΓτ = I + VΓτ . Let Snd (Γτ ), Snd (R+ ) be the spline spaces constructed in Section 2.5. In addition, let Pnτ and Pn+ be the orthogonal projections of L 2 (Γτ ) onto Snd (Γτ ) and L 2 (R+ ) onto Snd (R+ ), respectively. ˜τ can be defined Now algebra B Γτ and its ideal J Γτ together with the local ideal I analogously to the construction of B Γ , J Γ and Iτ . Similarly to Proposition 3.4, one can formulate the following result. Corollary 3.5. The sequence (Pnτ AΓτ Pnτ ) ∈ B Γτ is stable if and only if the operator AΓτ is invertible and the coset (Pnτ AΓτ Pnτ ) + J Γτ is invertible in the quotient algebra B Γτ /J Γτ . By Corollary 3.2, the operator AΓτ is invertible. Therefore, the coset (Pnτ AΓτ Pnτ ) + J Γτ is invertible in the corresponding quotient algebra if and only if the sequence (Pnτ AΓτ Pnτ ) is stable. Let us now consider this stability problem in more detail. By L22 (R+ ) we denote the product of two copies of L 2 (R+ ) provided with the norm 1/2 k(ϕ1 , ϕ2 ) T k L22 (R+ ) = kϕ1 k2L 2 (R+ ) + kϕ2 k2L 2 (R+ ) , and let η : L 2 (Γτ ) → L22 (R+ ) be the mapping defined by (3.4). This isometry generates an isometric algebra isomorphism Ψ : L (L 2 (Γτ )) → L (L22 (R+ )) defined by Ψ(A) = ηAη−1 .

(3.13)

In particular, for the operators Pnτ , I and VΓτ , one has Ψ(Pnτ ) = diag (Pn+ , Pn+ ) and Ψ(I) =

I

0

0 I

,

Ψ(VΓτ ) =

0

Nθτ

Nθτ

0

,

(3.14)

where Nθ is the Mellin convolution operator defined by 1 (Nθ (ϕ))(σ) = 2πi

Z 0

+∞

1 1 − ϕ(s) ds. s − σe iθ s − σe−iθ

(3.15)


38

The operator Nθ can also be written in another form reflecting its Mellin structureviz., Nθ (ϕ)(σ) =

+∞ Z

kθ

σ s

ϕ(s)

ds s

(3.16)

0

where

1 u sin θ . 2π |1 − ue iθ |2

kθ = kθ (u) =

(3.17)

Thus Ψ(AΓτ ) =

I

Nθτ

Nθτ

I

,

(3.18)

and an immediate consequence of the isomorphism (3.13) is that the sequence (Pnτ AΓτ Pnτ ) is stable if and only if so is the sequence (Ψ(Pnτ AΓτ Pnτ )). On the other hand, the study of the stability of the sequences (Ψ(Pnτ AΓτ Pnτ )), τ ∈ Γ can be reduced to the study of two main cases related to the nature of the points τ ∈ Γ . Thus if τ 6∈ MΓ , then θτ = π so that the operator Nπ = 0 and Ψ(Pnτ AΓτ Pnτ ) is just the diagonal sequence diag (Pn+ , Pn+ ) which is obviously stable. Therefore the corresponding coset (Pnτ ) + J Γτ ∈ B Γτ /J Γτ is invertible. Consider now the case where τ is a corner point of Γ , and θτ ∈ (0, 2π) is the opening angle of this corner. By l 2 we denote the set of sequences of complex numbers (ξk )+∞ such that k=0 ∞ X

|ξk |2 < ∞.

k=0

Moreover, let Λn be the operator acting from Snd (R+ ) into l 2 and defined by Λn

∞ X

en j ξjφ

= (ξ0 , ξ1 , . . .).

j=0

The operators Λn are continuously invertible and there is a constant m such that ||Λn || ||Λ−n || ≤ m for all

n = 1, 2, . . . ,

where Λ−n := Λ−1 , [20]. It implies that the sequence (Ψ(Pnτ AΓτ Pnτ )) is stable if and n only if the sequence (Rτn ), Rτn := diag (Λn , Λn ) Ψ(Pnτ AΓτ Pnτ ) diag (Λ−n , Λ−n ) : l 2 × l 2 → l 2 × l 2 is stable.


39

Lemma 3.6. The sequence (Pnτ AΓτ Pnτ ) is stable if and only if the operator Rτ := Rτ1 is invertible. Proof. According to the above considerations, the sequence (Pnτ AΓτ Pnτ ) is stable if and only if the sequence (Rτn ) is so. Consider now the matrix of the operator Rτn . For the sake of convenience, it is again denoted by Rτn . This matrix has the form Rτn

=

I

A12

A21

I

where A21 = A12 = (al j )∞ and l, j=0

al j =

ν2d

d+1 Z d+1 Z

kθτ 0

du Ò(d) v + l Ò(d) φ (u) φ (v)d v. u+ j u+ j

0

Observe that the entries of Rτn do not depend on n, i.e. (Rτn ) is a constant sequence. Therefore it is stable if and only if any of its members, say Rτ = Rτ1 , is invertible. Using the above results, one can obtain a stability criterion for the spline Galerkin method. Theorem 3.7. If operator AΓ is invertible, then the spline Galerkin method (3.11) is stable if and only if all the operators Rτ : l 2 × l 2 → l 2 × l 2 , τ ∈ MΓ are invertible. Proof. It follows from Proposition 3.4 that the spline Galerkin method is stable if and only if the coset (Pn AΓ Pn ) + J Γ ∈ B Γ /J Γ is invertible. By Theorem 2.10, this coset is invertible if and only if so are all the cosets Aτn , τ ∈ Γ . However, as we already know, for τ ∈ / MΓ the cosets Aτn are invertible. In the case τ ∈ MΓ , by an observation similar to one made in [24], the cosets Aτ and A˜τ := ((P τ AΓ P τ ) + J Γτ )/I˜τ are n

n

n

τ

n

simultaneously invertible or not. The invertibility of the last cosets involves the invertibility of the cosets (Pnτ AΓτ Pnτ ) + J Γτ in the corresponding quotient-algebra B Γτ /J Γτ . In turn, the invertibility of (Pnτ AΓτ Pnτ ) + J Γτ is equivalent to the stability of the approximation method (Pnτ AΓτ Pnτ ), which is shown by the Lemma 3.6 and the proof is completed.


3.2.3

40

Numerical experiments

Theorem 3.7 shows that the stability of the spline Galerkin method depends on the invertibility of the operators Rτ , τ ∈ MΓ . However, these operators belong to an algebra of Toeplitz operators generated by piecewise continuous matrix functions and at present there is no analytic tool to check their invertibility. On the other hand, a numerical approach to such a kind of problem has been proposed in [21, 24]. Thus one can consider stability of an approximation method on curves having corner points of the same magnitude. If this is the case, the stability of the corresponding method depends on the operator itself and on only one additional operator Rτ . More precisely, the following result is true. Proposition 3.8. If Γ is a piecewise smooth curve such that all corners τ ∈ MΓ have the same magnitude, then 1. For any τ1 , τ2 ∈ MΓ one has Rτ1 = Rτ2 . 2. The operator Rτ1 is invertible if and only if the spline Galerkin method (Pn (I + VΓ )Pn ) is stable. Proof. This result is an immediate consequence of Theorem 3.7. One only has to take into account that if Γ satisfies the conditions stated, then the corresponding operator I + VΓ is invertible on the space L 2 (Γ ), [83]. Thus in order to detect critical angles, i.e. the opening angles θτ where the operators Rτ are not invertible, one can compute the condition numbers of the method on families L (θ ), θ ∈ (0, 2π) of special contours having one or more corner points, all of the same magnitude. As a result, at any critical point of the method, the graph representing the condition numbers has to have an "infinite" peak regardless of the family of the curves used. In this Section, we employ the curves L1 (θ ), L2 (θ ) and L4 (θ ) constructed in Chapter 2. More precisely, our numerical experiments are designed as follows. First, we divide the interval [0.1π, 1.9π] by the points {θk }, where θk = π(0.1 + k × 0.01). For every family of test contours L j (θk ), θk ∈ [0.1, 1.9], j = 1, 2, 4 consider the spline Galerkin methods with n = 256 based on the splines of order 0, 1 or 2. Compute then the condition numbers of the corresponding linear algebraic systems described by (3.9). Should it appear any point θ ∗ in the vicinity of which the condition numbers become


41

2

2

2

1.6

1.6

1.6

1.2

1.2

1.2

0.8

0.8

0.4

0.4

0.8 0.4 0 0

0.4

0.8

θ/π

1.2

1.6

2

0 0

0.4

0.8

θ/π

1.2

1.6

2

0 0

2

2

2

1.6

1.6

1.6

1.2

1.2

1.2

0.8

0.8

0.8

0.4

0.4

0.4

0 0

0.4

0.8

θ/π

1.2

1.6

2

0 0

0.4

0.8

θ/π

1.2

1.6

2

0 0

2

2

2

1.6

1.6

1.6

1.2

1.2

1.2

0.8

0.8

0.8

0.4

0.4

0.4

0 0

0.4

0.8

θ/π

1.2

1.6

2

0 0

0.4

0.8

θ/π

1.2

1.6

2

0 0

0.4

0.8

0.4

0.8

0.4

0.8

θ/π

θ/π

θ/π

1.2

1.6

2

1.2

1.6

2

1.2

1.6

2

FIGURE 3.3: Logarithm log10 of condition numbers versus opening angles for Galerkin methods with n = 256 for various contours and spline spaces. Left: contour L1 ; Middle: contour L2 ; Right: contour L4 ; First row: d = 0; Second row: d = 1; Third row: d = 2

large, a neighborhood of θ ∗ is refined by a smaller step 0.001π, and condition numbers are recalculated with n changed to 512. The outcome of our computations is presented in Figure 3.3. In all cases, one can observe the absence of peaks in the graphs, which means that the Galerkin methods under consideration do not have "critical" angles in the interval [0.1, 1.9]. In other words, if the opening angles of all corners of the integration contour are located in the interval [0.1, 1.9], the spline Galerkin methods based on the splines of degree 0, 1 or 2 are always stable.

3.3

Nyström method for double layer potential equation

In this Section, we consider Nyström method for the equation (3.1) on closed piecewise smooth curves.


3.3.1

42

Nyström method: Description and examples

Let Γ be the curve described in Chapter 2 and Snd (Γ ) be the corresponding spline space on the curve. Consider the following set of points on Γ τl p = γ

l + "p

,

n

l = 0, 1, . . . , n − 1;

p = 0, 1, . . . , d − 1.

where 0 < "0 < "1 < . . . < "d−1 < 1 are real numbers. If the integral operator K, Z

k(t, τ)ϕ(τ) dτ

Kϕ(t) := Γ

has a sufficiently smooth kernel k and if ϕ is a Riemann integrable function, then we can approximate the above operator by the quadrature rule (2.11). Thus Z

k(t, τ)ϕ(τ) dτ =

Γ

Z

1

k(γ(σ), γ(s))ϕ(γ(s))γ0 (s) ds

0

≈K

(",n)

ϕ(t) =

n−1 X d−1 X

(3.19) w p k(t, τl p )ϕ(τl p )τ0l p /n,

l=0 p=0

where τ0l p = γ0 ((l + " p )/n). Let Q"n : L∞ (Γ ) → Snd (Γ ) denote the interpolation projection on the space Snd (Γ ) such that Q"n x(τl p ) = x(τl p ),

l = 0, 1, . . . , n − 1, p = 0, 1, . . . , d − 1.

for all x from the set R(Γ ) of all Riemann integrable functions on Γ . Note that if none of " p is equal to 0.5, such projection operators Q"n exist and the sequence (Q"n )n∈N : R(Γ ) → L 2 (Γ ) converges strongly to the corresponding embedding operator [78], viz., lim kQ"n f − f k L 2 (Γ ) = 0,

n→∞

f ∈ R(Γ ).

(3.20)

Let Pnd : L 2 (Γ ) → Snd (Γ ) be the orthogonal projection onto the spline space Snd (Γ ). Recall that on the space L 2 (Γ ) the sequence (Pnd ) converges strongly to the identity operator. Consider the Nyström method for the double layer potential equation (3.1). For simplicity, we drop the compact operator T and consider the equation AΓ x = (I + VΓ )x = f .

(3.21)


43

This simplifies the notation but does not influence the proof of the main result. An approximate solution x n of (3.21) can be derived from the equations (",n)

Q"n AΓ

(",n)

Pnd x n := Q"n Pnd x n + Q"n VΓ

Pnd x n = Q"n f , x n ∈ Snd (Γ ), n ∈ N.

(3.22)

These operator equations are equivalent to the following systems of linear algebraic equations, n−1 d−1 τ0l p τ0l p 1 XX x(τkr ) + w p x(τl p ) − 2πi l=0 p=0 τl p − τkr τl p − τkr

= f (τkr )

!

1 n

(3.23)

k = 0, 1, . . . , n − 1, r = 0, 1, . . . , d − 1.

Remark 3.9. Straightforward calculations show that if k = k(t, τ) is the kernel of the double layer potential operator VΓ , then for any τ ∈ / MΓ the limit lim t→τ k(t, τ) = ıIm (γ0 (s)γ00 (s)) γ0 (s)|γ0 (s)|2 , τ = γ(s). Therefore, for the indices l = k, p = r the expressions in ıIm (γ0 (skr )γ00 (skr )) round brackets of (3.23) are replaced by . |γ0 (skr )|2 Let us consider examples of approximate solution of the equation (3.21) for two different contours defined in Section 2.4, viz., L1 = L1 (0.3π) and L2 = L2 (0.3π) in the case of continuous and discontinuous right hand sides. Thus, let us recall the definition of two functions f1 and f2 in Section 3.2.1 f1 (z) = −z|z|, and f2 (z) =

 −1 + iz

if Im z < 0

 1 + iz

if Im z ≥ 0.

The right-hand side f1 (z) is continuous on both curves L1 and L2 , whereas f2 (z) is discontinuous on both curves. Moreover, one of the discontinuity points of f2 coincides with the angular point of L1 . Approximate solutions are obtained by the Nyström method (3.23) with d = 16 and various n. Their graphs are presented in Figure ( f ,L j )

3.4. In Table 3.3.1, the term En k

shows the relative error kx 2n −x n k2 /kx 2n k2 where

x n is the approximate solution of the equation (3.21) for the contour L j , j = 1, 2 with the right hand side f k , k = 1, 2. Note that after obtaining approximate values x n (τkr ), we construct the corresponding approximate solution x n as a spline in Snd (Γ ). The discrete L 2 −norm is then evaluate by using values of the spline at 1001 points γ(sk ), sk = k × 0.001, k = 0, 1, . . . , 1000.

Chapter 3. Double layer potential equation 0.25

44 0.1

0.2 0

0.15 −0.1

0.1 0.05

−0.2

0 −0.3

−0.05 −0.1

−0.4

−0.15 −0.5

−0.2 −0.25 −0.7

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

−0.6 −4

0.2

0.4

0.15

0.3

0.1

0.2

0.05

0.1

0

0

−0.05

−0.1

−0.1

−0.2

−0.15

−0.3

−0.2 −0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

−0.4 −1

−3

−0.8

−2

−0.6

−1

−0.4

−0.2

0

0

1

0.2

2

0.4

3

0.6

0.8

4

1

FIGURE 3.4: Approximate solutions of (3.21) with two dierent right hand sides and contours obtained by using method (3.23) with n = 512, d = 16. Left: Solutions in the case of continuous r.-h.s. f1 (z). Right: Solutions in the case of discontinuous r.-h.s. f2 (z). First row: Equations on L1 . Second row: Equations on L2 .

n 32 96 256

En( f1 ,L1 ) 2.5 × 10−3 8.3 × 10−4 3.1 × 10−4

En( f1 ,L2 ) 2.6 × 10−3 1.1 × 10−3 2.1 × 10−4

En( f2 ,L1 ) 1.5 × 10−2 7.5 × 10−3 4.0 × 10−3

En( f2 ,L2 ) 2.0 × 10−2 1.3 × 10−2 7.3 × 10−3

TABLE 3.2: Convergence of Nyström solutions for double layer potential equation

It is worth noting that a better convergence rate can be achieved by using certain modifications of the Nyström method [5, 6, 53] but our main concern is stability and on the angles the presence of which induces the instability of the Nyström method.

3.3.2

Stability of the Nyström method

Let us now show the stability of the method (3.23). Note that from now on we consider our operators as acting on the space L 2 := L 2 (Γ , 0). Therefore, according to Corollary 3.2, the operator AΓ of (3.21) is Fredholm. It follows from (3.20) that the (",n)

sequence of approximation operators (An )n∈N , where An = Q"n AΓ

Pnd corresponding

to the Nyström method converges strongly to the operator AΓ . Similar statement


45

is valid for the sequence of adjoint operators. Therefore, we can apply Proposition 3.4 to study the invertibility. Now, it remains to show the invertibility of the coset (An )+J Γ . Recall that J Γ is the set {(Pnd K Pnd )+(Gn )} where (Gn ) converges uniformly to 0 and K is compact. Localizing this invertibility problem via Allan’s local principle, we obtain local cosets Aτn := ((An ) + J Γ ) + Iτ , τ ∈ Γ , where the local ideal Iτ is defined similarly to one in Section 3.2.2. Note that if τ ∈ / MΓ and Uτ ⊂ Γ is a neighborhood of τ such that MΓ ∩ Uτ = ; and if fτ is a function continuous on Γ and such that ¨ fτ (t) =

1

if t = τ

0

if t ∈ Γ \ Uτ ,

then the operator fτ VΓ fτ is compact on L 2 (Γ )) [79, Corollary 4.6.3]). Therefore, the sequence (Aτn ) is locally equivalent to the sequence generated by the projections (Pnd ), so that the corresponding coset containing the sequence (Aτn ) is invertible. Thus one only has to identify and study the cosets associated with the corner points of Γ . To this end, for each corner point τ j ∈ Γ we consider the corresponding approximation R method for the operator AΓ j of (3.2) and approximate the integral Γ x(τ)dτ by a j

quadrature rule similar to (3.19), viz., −1 X d−1 X

Z x(τ)dτ ≈ Γj

wp x

l + "p n

l=−∞ p=0

+

+∞ X d−1 X

wp x

e

l + "p n

l=0 p=0

i(β j +θ j )

e

iβ j

e i(β j +θ j ) n (3.24)

e iβ j n

where w p and " p are as in (3.19). We also need spline spaces on the contours Γ j and R+ . Let Snd (Γ j ) and Snd (R+ ) be the spline spaces defined in Section 2.5 of Chapter 2. en and P bn denote the orthogonal projections from L 2 (Γ j ) onto S d (Γ j ) Moreover, let P n and from L 2 (R+ ) onto Snd (R+ ), respectively. Let R2 (Γ j ) denote the set of functions on Γ j which are Riemann integrable on each finite part of Γ j and satisfy the condition

k f kR = k f k L 2 (Γ j ) +

1/2

+∞ X

sup

k=0 t∈e

+

+∞ X k=0 t∈e

| f (t)|

2

i(β j +θ j )

[k,k+1]

1/2 sup iβ j

| f (t)|2

< +∞.

[k,k+1]

Consider the integral equation AΓ j x = f ,

f ∈ R2 (Γ j ).


46

As before, replace x by an element x n ∈ Snd (Γ j ), apply quadrature formula (3.24) to e " : R(Γ j ) → S d (Γ j ) the corresponding integrals and use the interpolation projections Q n

n

defined by e " x(τl p ) = x(τl p ), l ∈ Z, p = 0, 1, . . . , d − 1; Q n  l + " p i(β +θ )   e j j if l < 0,  n τl p = l + " p iβ    e j if l ≥ 0. n As the result, we obtain the following operator equations e " A(",n) P e" f , en x n = Q Q n Γj n

x n ∈ Snd (Γ j ), n ∈ N.

(3.25)

These equations are equivalent to the following infinite systems of linear algebraic equations, i(β +θ ) −1 d−1 e j j e−i(β j +θ j ) 1 1 X X w p x n (τl p ) − x n (τkr ) + 2πi l=−∞ p=0 τl p − τkr τl p − τkr n ∞ d−1 e iβ j 1 1 XX e−iβ j + w p x n (τl p ) − 2πi l=0 p=0 τl p − τkr τl p − τkr n = f (τkr ),

k ∈ Z, p = 0, 1, . . . , d − 1,

with the note that when k = l, r = p the expressions in the round bracket should be replaced by 0, cf. Remark 3.9. If one now uses the integral representation (2.4) of the Mellin convolution operator M 0 (nθ j ) with the symbol nθ j defined by (3.5) and let M (",n) (kθ j ) denotes the approximate operator of M 0 (kθ j ) by the quadrature formula (3.24), one can write the b " , n ∈ N be the interpolation operator (3.25) in a different form. More precisely, let Q n

operators defined on the positive semi-axis. e " A(",n) P en )n∈N Lemma 3.10. If kθ is the function defined in (3.7), then the sequence (Q n Γj b",n diag ( P bn , P bn )), is stable if and only if the sequence (A θj

b",n diag ( P bn , P bn ) A θj is so.

=

bn P

b 1−" M (1−",n) (kθ ) P bn Q n j

b " M (",n) (kθ ) P bn Q n j bn P


47

Proof. Let η : L 2 (Γ j ) → L 2 (R+ )2 be the isomorphism defined in Section 3.3.2. It is easily seen that e " η−1 = diag (Q b" , Q b 1−" ), η P en η−1 = diag ( P bn , P bn ) ηQ n n n and

(",n) M (k ) θj (",n) en η−1 + ηVΓ(",n) η−1 = ηAΓ j η−1 = η P . j (1−",n) b M (kθ j ) Pn

(",n) e Pn )η−1

bn P

e " η−1 )(ηA(",n) η−1 )(η P en η−1 ) completes = (ηQ Γj n

e" A The obvious identity η(Q n Γj the proof.

Let l2 denote the space of sequences (ξ j )∞ of complex numbers ξ j , j = 0, 1, . . . such j=0 ∞ 1/2 P that |ξ j |2 < +∞. Let Λn and Λ−n be the operators defined in Section 3.2.2. j=0

e δ A(",n) P en )n∈N now can be rewritten The conditions of the stability of the sequence (Q n Γj b n and Λ b −n we, respectively, denote the diagonal in a more convenient form. By Λ operators, b n := diag (Λn , Λn ), Λ

b −n := diag (Λ−n , Λ−n ). Λ

e " A(",n) P en )n∈N is stable if and only if the operator Bθ ," = Corollary 3.11. The sequence (Q n Γj j ",1 b b b b b Λ1 A diag ( P1 , P1 )Λ−1 is invertible. θj

Proof. Straightforward calculations show that the entries of the approximation opb",n diag ( P bn , P bn )Λ b nA b −n do not depend on n. Indeed, consider for example, the erator Λ θj

b " M (",n) (kθ ) P bn Λ−n ). If x n ∈ im P bn , then sequence (ΛnQ n j (M

(",n)

(kθ j )x n )(σ) =

+∞ X d−1 X

w p kθ j

l=0 p=0

=

d−1 X p=0

wp

∞ X l=0

kθ j

!

l + "p 1 1 x l+" p l+" p n n n n ! n l + "p σ 1 xn , l+" p l + "p n σ

n

e " leads to the relation and application of the interpolation operators Q n k+" r n l+" p n

!

l + "p 1 xn l + "p n d−1 ∞ X X l + "p k + "r 1 = wp kθ j xn . l + "p l + "p n p=0 l=0

d−1 ∞ X k + " X r " (",n) e M Q (k )x = w kθ j θj n p n n p=0 l=0


48

b",n diag ( P bn , P bn )Λ b nA b −n do not depend on n. ThereThus the entries of the operator Λ θj fore, the sequence in question is constant and one concludes that it is stable if and b",1 diag ( P b1 , P b1 )Λ−1 , is invertible. This completes only if one of its members, say Λ1 A θj

the proof. Theorem 3.12. Let n = qm, m ∈ N. Suppose that the operator A is invertible. The Nyström method for the operator AΓ : L 2 (Γ ) → L 2 (Γ ) is stable if and only if all the operators Bθ j ," , j = 0, 1, . . . , q − 1 are invertible. Proof. Let A Γ and J Γ be the algebra and closed ideal defined Section 3.2.2. Let C Γ denote the smallest closed C ∗ -subalgebra of A Γ that contains the sequences (",n)

(Q"n SΓ

Pnd ), (Q"n M Pnd ) and (Q"n f Pnd ) where f ∈ C(Γ ). Then similarly to the case (",n)

of Galerkin method considered (Q"n AΓ (",n)

A /J . Therefore, the coset (AΓ

Pnd ) ∈ C and C /J is a C ∗ -subalgebra of

Pnd ) + J is invertible in A Γ /J Γ if and only if it

is invertible in C Γ /J Γ . However, the algebra C Γ /J Γ has a nice center and the (",n)

invertibility of the coset (AΓ

Pnd ) + J in C Γ /J Γ can be established by Allan’s local

principle. Thus following the consideration in Section 3.2.2, cf. also Theorem 3.4 of [24] one can show that for any τ = τ j ∈ MΓ this coset is invertible if and only if the corresponding operator Bθ j ," is invertible. On the other hand, it was already mentioned that for τ ∈ / MΓ , the corresponding coset is always invertible.

3.3.3


Due to Theorem 3.12, the stability of the Nyström method depends on the invertibility of the operators Bθ j ," , j = 0, 1, . . . , q − 1. A more detailed study of these operators shows that they belong to an algebra of Toeplitz operators with matrix symbols. Unfortunately, at present there is no efficient criterion to check whether such operators are invertible or not. Recall that before we have considered approximation methods on special curves L1 , L2 in order to study the invertibility of the local operators. Here, we proceed in the same way. Theorem 3.13. Let L = L (θ ) denote any of the curves L1 (θ ) or L2 (θ ), θ ∈ (0, 2π) defined in Section 2.4 and let AL (θ ) be the operator (3.21) considered on the contour (",n)

L (θ ). The operator Bθ ," is invertible if and only if the Nyström method (Q"n AL (θ ) Pnd ) is stable.


49

of Condition numbers

16

14 12 10 8 6

10

4

log

log10 of Condition numbers

16

2 0 0

0.2

0.4

0.6

0.8

1

ω/π

1.2

1.4

1.6

1.8

2

14 12 10 8 6 4 2 0 0

0.2

0.4

0.6

0.8

1

ω/π

1.2

1.4

1.6

1.8

2

FIGURE 3.5: Condition numbers vs. opening angles in case n = 128, d = 16. Left:

for one-corner curve, right: for two-corner curve

This statement is a reformulation of Theorem 3.12 with respect to special contours and operator. Of course one has to take into account that the operator AL (θ ) : L 2 (L (θ )) → L 2 (L (θ )) is invertible [83]. It is worth noting that the curves L1 and L2 have distinct shapes and the number of corner points. However, if their corner points have the same opening angle, the corresponding numerical experiments shall produce the same results. In what follows we are varying parameter θ in the interval (0.1π, 1.9π) and obtain two families of contours with one and two corner points, respectively. In order to find the instability angles, we divide the interval [0.1π, 1.9π] by the points θk = π ∗ (0.1 + 0.001k). Furthermore, for each point θk we compute the condition numbers for the Nyström method in the case where n = 128 and the Gauss–Legendre quadrature with d = 16 is used, viz. there are 128 panels on the curves and each panel has 16 discretized points. Calculating the corresponding condition numbers at the points θk , we detect "suspicious" points, in the neighbourhoods of which condition numbers grow rapidly. Thereafter, in neighborhoods of such points the initial mesh has been refined and condition numbers are recalculated. The procedure is repeated until condition numbers reach the point 1016 . The outcome of these computations is presented in Figure 3.5. Thus using both contours we found that the corresponding graphs have four peaks in the interval (0.1, 1.9), and approximate value for the critical "angles are: The case of one corner geometry, curve L1

0.11781222π,

0.25164815π,

1.74949877π,

1.88430019π

The case of two corner geometry, curve L2


50

9

log

10

of Condition numbers

ω=π/4 8

ω=0.25183π

7

ω=π/3 ω=π/2

6 5 4 3 2 1 0 0

500

1000

1500

2000

2500

3000

3500

4000

4500

FIGURE 3.6: Condition numbers for some opening angles. The numbers of discretization points is 16n.

0.11780844π,

0.25164706π,

1.74840993π,

1.88390254π.

Let us emphasize that for both the curve L1 and L2 , the results obtained coincide up to three significant numbers. The peaks obtained are connected with four possible critical angles in the interval (0.1π, 1.9π). Remark 3.14. In order to have a clearer view of critical angles, we fix some opening angles, vary the numbers of discretization points and calculate the condition numbers in those cases. The graphs in Figure 3.6 show that the Nyström method for the double layer potential operator considered on the contours L1 (ω), ω = 0.25183π, π/3, π/4, π/2 is stable. An abnormality of the graph in the case ω = 0.25183π is caused by the proximity of this point to one of the critical angle.

Chapter 4 Spline Galerkin method for Sherman-Lauricella equation In this Chapter, we consider the Sherman–Lauricella equation 1 ω(t) + 2πi

Z

ζ− t

1 − ω(ζ) d ln 2πi ζ− t Γ

Z

ω(ζ) d

Γ

ζ− t ζ− t

= f (t),

t = x +iy ∈ Γ, (4.1)

where the bar denotes the complex conjugation and ω is an unknown function. We describe the spline Galerkin method used and illustrate it by some numerical examples. The necessary and sufficient conditions for stability in L 2 are then established and verified by a numerical approach.

4.1

Spline Galerkin method: Description and examples

Let AΓ : L 2 (Γ ) → L 2 (Γ ) be the operator corresponding to the Sherman-Lauricella equation (4.1). It is well known that the operator AΓ is not invertible on the space L 2 (Γ ) [68]. Therefore, for the approximate solution of the equation (4.1) we use the equation with an operator BΓ instead of AΓ and choose the right-hand sides f of the initial equation (4.1) from a suitable subspace of L 2 (Γ ). More precisely, let W21 (Γ ) denote the closure of the set of all functions f with bounded derivatives in the norm Z || f ||W21 :=

| f (t)|2 ds +

Γ

Z Γ

51

| f 0 (t)|2 ds

1/2 ,

Chapter 4. Sherman - Lauricella equation

52

and let TΓ : L 2 (Γ ) → L 2 (Γ ) refer to the operator defined by 1

1 TΓ ω(t) := (t − a) 2πi

Z Γ

ω(ζ) ω(ζ) dζ , dζ + (ζ − a)2 (ζ − a)2

(4.2)

where a is a point in D. Theorem 4.1 (cf. [22]). If Γ is a simple closed piecewise smooth contour and if f ∈ W21 (Γ ) and satisfies the condition (1.14), then the equation (4.1) has a unique solution, which can be found from the equation BΓ ω = f

(4.3)

where the operator BΓ := AΓ + TΓ is invertible on both spaces W21 (Γ ) and L2 (Γ ). Note that all results concerning the behavior of the operators AΓ and BΓ in the space W21 (Γ ) and similar operators connected with the Muskhelishvili equation follow from [37] where they proved in a more general situation of the spaces Wp1 (Γ , ρ), 1 < p < ∞ with Khvedelidze weights ρ. The same operators but acting in L p (Γ , ρ) have been proved to be invertible by [28, Theorem 14], in the special case of the space L2 (Γ ) in Corollary 15 of the same paper or [22, 24] . Thus if the right hand sides f ∈ W21 (Γ ), an exact or an approximate solution of the equation (4.1) can be derived from the corrected Sherman-Lauricella equation (4.3). In the present work, we employ spline based Galerkin methods to the equation (4.3) and study their stability and convergence. Let us describe these methods in more d detail. Let Sn,∗ and the index set I (n, d) be as in Section 2.5. An approximate

solution of the equation (4.3) is sought in the form ωn (t) =

X

ak φnk (t),

(4.4)

d (Γ ) φnk ∈Sn,∗

the coefficients ak of which are obtained from the following system of algebraic equations (BΓ ωn , φn j ) = ( f , φn j ),

j ∈ I (n, d).

(4.5)

An important problem now is to study the solvability of the equations (4.5) and convergence of the approximate solutions to an exact solution of the original Sherman– Lauricella equation (4.1). In Section 4.2, this problem is discussed in a more detail but, at the moment, we would like to illustrate the method by a few numerical examples. Thus we present Galerkin solutions of the equation (4.3) with the right-hand


53

side f = f1 , f1 (z) = f (x, y) = 4x 3 − 12x y 2 + i(4 y 3 − 12x 2 y);

z = x + iy ∈ Γ,

(4.6)

on the unite square and rhombuses, and trace the evolution of the solution when the initial contour is transformed from the unit square into rhombuses with various opening angle α. Some of these contours have been used in [22] in order to illustrate the behavior of the Nyström method. Note that in the corresponding examples from [22], approximate solutions of the equation (4.1) with the right-hand side f2 (z) = |z| have been determined. We apply the spline Galerkin method to the equations with such right-hand side, too. The results obtained have a very good correlation with fi [22] and the error evaluation for both cases are reported in Table 4.1, where En,α

denotes the relative error kω2n − ωn k2 /kω2n k2 computed for the righthand side f i and equation (4.1) is considered on the rhombus with the opening angle α. In TABLE 4.1: Convergence of the approximate solutions for Sherman–Lauricella equation by spline Galerkin method with d = 0.

n 128 256 512

f

1 En,π/2 0.0373 0.0198 0.0096

f

1 En,π/3 0.6194 0.0268 0.0059

f

1 En,π/4 1.3577 0.2046 0.0616

f

1 En,π/5 2.1716 0.6169 0.1888

f

2 En,π/2 0.0121 0.0067 0.0045

f

2 En,π/3 0.0217 0.0112 0.0102

f

2 En,π/6 0.0205 0.0245 0.0193

addition, Figures 4.1–4.4 show the convergence of the approximate solutions of the equation (4.1) with the right-hand side (4.6) obtained by the Galerkin method based on the splines of degree d = 0 and the transformation of these approximate solutions when n increases. Let us mention a few technical details related to the examples below. Thus the rhombus with an opening angle α is parameterized as follows,  α   4s − cos e iα/2   2  α    iα (4s − 1)e − i sin e iα/2 2 γ(s) = α   −(4s − 2) + cos e iα/2   2   α   −(4s − 3)e iα + i sin e iα/2 2

if 0 ≤ s < 1/4, if 1/4 ≤ s < 1/2, (4.7) if 1/2 ≤ s < 3/4, if 3/4 ≤ s ≤ 1.


54

We also have to compute the scalar products (BΓ ωn , φn j ). Recall that supp φn j ⊂ [ j/n, ( j + d + 1)/n] and use the Gauss-Legendre quadrature rule with quadrature points which coincide with the zeros of the Legendre polynomial P24 (x) on the canonical interval [−1, 1], scaled and shifted to the interval [ j/n, ( j + d + 1)/n]. More specifically, the corresponding formula is (BΓ ωn , φn j ) =

Z

( j+d+1)/n

BΓ ωn (γ(s))φn j (γ(s))ds ≈ j/n

24 X

w k BΓ ωn (γ(sk ))φn j (γ(sk )),

k=1

(4.8)

where w k , sk are the Gauss-Legendre weights and the Gauss-Legendre points on the interval [ j/n, ( j + d + 1)/n]. In order to find the values of the corresponding line integrals at the Gauss-Legendre points, the composite Gauss-Legendre quadrature (2.11) with m = 40 and r = 24 is used. Table 4.1 and Figures 4.1-4.2 show a good convergence of approximate solutions if the corner point of the contour has an opening angle close or equal to π/2. On the other hand, the presence of opening angles of a small magnitude can cause problems and leads to a convergence slowdown (see Figures 4.3-4.4). Note that although the focus of this work is on the stability, the error estimates presented in Table 4.1 are comparable with estimates of the recent work [56] for fast Fourier–Galerkin method 1

1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0

0

−0.2

−0.2

−0.4

−0.4

−0.6

−0.6

−0.8 −1 −1

−0.8 −0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

−1 −1

1

1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0

0

−0.2

−0.2

−0.4

−0.4

−0.6

−0.6

−0.8

−0.8

−1 −1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

−1 −1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

FIGURE 4.1: Approximate solution ωn (t) of the ShermanLauricella equation (4.1) on the unit square Γ with f := f1 dened by (4.6) and d = 0. From the left to the right: n = 128, 256, 512, 1024


55

4

4

3

3

2

2

1

1

0

0

−1

−1

−2

−2

−3

−3

−4 −4

−3

−2

−1

0

1

2

3

4

2.5

−4 −4

−2

−1

0

1

2

3

4

2.5

2

2

1.5

1.5

1

1

0.5

0.5

0

0

−0.5

−0.5

−1

−1

−1.5

−1.5

−2 −2.5 −2.5

−3

−2

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

−2.5 −2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

FIGURE 4.2: Approximate solution ωn (t) of the ShermanLauricella equation (4.1) on the rhombus Γ , α = π/3 with f := f1 dened by (4.6) and d = 0. From the left to the right: n = 128, 256, 512, 1024 3

2.5 2

2

1.5 1

1

0.5 0

0 −0.5

−1

−1 −1.5

−2

−2 −3 −5

−4

−3

−2

−1

0

1

2

3

4

5

2.5

−2.5 −2

−1

−0.5

0

0.5

1

1.5

2

2.5

2

2

1.5

1.5

1

1

0.5

0.5

0

0

−0.5

−0.5

−1

−1

−1.5

−1.5

−2 −2.5 −2.5

−1.5

−2 −2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

−2.5 −2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

FIGURE 4.3: Approximate solution ωn (t) of the ShermanLauricella equation (4.1) on the rhombus Γ , α = π/4 with f := f1 dened by (4.6) and d = 0. From the left to the right: n = 128, 256, 512, 1024

Chapter 4. Sherman - Lauricella equation 2.5

56 2.5

2

2

1.5

1.5

1

1

0.5

0.5

0

0

−0.5

−0.5

−1

−1

−1.5

−1.5

−2

−2

−2.5 −4

−3

−2

−1

0

1

2

3

4

2.5

−2.5 −2

−1.5

−1

−0.5

0

0.5

1

1.5

2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

2

2

1.5

1.5

1

1

0.5

0.5

0

0

−0.5

−0.5

−1

−1

−1.5

−1.5

−2

−2

−2.5 −2.5

−2.5 −2

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

FIGURE 4.4: Approximate solution ωn (t) of the ShermanLauricella equation (4.1) on the rhombus Γ , α = π/5 with f := f1 dened by (4.6) and d = 0. From the left to the right: n = 128, 256, 512, 1024

for an integral equation used to solve boundary value problem (1.11) in smooth domains. Moreover, further improvement of the convergence rate is possible if for the approximations of singular integrals and inner products arising in the Galerkin method one employs graded meshes of various kind, see [13, 52].

4.2

Stability of Galerkin method

Our next task is to find conditions of applicability of the spline Galerkin methods to the equation (4.3). It is worth mentioning that for smooth contours Γ , the methods considered here are always applicable and provide satisfactory results, see [31] or [29, Chapter 6] where similar methods for the Muskhelishvili equation on smooth contours are considered. On the other hand, the presence of corners changes the situation drastically, and the applicability of the approximation method is not always guaranteed. d Let Pnd be the orthogonal projection from L 2 (Γ ) on the subspace Sn,∗ (Γ ). Then the

systems (4.5) are equivalent to the following operator equations Pnd BΓ Pnd ωn = Pnd f ,

n ∈ N.

(4.9)


57

Let us study the stability of the corresponding sequence (Pnd BΓ Pnd ). Let Ladd (L 2 (Γ )) refer to the real C ∗ −algebra of all additive continuous operators acting on the space L 2 (Γ ). According to [29], every operator A ∈ Ladd (L 2 (Γ )) admits the unique representation A = A1 + A2 M , where A1 , A2 are linear operators and M is the operator of complex conjugation. This representation allows one to introduce the operation of involution on Lad d (L 2 (Γ )) as follows A∗ := A∗1 + M A∗2 ,

(4.10)

with A∗1 , A∗2 being the usual adjoint operators to the linear operators A1 , A2 , cf. [29, Theorem 1.3.8 and Example 1.3.9]. By A Γ we denote the set of all bounded sequences (An ) of bounded additive operators An : im Pnd → im Pnd such that there is an operator A ∈ Ladd (L 2 (Γ )) with the property s − lim An Pnd = A,

s − lim (An Pnd )∗ Pnd = A∗ ,

where s − lim An denotes the strong limit of the operator sequence (An ). Endowed by the natural operations of addition, multiplication, multiplication by scalars λ ∈ C, by an involution introduced according to (4.10), and by the norm ||(An )|| := sup ||An ||, n∈N

the set A Γ becomes a real C ∗ -algebra. Consider also the subset J Γ ⊂ A Γ consisting of all sequences (Jn ) of operators Jn : im Pnd → im Pnd which can be represented in the form Jn = Pnd K Pnd + Gn ,

n ∈ N,

where the operator K belongs to the ideal Kadd (L 2 (Γ )) ⊂ Ladd (L 2 (Γ )) of all compact operators and the sequence tends to zero uniformly. The stability form the real C ∗ −algebra A Γ is characterized as follows: Theorem 4.2 (cf. Proposition 3.4 and Theorem 2.17). A sequence (An ) ∈ A Γ such that A := s − lim An Pnd is stable if and only if the operator A is invertible in Ladd (L 2 (Γ )) and the coset (An ) + J Γ is invertible in the quotient algebra A Γ /J Γ . Consider now the sequence (Pnd BΓ Pnd ) of the Galerkin operators defined by the projection operators Pnd . Recall that on the space L 2 (Γ ) the sequence of the orthogonal projections (Pnd ) strongly converges to the identity operator I and (Pnd )∗ = Pnd , n ∈ N.


58

This implies that for any operator A ∈ Ladd (L 2 (Γ )) the following relations s − lim Pnd APnd = A,

s − lim(Pnd APnd )∗ Pnd = A∗

hold [78]. Corollary 4.3. Let Γ be a simple closed piecewise smooth curve. The spline Galerkin method (4.9) is stable if and only if the coset (Pnd BΓ Pnd )+J Γ is invertible in the quotient algebra A Γ /J Γ . The invertibility problem of (Pnd BΓ Pnd ) + J Γ in A Γ /J Γ can be tackled more efficiently in a smaller algebra containing the coset (Pnd BΓ Pnd ) + J Γ . More precisely, let us consider the smallest closed real C ∗ -subalgebra B Γ of the algebra A Γ which contains all operator sequences of the form (Pnd M Pnd ), (Pnd SΓ Pnd ) and also the sequences (Pnd f Pnd ), f ∈ CR (Γ ) and (Gn ), where CR (Γ ) is the set of all continuous real-valued functions on the contour Γ . Remark 4.4. It follows from [28, 36, 70, 78] that J Γ ⊂ B Γ and that the sequence (Pnd BΓ Pnd ) belongs to B Γ . Therefore, B Γ /J Γ is a real C ∗ -subalgebra of A Γ /J Γ , and by Corollary 2.8 the coset (Pnd BΓ Pnd ) + J Γ is invertible in A Γ /J Γ if and only if it is invertible in B Γ /J Γ . Therefore, one can now study the invertibility of the coset (Pnd BΓ Pnd ) + J Γ in the smaller algebra B Γ /J Γ . To this end we will employ a localizing principle and the approach is similar to those made in Chapter 3. Let Γτ , τ ∈ Γ be the model curve (2.7). On the curve Γτ consider the corresponding Sherman-Lauricella operator Aτ = I + Lτ − Kτ M ,

(4.11)

where 1 Lτ ω(t) := 2πi

Z Γτ

ω(ζ) d ln

ζ− t ζ− t

,

1 Kτ ω(t) := 2πi

Z Γτ

ζ− t . ω(ζ) d ζ− t

Analogously to the algebra B Γ and to the ideal J Γ one can introduce algebras B Γτ and ideals J Γτ ⊂ B Γτ , τ ∈ Γ , which allow to establish conditions of the applicability of the corresponding Galerkin method for the operator (4.11). For this we also need appropriate spline spaces on both the contour Γτ and the positive semi-axis R+ := R++ . These spline spaces have been constructed earlier in Section 2.5 of Chapter 2.


59

Let us use the same notations, viz., Snd (Γτ ) and Snd (R+ ). Moreover, let Pnτ , n ∈ N and Pn+ denote the orthogonal projections onto the subspaces Snd (Γτ ) and Snd (R+ ), respectively. For the stability of (Pnτ Aτ Pnτ ) ∈ B Γτ one has Corollary 4.5 (cf. Corollary 3.5). The sequence (Pnτ Aτ Pnτ ) ∈ B Γτ is stable if and only if the operator Aτ is invertible in B τ and the coset (Pnτ Aτ Pnτ ) + J Γτ is invertible in the quotient algebra B Γτ /J Γτ . Further, as before, let L22 (R+ ) be the space of all pairs (ϕ1 , ϕ2 ) T , ϕ1 , ϕ2 ∈ L 2 (R+ ) endowed by the norm ||(ϕ1 , ϕ2 ) T || := (| |ϕ1 ||2 + ||ϕ2 ||2 )1/2 , and let η : L 2 (Γτ ) → L22 (R+ ) be the mapping defined by (3.4). Recall that η is a linear isometry from L 2 (Γτ ) onto L22 (R+ ) and the mapping Ψ : Ladd (L 2 (Γτ )) → Lad d (L22 (R+ )) defined by Ψ(A) = ηAη−1 ,

(4.12)

is an isometric algebra isomorphism. In particular, straightforward calculations show that

Ψ(Pnτ ) = diag (Pn+ , Pn+ ),

(4.13)

e,M e ), Ψ(M ) = diag ( M

(4.14)

 Ψ(Lτ ) = 

0 Nθτ

 Ψ(Kτ ) = 

Nθτ 0

 ,

(4.15)

0

e i2βτ M2π−θτ

−e i2(βτ +θτ ) Mθ j

0

 ,

where 1 1 Nθτ ϕ(σ) = 2 2πi 1 Mθτ ϕ(σ) := π

Z 0

Z

∞

0 ∞

ds 1 1 ϕ(s) , − iθ i(2π−θ ) τ τ 1 − (σ/s)e 1 − (σ/s)e s

sin θτ σ ds ϕ(s) , iθ 2 s (1 − (σ/s)e τ ) s

(4.16)


60

e in the right-hand side of (4.14) refers to the operator of the comand the symbol M plex conjugation on the space L 2 (R+ ). Moreover, one can observe that the operators Nθτ and Mθτ have a special integral form – viz. ∞

Z

kθτ

Kϕ(σ) := 0

σ s

ϕ(s)

ds s

(4.17)

and

kθτ = kθτ (u) := nθτ (u) =

1 u sin θτ , 2π |1 − ue iθτ |2

if

K = Nθτ ,

(4.18)

kθτ = kθτ (u) := mθτ (u) =

1 u sin θτ , π (1 − ue iθτ )2

if

K = Mθτ .

(4.19)

On the space l 2 of the sequences (ξk ) of complex numbers ξk , k = 0, 1, . . ., l 2 := {(ξk )∞ : k=0

∞ X

|ξk |2 < ∞},

k=0

the function kθτ defines a bounded linear operator A(kθτ ) with the matrix representation A(kθτ ) = ν2d

Z

d+1

Ò φ(t)

0

Z

d+1

kθτ 0

∞ du u+l Ò φ(u) dt t +q u+q

,

(4.20)

q,l=0

where νd is the constant (2.8). Theorem 4.6. Let nθτ and mθτ be the functions defined by (4.18) and (4.19), respectively. The spline Galerkin method (4.9) is stable if and only if the operators Rτ : l 2 × l 2 → l 2 × l 2 , Rτ :=  

I A(nθτ )

A(nθτ ) I

  +

0

e iβτ A(m2π−θτ )

−e−i(βτ +θτ ) A(mθτ )

0

 

M 0

0 M



(4.21)



are invertible for all τ ∈ MΓ . Proof. By Corollary 4.3 the sequence (Pnd BΓ Pnd ) is stable if and only if the coset (Pnd BΓ Pnd ) + J Γ is invertible. Moreover, since TΓ of (4.2) is a compact operator, the sequences (Pnd AΓ Pnd ) and (Pnd BΓ Pnd ) belong to the same coset (Pnd AΓ Pnd ) + J Γ of the


61

quotient algebra B Γ /J Γ . However, by Theorem 2.10, the coset (Pnd AΓ Pnd )+J Γ is invertible if and only if for every τ ∈ Γ the coset (Pnτ AΓτ Pnτ )+J Γτ is invertible in the corresponding algebra B Γτ /J Γτ . Therefore, the stability of our operator sequence will be established if we manage to show the invertibility of all cosets (Pnτ AΓτ Pnτ ) + J Γτ , τ ∈ Γ . Let us start with the case where τ is not a corner point of Γ . If τ ∈ / MΓ , then θτ = π, and straightforward calculations show that Lτ and Kτ are the zero operators. Hence, Aτ is just the identity operator I in the corresponding space, so that Pnτ Aτ Pnτ = Pnτ . The sequence (Pnτ ) is obviously stable so that the corresponding coset (Pnτ ) + J τ is invertible. Consider next the case where τ ∈ MΓ . The operator Aτ is invertible on the space L 2 (Γτ ), [22, Theorem 2.2]. Note that the invertibility of the operator Aτ in L2 (Γ ) also follows from [28, Corollary 11] and from [24, Lemma 2.3]. Therefore, by Corollary 4.5 the coset (Pnτ Aτ Pnτ ) + J τ is invertible in B Γτ /J Γτ if and only if the sequence (Pnτ Aτ Pnτ ) is stable. However, the stability of this sequence is equivalent to the stability of the sequence (Ψ(Pnτ Aτ Pnτ )), where the mapping Ψ is defined by (4.12). Consider also the operators Λn : Snd (R+ ) → l 2 defined by Λn

∞ X

ξ j φn j

= (ξ0 , ξ1 , . . . , ).

j=0

It follows from [20] that these operators are bounded and continuously invertible. Set Λ−n := Λ−1 and note that the sequence (Ψ(Pnτ Aτ Pnτ )) is stable if and only if so is n the sequence (Rτn ), where Rτn = diag (Λn , Λn ) · Ψ(Pnτ Aτ Pnτ ) · diag (Λ−n , Λ−n ) : l 2 × l 2 → l 2 × l 2 . From the definition of the mappings Ψ and Λ±n one obtains that the operators Rτn have the form (n,τ)

(n,τ) 2 )l,p=1 diag (M , M ),

Rτn = (Al p )2l,p=1 + (Dl p (n,τ)

(n,τ)

with the operators Al p , Dl p

: l 2 → l 2 defined according to the relations (4.13)-

(4.16), (4.20)-(4.21). However, these operators do not depend on the parameter n (n,τ)

(n,τ)

(n,τ)

at all. Really, consider the matrix representations of the operators A12 , A21 , D12 (n,τ)

and D21 . It follows from (4.17) that the entries alq of the corresponding matrices


62

(alq )∞ are l,q=0 a pq =

Z

Kφqn (σ)φl n (σ) dσ =

1 = n

Z

1 = n

Z

=

Z

R+

R+

Z

R+

ν2d

Z

R+

kθτ

Z

R+

Z 0

R+

kθτ

d+1

Ò φ(t)

R+

kθτ

σ s

φ(ns − q)

ds φ(nσ − l) dσ s

du u+l φ(u) φ(t) d t t +q u+q

p p du u+l Ò Ò (νd nφ(u)) (νd nφ(t)) dt t +q u+q

Z

d+1

kθτ 0

u+l Ò du φ(u) d t, t +q u+q (n,τ)

(n,τ)

hence these operators are independent of n. Moreover, D11 , D22 (n,τ) A22 = I. Combining all the above Rτn do not depend on the parameter

(n,τ)

= 0 and A11

=

representations, one obtains that the operators n. Therefore, (Rτn ) is a constant sequence and it

is stable if and only if any of its members, say Rτ1 , is invertible. It remains to observe that Rτ = Rτ1 , which completes the proof.

4.3


As was already mentioned, there is no efficient analytic method to verify the invertibility of the local operators Rτ . On the other hand, numerical approaches turn out to be surprisingly fruitful. Recall that the operators Rτ , τ ∈ MΓ do not depend on the shape of the contour Γ but only on the relevant angles θτ and βτ . Therefore, for contours having only one corner point, Theorem 4.6 can be reformulated as follows. Corollary 4.7. If τ is the only corner point of the contour Γ , then the operator Rτ is invertible if and only if the Galerkin method (Pnd BΓ Pnd ) is stable. Thus in order to determine the critical angles, i.e. the opening angles θ for which the operators Rτ are not invertible, one can consider the behaviour of the spline Galerkin methods on special contours. A family of such contours Γ1θ , θ ∈ (0, 2π), Γ1θ := {t ∈ C : t = γ1 (s) = sin(πs) exp(iθ (s − 0.5)), s ∈ [0, 1]} has been used in [21, 24] to study the local operators of the Nyström method for Sherman–Lauricella and Muskhelishvili equations. Changing the parameter θ in


63

the interval (0, 2π), one obtains contours located at the origin and having only one corner of various magnitude. In the present paper, we use the same contours to detect the critical angles of the spline Galerkin methods. It is worth mentioning that the operator Rτ depends not only on θτ but also on the angle βτ between the right semi-tangent to the contour Γ1θ at the point τ and the real line R. However, numerical experiments conducted for both the Nyström and spline Galerkin methods show that, in fact, the angle βτ does not influence the invertibility of the operator Rτ (see Figure 4.8 below and Remark 4.8). This opens a way for verifying the results obtained for contour Γ1θ by conducting similar tests for equations on contours with two or more corners, all of the same magnitude. To this end, we will use another contour Γ2θ , which is the union of two circular arcs with the parametrization γ1 (s) = −0.5 cot(0.5θ ) + 0.5/ sin(0.5θ ) exp(iθ (s − 0.5)),

0 ≤ s ≤ 1,

γ2 (s) = 0.5 cot(0.5θ ) − 0.5/ sin(0.5θ ) exp(iθ (s − 0.5)),

0 ≤ s ≤ 1.

To find the angles of instability, the interval [0.1π, 1.9π] has been divided by the points θk := π(0.1 + 0.01k) and for each opening angle θk we constructed the matrices of the corresponding approximation operators for the Galerkin methods based on the splines of degree d = 0, d = 1 and d = 2. Note that we consider Galerkin methods for two choices of n, namely for n = 128 and n = 256, and the integrals arising in the equation (4.3) and in the method (4.5) have been approximated by quadrature formulas (4.8) and (2.11) of Chapter 2. Further, to verify the stability of the method, for each angle θk we compute the condition numbers of the corresponding matrices and the results of these computations are presented in Figures 4.5-4.7, where possible presence of peaks might indicate critical angles. Thus it seems that inside of the interval (0.1π, 1.9π) neither of the Galerkin methods based on splines of degree 0, 1 or 2 has critical angles. This differs from the Nyström method, where critical angles have been discovered for both Sherman–Lauricella and Muskhelishvili equations [21, 24]. Contrariwise, information about the critical angles at the interval ends is not so conclusive. Thus in the case n = 256, the computation of the condition numbers for both one and two corner geometry shows that for the Galerkin method based on the splines of degree zero there can be a critical angle at the right end of the interval mentioned. For splines of the degree d = 0 and d = 1, the one and two corner geometries give contradictory results (see Figure 4.6). To clarify the situation one has to refine the mesh {θk } and essentially increase the dimension of the matrices used. Note that

64

8

8

7

7 log10 Condition Numbers

log10 Condition Numbers


6 5 4 3 2 1

0.2

0.4

0.6

0.8

1

θ/π

1.2

1.4

1.6

1.8

4 3 2

0 0

2

8

8

7

7

6 5 4 3 2 1 0 0

0.4

0.6

0.8

0.2

0.4

0.6

0.8

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

1

1.2

1.4

1.6

1.8

2

1

1.2

1.4

1.6

1.8

2

θ/π

6 5 4 3 2

0.2

0.4

0.6

0.8

1

θ/π

1.2

1.4

1.6

1.8

0 0

2

8

8

7

7

6 5 4 3 2 1 0 0

0.2

1



5

1



0 0

6

θ/π

6 5 4 3 2 1

0.2

0.4

0.6

0.8

1 θ/π

1.2

1.4

1.6

1.8

2

0 0

θ/π

FIGURE 4.5: Condition numbers vs. opening angles in case n = 128. From row 1 to row 3: splines of degree 0, 1 and 2, respectively. Left column: one-corner geometry, right column: two-corner geometry.

while discovering a suspicious critical angle for n = 256, we refined the mesh {θk } in a neighbourhood of that angle by reducing its step to 0.001π, and calculated the condition numbers for the corresponding Galerkin methods with n changed to 512. This allows us to show that, in fact, there are no critical angles in the interval mentioned. However, the computing time increases drastically. Remark 4.8. As was already mentioned, numerical experiments do not show that the stability depends on the angle β. The results obtained for both curves show the same behavior of condition numbers even if the parameter β is different for Γ1θ and Γ2θ . Moreover, we rotate the curve Γ1θ by various angles and compute the corresponding condition numbers. Although the condition numbers are different the same angles θ , the graphs do not have any peaks. Note that the mesh has again been refined in neighborhoods of suspicious points.

16

14

14

12

12

10

8

10

8

10

6

log

6 4

4

2

2

0

0.5

1 0.1π ≤ θ ≤ 1.9π

1.5

0

2

16

14

14

12

12

Cond.Num(θ)

16

10

0.5

1 0.1π ≤ θ ≤ 1.9π

1.5

2

0

0.5

1 0.1π ≤ θ ≤ 1.9π

1.5

2

8

10

8

0

10

6

6

log

log 10Cond.Num(θ)

Cond.Num(θ)

16

0

4

4

2

2

0

log 10Cond.Num(θ)

65

0

0.5

1 0.1π ≤ θ ≤ 1.9π

1.5

0

2

16

16

14

14

12

12 log10 CondNum(θ)

log 10Cond.Num(θ)


10 8 6

10 8 6

4

4

2

2

0

0

0.5

1 0.1π ≤ θ ≤ 1.9π

1.5

2

0 0

0.2

0.4

0.6

0.8

1 1.2 0.1π ≤ θ ≤ 1.9π

1.4

1.6

1.8

2

FIGURE 4.6: Condition numbers vs. opening angles in case n = 256. From row 1 to row 3: splines of degree 0, 1 and 2, respectively. Left column: one-corner geometry, right column: two-corner geometry.

66

8

8

7

7 log10 Condition Numbers



6 5 4 3 2 1

0.2

0.4

0.6

0.8

1

θ/π

1.2

1.4

1.6

1.8

4 3 2

0 0

2

8

8

7

7

6 5 4 3 2 1 0 0

0.4

0.6

0.8

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

1

1.2

1.4

1.6

1.8

2

1.2

1.4

1.6

1.8

2

θ/π

6 5 4 3 2

0.2

0.4

0.6

0.8

1

θ/π

1.2

1.4

1.6

1.8

0 0

2

θ/π

8

8

7

7

6 5 4 3 2 1 0 0

0.2

1



5

1



0 0

6

6 5 4 3 2 1

0.2

0.4

0.6

0.8

1 θ/π

1.2

1.4

1.6

1.8

2

0 0

0.2

0.4

0.6

0.8

1

θ/π

FIGURE 4.7: Condition numbers vs. opening angles in case n = 256 and n = 512 in

neighbourhoods of suspicious points. From row 1 to row 3: splines of degree 0, 1 and 2, respectively. Left column: one-corner geometry, right column: two-corner geometry.

FIGURE 4.8: Condition numbers vs. opening angles in case n = 256 and the curve Γ1θ rotated by 0.3π, 0.5π, 1.2π and 1.4π. The mesh has been rened in

neighbourhoods of suspicious points and no "innite" peaks are discovered.

Chapter 5 Application to the approximate solutions of boundary value problems for biharmonic equation In this Chapter, we present the transition between boundary value problems (BVP) for homogeneous and nonhomogeneous equations and boundary conditions. A numerical scheme for a class of general boundary value problem for biharmonic equation is proposed. This scheme relies only on the boundary integral equation method.

5.1

Boundary value problems with homogeneous and nonhomogeneous boundary data

Let Ω ⊂ R2 be a planar domain with the Lipschitz boundary Γ := ∂ Ω. Let ν(t) = (ν1 (t), ν2 (t))> be the unit normal vector field on the boundary t ∈ Γ and ∂ν = 2 X ν j ∂ j be the normal derivative. j=1

Consider the boundary value problem (BVP) for the biharmonic equation  2 ∆ U (x) = f (x),      

x ∈ Ω,

U + (t) = g(t),

t ∈ Γ,

(∂ν U )+ (t) = h(t),

t ∈ Γ,

67

(5.1)

Chapter 5. Applications

68

where the superscript ϕ + (t) denotes the trace of the function ϕ(x), defined on the domain Ω, on the boundary Γ . Lax-Milgram Lemma (cf. [63, 67]) applied to the BVP (5.1) gives the following result (see [64] for the Laplace equation and the recent [39] for a similar result for mixed BVP’s on hypersurfaces for the Laplace-Beltrami equation). Theorem 5.1. The BVP (5.1) has a unique solution U ∈ W2 (Ω) in the classical weak setting: e −2 (Ω), f ∈W

g ∈ W3/2 (Γ ),

h ∈ W1/2 (Γ ),

(5.2)

e m (Ω), Wm (Ω) and Ws (Γ ) are the standard Sobolev spaces on the domain Ω where W and the boundary Γ . e m (Ω) and Wm (Ω) is that the functions from The difference between the spaces W e m (Ω) have vanishing traces ϕ + , . . . , (∂ m−1 ϕ)+ on the boundary Γ , while Wm (Ω) is W the restriction of the space Wm (R2 ) to the subdomain Ω ⊂ R2 (see [82] for precise definitions). Let us note that instead of full BVP (5.1), it is convenient to solve particular cases, namely, homogeneous equation with nonhomogeneous boundary conditions and another nonhomogeneous one with homogeneous boundary conditions. Thus let us consider the BVPs  2 ∆ V (x) = 0      

x ∈ Ω,

V + (t) = g(t),

t ∈ Γ,

(∂ν V )+ (t) = h(t),

t ∈Γ

(5.3)

and  2 ∆ W (x) = f (x)      

x ∈ Ω,

W + (t) = 0,

t ∈ Γ,

(∂ν W )+ (t) = 0,

t ∈Γ

(5.4)

where known functions g, h and f have the same properties as in (5.2). For the BVP (5.3) it is convenient to apply the potential method and reduce it to boundary integral equation, which can be solved approximately by boundary element method, while to the BVP (5.4) it is convenient to apply the discretization directly in the domain.


69

It is obvious, that the solutions U , V and W of the respective BVPs (5.1), (5.3) and (5.4) are related by the equality: U (x) = V (x) + W (x),

x ∈Ω

(5.5)

and knowing the solutions of two of them, we easily find the solution of the third one. Let us discuss the well known fact: how solutions of BVP’s (5.3) and (5.4) are related to each-other. Knowing the solution of one of them, how we can get the solution of the other one. For this we need to recall the so-called potential operators, viz., the Newton potential for the biharmonic equation (5.1) N ∆2 ϕ(x) :=

Z Ω

K∆2 (x − y)ϕ( y) d y

x ∈ Ω.

(5.6)

In (5.6) the kernel K∆2 (x) is the fundamental solution to the biharmonic equation ∆2 K∆2 (x) = δ(x),

x ∈ Ω.

(5.7)

The fundamental solution K∆ (x) to the harmonic (Laplace) equation ∆K∆ (x) = δ(x),

x ∈Ω

(5.8)

is well known and is exposed in many textbooks. Thus: K∆ (x) :=

1 ln |x|. 2π

(5.9)

In contrast to the fundamental solution K∆ (x) the fundamental solution K∆2 (x) is less popular and we present some results here. Lemma 5.2. The fundamental solution of biharmonic equation has the form K∆2 (x) =

1 |x|2 ln |x|, 8π

and satisfies the equality (5.7).

∆ := ∂12 + ∂22 ,

(5.10)


70

Proof. First, using routine calculations, we check that the following equalities hold: ∆[|x|2 ] = ∂12 + ∂22

x 12 + x 22 = 4,

∆[ln |x|] = 2πδ(x), ∆[ϕ(x)ψ(x)] = ϕ(x)∆ ψ(x) + ψ(x)∆ ϕ(x) + 2〈∇ ϕ(x), ∇ ψ(x)〉, (5.11) ∇[|x|2 ] = ∇(x 12 + x 22 ) = 2(x 1 , x 2 ) =: 2x, 1 2x 2 ∇[ln |x|] = ∇[ln(x 12 + x 22 )] = 2 , (x 1 , x 2 ) = 2 2 |x| x1 + x2 where ∇ϕ := (∂1 ϕ, ∂2 ϕ)> denotes the gradient vector. Now, using the equalities (5.11) we proceed as follows: ∆2 [|x|2 ln |x|] = ∆ ln |x|∆[|x|2 ] + |x|2 ∆[ln |x|] + 〈∇[|x|2 ], ∇[ln |x|]〉 § ª 4 2 = ∆ 4 ln |x| + 2π|x| δ(x) + 2 〈x, x〉 = 4∆[ln |x| + 1] = 8πδ(x), |x| since |x|γ δ(x) = 0 for arbitrary γ > 0 and (5.7) is proved. Lemma 5.3. The function W0 = N ∆2 f ∈ W2 (Ω) is a solution of the first equation in (5.4), but does not satisfy the boundary conditions (the function and its normal derivative do not necessarily vanish on the boundary). If V ∈ W2 (Ω) is a (unique) solution to the BVP (5.3) with the following data g := −W0+ ∈ W3/2 (Γ ),

h = −[∂ν W0 )]+ ∈ W1/2 (Γ ),

(5.12)

then the function W := V + W0 belongs to the Sobolev space and solves the BVP (5.4) e −2 (Ω). In particular, the solution and its normal with the original right-hand side f ∈ W derivative vanish on the boundary, W + (t) = (∂ν W )+ (t) ≡ 0 on Γ . Proof. Using equality (5.9), it is easy to show that ∆2 W0 = ∆2 N ∆2 f (x) = f (x)

for all

and, therefore, W0 solves the first equation in (5.4).

x ∈Ω

(5.13)


71

It follows from the standard boundedness property of the Newton’s potential operator (see [38]) e s (Ω) −→ Ws+4 (Ω) N ∆2 : W

∀s ∈ R

e −2 (Ω) that W0 ∈ W2 (Ω). Now one can apply and from the initial condition f ∈ W the celebrated trace theorem: If ϕ ∈ Wm (Ω), m ¾ 2, then the traces of ϕ on the boundary belong to the spaces ϕ + ∈ Wm−1/2 (Γ ),

[∂ν ϕ)]+ ∈ Wm−3/2 (Γ )

(see [82] for details and proofs). Thus, the traces of W0 fall into the proper spaces: g := −W0+ ∈ W3/2 (Ω), h = −[∂ν W0 )]+ ∈ W1/2 (Ω). All conditions of Theorem 5.1 hold and the BVP (5.3) has a unique solution V ∈ W2 (Ω). Then the function W := V + W0 solves the BVP (5.4): ∆2 W (x) = ∆2 V (x) + ∆2 W0 (x) = ∆2 W0 (x) = f (x)

x ∈ Ω,

W + (t) = V + (t) + W0+ (t) = −W0+ (t) + W0+ (t) = 0,

t ∈ Γ,

(∂ν W )+ (t) = (∂ν V )+ (t) + (∂ν W0 )+ (t) = −(∂ν W0 )+ (t) + (∂ν W0 )+ (t) = 0,

t ∈Γ

and the lemma is proved. Lemma 5.4. For any pair of functions g ∈ W3/2 (Γ ) and h ∈ W1/2 (Γ ) (cf. the BVP (5.3)) there exists a function V0 ∈ W2 (Ω) such, that V0+ = g, (∂ν V0 )+ = h and ∆2 V0 ∈ e −2 (Γ ). W If W ∈ W2 (Ω) is a (unique) solution to the BVP (5.4) with the following right-hand side e −2 (Γ ), f := −∆2 V0 ∈ W

(5.14)

then the function V := W + V0 belongs to the Sobolev space W2 (Ω) and solves the BVP (5.3). Proof. The first assertion is a theorem on coretraction (the inverse theorem on traces) and is proved in [38, Lemma 4.8] (see also [82]). All conditions of Theorem 5.1 hold and the BVP (5.4) with the right-hand side f := −∆2 V0 has a unique solution W ∈ W2 (Ω). Then the function V := W + V0 solves


72

the BVP (5.3): ∆2 V (x) = ∆2 W (x) − ∆2 V0 (x) = f (x) − f (x) = 0

x ∈ Ω,

V + (t) = W + (t) + V0+ (t) = V0+ (t) = g(t),

t ∈ Γ,

(∂ν V )+ (t) = (∂ν W )+ (t) + (∂ν V0 )+ (t) = (∂ν V0 )+ (t) = h(t),

t ∈Γ

and the lemma is proved. Remark 5.5. There exists no explicit formula for the function V0 ∈ W2 (Ω) which has the prescribed traces V0+ = g ∈ W3/2 (Γ ) and (∂ν V0 )+ = h ∈ W1/2 (Γ ), but in the Example 5.2 below we will solve this problem for a particular case of square and particular polynomial boundary data. Example 5.1. Let Ω = Q1 := [0, 1] × [0, 1] be the unit square and f (x) := [x 1 (1 − x 1 )x 2 (1 − x 2 )]2 ,

x := (x 1 , x 2 )> ∈ Q1 .

(5.15)

Then the function W0 = N ∆2 f (see Lemma 5.3) has the following explicit form W0 (x) = N ∆2 f (x) :=

Z Q1

1 K∆2 (x − y) f ( y)d y = 16π

Z 1Z 0

1

(x 1 − y1 )2 + (x 2 − y2 )2

0 2

× ln[(x 1 − y1 ) + (x 2 − y2 ) ] [ y1 (1 − y1 ) y2 (1 − y2 )] d y2 d y1 , 2

2

(5.16)

x = (x 1 , x 2 )> ∈ Q1 . and can be used to reduce the BVP (5.4) to the BVP (5.3) with non-zero boundary conditions but zero right hand side (cf. Lemma 5.3). The integral in (5.16) can easily be calculated approximately, since the function F (x, y) := (x 1 − y1 )2 + (x 2 − y2 )2 [ y1 (1 − y1 ) y2 (1 − y2 )]2 has second order zero both on the boundary F (0, x 2 ) = F (1, x 2 ) = F (x 1 , 0) = F (x 1 , 1) = 0 and on the diagonal F ((x 1 , x 1 ), ( y1 , y1 )) = 0 (where the logarithmic function ln[(x 1 − y1 )2 + (x 2 − y2 )2 ] has a weak singularity). Example 5.2. Let, as in the foregoing example, Ω = Q1 := [0, 1] × [0, 1] be the unit square and represent 4 sides of its boundary as follows Γ·,0 := [0, 1] × {0},

Γ·,1 := [0, 1] × {1},

Γ0,· := {0} × [0, 1]×,

Γ1,· := {1} × [0, 1].


73

Let us define the boundary data as follows g(t) = t 12 (1 − t 12 ), g(t) = t 1 + t 12 (1 − t 12 ), g(t) = −t 22 (1 − t 22 ), g(t) = t 2 − t 22 (1 − t 22 ),

h(t) = −t 1 , h(t) = −2 + t 1 , h(t) = −t 2 , h(t) = −2 + t 1 ,

t = (t 1 , 0)> ∈ Γ·,0 , t = (t 1 , 1)> ∈ Γ·,1 , t = (0, t 2 )> ∈ Γ0,· ,

(5.17)

t = (1, t 2 )> ∈ Γ1,· .

Noting, that ∂ν = (−1)k+1 ∂2 on the parts of the boundary Γ·,k , k = 0, 1 and ∂ν = (−1)k+1 ∂1 on the parts of the boundary Γk,· , k = 0, 1, it is easy to check, that the function V0 (x) = x 12 (1 − x 12 ) − x 22 (1 − x 22 ) + x 1 x 2 ,

x = (x 1 , x 2 )> ∈ Q1 ,

(5.18)

defined on the entire square, has the property V0+ (t) = g(t) and

(∂ν V0 )+ (t) = h(t) for all

t ∈ Γ = Γ·,0 ∪ Γ·,1 ∪ Γ0,· ∪ Γ1,· ,

∆2 V0 (x) = ∆[2 − 12x 12 − 2 + 12x 22 ] = −24 + 24 = 0

for all

x ∈ Q1

and can be used to reduce the BVP (5.3) to the BVP (5.4) with non-zero right hand side, but zero boundary conditions (cf. Lemma 5.4).

5.2

Boundary integral equation method for approximate solutions of boundary value problems for biharmonic equation

By Lemma 5.3, in order to find an approximate solution Wn of (5.4), one needs to find the Newton potential W0 , get the boundary data g, h by tracing the potential to the boundary, solve the Sherman–Lauricella equation (1.13) by using those data and finally recover the corresponding solution V0 of (5.3) . On the other hand, the discussion above shows that one can solve the full BVP (5.1) by using boundary integral equation method only, viz., first by separating (5.1) into two BVPs (5.3) and (5.4) and second by solving (5.3) and (5.4) by BIE method and then sum up the results according to (5.5).


74

In order to illustrate the boundary integral method, we consider the equation  2 ∆ U (x) = f (x),      

x ∈ Q1 ,

U + (t) = g(t),

t ∈ Γ,

(∂ν U )+ (t) = h(t),

t ∈ Γ,

(5.19)

where the right hand side f (x) = f (x 1 , x 2 ) = 4π4 sin(πx 1 ) sin(πx 2 ),

(5.20)

and the boundary data

g(t) =

h(t) =

  −x 12 ,      x − x 2 − 1, 2 2

x 2 = 0, 0 ≤ x 1 ≤ 1, x 1 = 1, 0 ≤ x 2 ≤ 1,

  x 1 − x 12 − 1, x 2 = 1, 0 ≤ x 1 ≤ 1,     2 −x 2 , x 1 = 0, 0 ≤ x 2 ≤ 1,   −x 1 − π sin(πx 1 ), x 2 = 0, 0 ≤ x 1 ≤ 1,      x − 2 − π sin(πx ), x = 1, 0 ≤ x ≤ 1, 2 2 1 2   x 1 − 2 − π sin(πx 1 ),     −x 2 − π sin(πx 2 ),

(5.21)

x 2 = 1, 0 ≤ x 1 ≤ 1, x 1 = 0, 0 ≤ x 2 ≤ 1.

The exact solution of this equation is U (x) = sin(πx 1 ) sin(πx 2 ) + x 1 x 2 − x 12 − x 22 .

5.2.1

(5.22)

Approximation method for biharmonic problem with homogeneous boundary data

Let us illustrate the result in Lemma 5.3 by finding approximate solutions Wn for the BVP (5.4) with right hand side f (x) of (5.20). Thus, substituting f (x) into (5.6),


75

one gets the Newton potential W0 (x) by the following double integral, W0 (x 1 , x 2 ) = 1 = 16π

Z1 Z1 0

[(x 1 − y1 )2 + (x 2 − y2 )2 ] ln[(x 1 − y1 )2 + (x 2 − y2 )2 ] f ( y1 , y2 )d y1 d y2 .

0

(5.23) This double integral (5.23) is then calculated approximately by using the builtin routine dblquad of MATLAB with the default tolerance of 10−6 . According to Lemma 5.3, the boundary data g, h are found by taking the limits of W0 (x 1 , x 2 ) and ∂n W0 (x 1 , x 2 ) as (x 1 , x 2 ) → t ∈ Γ . The partial derivatives involved are calculated by a standard formula using central differences with the step length 10−4 . After finding an approximate solution ωn of equation (1.13), we construct the approximate Goursat’s functions φn (t), ψn (t) by equations (1.12). In order to use Goursat’s representation, we need to integrate ψn (t) to obtain χn (t). Since the function ψ(t) is analytic in the unit square, χ(t) is an anti-derivative of ψ(t) and formulated as follows: χ(t) = χ(0) +

Z

t

ψ(ζ)dζ.

(5.24)

0

This line integral is independent on the integral path so we adopt the line segment connecting 0 and t. Making the natural parametrization, this integral is then approximated by Gauss – Legendre quadrature (2.10) with r = 20. Since we are only interested in ℜ (χ(t)), the initial value ℜ (χ(0)) is derived from Goursat’s representation as ℜ (χ(0)) = W0 (0, 0). Our recovery scheme can be summarized as follows: After obtaining approximate solution ωn (t) of (1.13), for each z = x 1 + i x 2 ∈ D, the function φn (z) is calculated approximately from the first equation of (1.12). All the auxiliary values of ψ(ζ) in (5.24) are approximated by the second equation of (1.12) using ωn (t). Finally, approximate value Vn (x 1 , x 2 ) of V (x 1 , x 2 ) is calculated by Goursat’s representation and added to W0 (x 1 , x 2 ) to obtain Wn (x 1 , x 2 ). The L 2 −errors kW2n − Wn k L 2 are considered only on the subdomain [0.1, 0.9] × [0.1, 0.9], calculated using the mesh grid [0.1 : 0.01 : 0.9] × [0.1 : 0.01 : 0.9]


76 d,n 0 1 2

128 0.0121 0.0192 0.0240

256 0.0099 0.0105 0.0119

512 0.0031 0.0045 0.0068

TABLE 5.1: Convergence of L 2 −error of Galerkin solutions of BVP for biharmonic

equation with homogeneous boundary conditions.

d,n 0 1 2

128 0.0131 0.0178 0.0208

256 0.0098 0.0103 0.0129

512 0.0045 0.0055 0.0080

1024 0.0038 0.0039 0.0048

TABLE 5.2: Convergence of L 2 −error of Galerkin solutions of general BVP for

biharmonic equation.

and presented in Table 5.1. For the points near the boundary, the accuracy deteriorates drastically. The minimum point-wise absolute error obtained is of the magnitude 10−5 for all three cases d = 0, 1, 2. Note that all the line integrals involved are calculated by a composite Gauss–Legendre quadrature with 16 nodes on each of 40 subintervals.

5.2.2

Approximation method for a class of general boundary value problems for biharmonic equation

It is worth noting that the Sherman–Lauricella equations arising from (5.3) and (5.4) only differ in their right-hand sides. Moreover, the same matrix (BΓ φni , φn j )i, j∈I (n,d) which requires the most time-consuming computation can be used for both (5.3) and (5.4). The approximate solution Wn obtained in the previous Section can be used as an adequate approximation for W in (5.5). Thus, it remains to find approximate solution Vn for (5.3) with boundary data g(t), h(t). We solve the Sherman– Lauricella equation arising from (5.3) by Galerkin methods based on splines of degree d = 0, 1, 2 and n = 128, 256, 512, 1024. The L 2 −errors kUn − U k L 2 are considered and calculated in the same subdomain as the previous subsection. The convergence rate is presented in Table 5.2. The relative pointwise errors |Un (x 1 , x 2 ) − U (x 1 , x 2 )|/|Un (x 1 , x 2 )| presented in Figure 5.1 are considered on the same subdomain and mesh grid as above.


77

0.9

0.9 1

1.5 0.8

0.8

0.5

1 0.7

0

0.7 0.5

0.6

−0.5

0.6

−1

0 0.5

−0.5

−1.5

0.5

−2

0.4

−1

0.4

0.3

−1.5

0.3

−2.5

−2

0.2

−3

−3.5

0.2 −4

−2.5 0.1 0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.1 0.1

0.9

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0.9 1 0.8 0

1

0.7 −1 0.6

0.5

0.5

−2

0.4

−3

0

−0.5 0.3 −4

−1 1

0.2 −5 0.1 0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1 0.5

0.5 0

0

FIGURE 5.1: The relative point-wise error in the domain [0.1, 0.9] × [0.1, 0.9]. Absolute errors: 1st -row left: d = 0,right: d = 1; 2nd -row left: d = 2. 2nd -row right: the approximate solution U1024 obtained with d = 0.

Chapter 6 Conclusions 6.1

Approximation methods for double layer potential equation

The materials presented in Chapter 3 are mostly from the papers [32, 33, 34, 35]. In this Chapter, we study Nyström method and Galerkin method using splines for the double layer potential equation (1.3) on L 2 spaces. Necessary and sufficient conditions for the stability of the spline Galerkin method and the Nyström method have been established. In particular, numerical experiments show that for the curves all angles of which are located in the interval [0.1π, 1.9π], the Galerkin methods based on splines of degree 0, 1, 2 are always stable. Moreover, the observed condition numbers of the resulting linear systems are relatively small. Another remarkable feature is that for the Galerkin methods based on the splines of the same degree 0, 1 or 2, the graphs of condition numbers are of the same shape and the corresponding condition numbers are very close. This suggests that the condition numbers of the Galerkin methods possess certain "locality" properties. They rather depend on the value of the critical angles present than on the shape of the curves used. This conjecture is closely related to Theorem 2.29 of [51] and may be an interesting topic for future work. On the other hand, the numerical experiments for the Nyström method show that there are four critical angles causing instability of the method. This means that the Nyström method needs some appropriate modifications for a few neighboring panels of the corner when dealing with the critical angles. Although in this work we consider only the original Nyström method without any modification, the approximate solutions obtained for some concrete examples demonstrate a good accuracy. 78

Chapter 5. Conclusions

6.2

79

Spline Galerkin method for Sherman–Lauricella equation

The presentation in Chapter 4 essentially follows the paper [30]. Here we consider spline Galerkin methods for the equation (4.1) and study their stability. It is shown that the corresponding method is stable if and only if certain operators Rτ from an algebra of Toeplitz operators are invertible. These operators depend on the spline space used and on the opening angles of the corner points τ ∈ MΓ . We propose a numerical approach to handle the invertibility problem for Rτ . Thus spline Galerkin methods are applied to the Sherman–Lauricella equation on simple model curves and the behavior of the corresponding approximation operators provide an information about the invertibility of the operators Rτ , τ ∈ Γ . Note that in comparison to the Nyström method, the implementation of spline Galerkin methods requires more preparatory work. On the other hand, numerical experiments suggest that these methods have no "critical" angles located in the interval [0.1π, 1.9π], i.e. if the boundary Γ possesses only corners with opening angles from the interval mentioned, then these methods are stable. In a sense, this is similar to the behavior of the corresponding approximation methods for Sherman–Lauricella and Muskhelishvili equations in the case of smooth curves which always converge [22, 28, 31]. This fact is in strong contrast with the behaviour of the Nyström method [21, 22] applied to the same equation. The Nyström method has 8 critical angles causing instability. Of course, one also has to study the opening angles in the intervals (0, 0.1π) and to (1.9π, 2π) but this is the case relating to "cusps" and there are many difficulties both in theory and numerical computations.

6.3

Application to approximate solutions of boundary value problems for biharmonic equation

The materials presented in this Chapter follows a joint work by the author, Dr Yeo Wee Ping and Prof R. Duduchava. We propose a numerical scheme for the approximate solution for a class of general boundary value problems for biharmonic equation. The method demonstrates a fair accuracy. The numerical results can be improved by using other approximation methods, such as Nyström method [22]. Note that all numerical experiments are performed in MATLAB environment(version

Chapter 5. Conclusions

80

7.9.0) and executed on an Acer Veriton M680 workstation equipped with a Intel Core i7 vPro 870 processor and 8GB of RAM and on an Alienware Multimedia machine with similar configuration.

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