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A general framework for multigrid methods for mortar finite elements Christian Wieners and Barbara I. Wohlmuth A BSTRACT. In this paper, a general framework for the analysis of multigrid methods for mortar finite elements is considered. The numerical realization is based on the algebraic saddle point formulation arising from the discretization of second order elliptic equations on nonmatching grids. Suitable discrete Lagrange multipliers on the interface guarantee weak continuity and an optimal discretization scheme. In particular, the mortar method is applied on the coupling of conforming and nonconforming discretizations in case of scalar diffusion problems and linear elasticity in 2D. In contrast to earlier works, no exact solver for a modified Schur complement is required. Numerical results demonstrate the efficiency and reliability of the multigrid method.

1. Introduction Domain decomposition techniques form the starting point for the mortar approach. It allows the coupling of different models, subdomains with nonmatching grids, and discretization techniques within a unified framework. Originally introduced for spectral methods by B ERNARDI M ADAY-PATERA [9, 10], the mortar method is meanwhile extended to the coupling of spectral and finite elements as well as to the coupling of primal and dual methods [4, 7, 8, 11, 30, 33]. The basic idea is to replace the strong continuity condition at the interfaces between the different subdomains by a weaker one. Suitable constraints at the interfaces guarantee optimal discretization schemes in the sense that the global discretization error is bounded by the sum of the best approximation errors on the different subdomains. Here, the weak continuity is realized by a L2 -orthogonality between the jump on the interface and an adequate discrete space. In the last couple of years, a lot of work has been done on the construction of efficient iterative solvers for the arising algebraic linear system [1, 2, 3, 11, 12, 13, 19, 21, 22, 23, 27, 28, 33]. Taking the matching condition at the interface as starting point, there are two different approaches to obtain the discrete mortar solution. Either the weak continuity condition is imposed on the global discrete space or it is satisfied by means of Lagrange multipliers. The first technique results in a nonconforming method, where the discrete space is not contained in H 1 ( ). The characteristic feature of nonconforming methods is that the discrete spaces associated with a nested sequence of triangulations are not nested. However, the resulting linear system is symmetric and positive definite. Following the second approach [7], a saddle point problem is obtained. In particular, additive Schwarz methods and substructuring preconditioners are given for the positive definite system as well as Schur complement methods for the solution of the saddle 1991 Mathematics Subject Classification. 65N30,65N55. Key words and phrases. Mortar finite elements, nonconforming finite elements, multigrid methods. This work was supported in part by the Deutsche Forschungsgemeinschaft, SFB 404.

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CHRISTIAN WIENERS AND BARBARA I. WOHLMUTH

point problem are proposed. For many algorithms, the solution of local problems in each subdomain is supposed to be simple, and is required in each iteration step. In the multilevel context, it has been observed that smoothing of the local problems is sufficient even for indefinite problems to get good global convergence results. First algorithmic results on multigrid methods based on the saddle point problem approach for mortar finite elements were presented in [33]. Special multigrid methods for mortar finite elements are analyzed in [11, 12, 13, 23]. The characteristic idea in case of the saddle point formulation is to guarantee within each smoothing step that the iterate is contained in a subspace where the saddle point problem is positive definite. Then, the analysis of the positive definite case carries over. In this case, no approximation property for the Lagrange multiplier is necessary. This approach requires the exact solution of a modified Schur complement system within each smoothing step which might be too expensive in the multilevel context. In our approach, we solve the Schur complement system only iteratively, and we apply the multigrid method to far more general situations. The purpose of this paper is the construction of a flexible framework for multigrid algorithms which can be applied to many different mortar situations, e. g. to the coupling of different discretizations or to linear elasticity. Moreover, numerical experiments show that the method is applicable for more general situations than covered by the analysis. The paper is organized as follows: In Section 2, a mortar setting is given, and we formulate the restrictive assumptions required for a rigorous multigrid analysis. Section 3 is devoted to the analysis of a multigrid W-cycle for the saddle point problem. Following the approach of B RAESS -S ARAZIN [14], we consider in a first step a smoother having the same saddle point structure as the original problem. To avoid the exact solution of a modified Schur complement system in each smoothing step, we use a modification proposed by Z ULEHNER [36]. Finally in Section 4, we discuss numerical experiments which demonstrate the wide range of applications where the multigrid algorithm solves the arising equations within optimal complexity. 2. Mortar finite elements Let be a bounded polygonal domain in IR2 . We consider a geometrical conforming nonoverlapping decomposition into K subdomains k such that

=

K [ k=1

k :

Let kl := k \ l , 1 k; l K be the possibly empty intersection of the boundaries of two subdomains, and define the skeleton as the union of all non-empty interfaces kl

S :=

[

k;l=1;:::;K; kl 6=;

kl :

Then, the basic idea of the mortar method is the following: The skeleton is decomposed into non-overlapping disjoint open sets, the so-called mortars. Each mortar is associated with an entire open edge of one subdomain, the corresponding edge of the adjacent subdomain is called non-mortar side. Associated with each non-mortar side is a discrete Lagrange multiplier space. Very often, it is inherited from the discretization on the corresponding subdomain, and is a trace space of codimension 2. However, this is not necessary and we refer to [30, 34] for different alternatives. By means of this Lagrange multiplier space, suitable weak continuity constraints are imposed on the global discretization. In general, they can be stated as a L2 -orthogonality between the jump across the interfaces and the Lagrange multiplier space.

MULTIGRID METHODS FOR MORTAR FINITE ELEMENTS

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We consider an elliptic, second order boundary problem: Find u 2 V such that

a(u; v ) = (f; v )0; ; v 2 V; where a(; ) is a symmetric bilinear form on V V , H01 ( k )m V H 1 ( k )m , and f 2 L2 ( )m . In particular, we consider two different types of examples for a(; ). The first one is a standard diffusion problem (m = 1) with locally constant coefficients k > 0 Z K X a(u; v ) := k ru rv dx: (2) (1)

k=1

k

In the second one, the bilinear form describes a linear elasticity problem in 2D, (m = 2),

a(u; v ) :=

(3)

K Z X k=1

(2k "(u) : "(v ) + k div(u) div(v )) dx;

k

for given Lamé constants k ; k > 0. Each subdomain k , 1 k K is associated with a regular family of triangulations Tk;h , h < h0 and a finite element space Xk;h L2 ( k )m associated with the triangulation. For the discretization, two different global spaces are considered. The first one is the product space where no continuity condition at the skeleton is imposed

Xh := fu 2 L2 ( )m j uj k 2 Xk;h for k = 1; :::; K g: It is well known that Xh is no suitable discrete space for the approximation of (1). The bilinear forms given by (2) and (3) are extended to the finite element space Xh in a natural way. The coercivity error would be not bounded by the mesh size h. On the other hand, Xh \ H 1 ( )m

does in general not provide an approximation property. To define an appropriate global discrete space, we have to impose weak continuity conditions at the interfaces of the decomposition. By means of a Lagrange multiplier space Mh L2 (S )m living on the skeleton, we get the global space V := fvh 2 Xh j b(vh ; h ) = 0 for all h 2 Mh g; R h where b(vh ; h ) := S [vh ℄J h ds. For the explicit definition of Mh , we need the disjunct decomposition of the interface into nonmortars Æl , 1 l L, where Æl is an entire edge of one subdomain N (l) . Then, Mh will be given as the product space on the non-mortars

Mh :=

L Y

Ml;h;

l=1 2 m where Ml;h L (Æl ) stands for the local discrete space on the non-mortar Æl which inherits its 1D triangulation from the triangulation on N (l) . To obtain optimal a priori estimates for

the discretization error, Mh has to be chosen properly according to Xh . If Mh is too “large”, we would loose the approximation property of Vh and on the other hand, if Mh is too “small”, the consistency error would be unbounded or of smaller order as the best approximation error of Vh . In this paper, we consider two cases: The first one realizes the coupling of lowest order conforming elements on nonmatching triangulations. The Lagrange multiplier space Ml;h is a modified trace space of codimension two. We refer to [7, 9, 10] for the precise definition and for the modification at the endpoints of Æl . Secondly, we consider the coupling between conforming and nonconforming P1 and/or Q1 elements [33]. Here, we work with piecewise constant Lagrange multipliers living on the side where the nonconforming discretization is used. Again the Lagrange multiplier space satisfies an appropriate approximation property and a discrete

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Babu˘ska-Brezzi condition can be established. For other modifications of the Lagrange multiplier, we refer to [30, 34]. Now, a discrete approximation of (1) can be given by the following nonconforming variational problem on Vh : Find uh 2 Vh such that (4)

vh 2 Vh :

a(uh ; vh ) = (f; vh )0; ;

Working with (4) has the drawback that in the standard mortar finite element formulation, basis functions with local support cannot be defined. Thus very often, the implementation is based on the product space Xh and a local nodal basis. In our work, we use the equivalent saddle point problem as starting point for the multigrid approach: Find (uh ; h ) 2 Xh Mh such that (5)

vh 2 Xh h 2 Mh :

a(uh; vh ) + b(vh ; h ) = (f; vh )0;

b(uh ; h) = 0;

The multigrid analysis is based on the following two assumptions. We assume that the global triangulation on is quasi-uniform and that the triangulation on level l is obtained by uniform refinement from Tl 1 with hl 1 = 2hl . Let us further assume that we have the following approximation property: For each right hand side (f; ) 2 L2 ( )m L2 (S )m , we denote by (uh; h ) 2 Xh Mh the solution of

a(uh ; vh ) + b(vh ; h) = (f; vh )0;

vh 2 Xh b(uh ; h) = ( ; h)S ;h h 2 Mh and the by (u2h ; 2h ) 2 X2h M2h the solution of a(u2h ; v2h ) + b(v2h ; 2h ) = (f; v2h )0;

v2h 2 X2h b(u2h ; 2h ) = ( ; 2h )S ;h 2h 2 M2h : Then, there is a constant independent of h such that (6) jj(uh; h) (u2h; 2h)jj0;h C h2 jj(f; )jj0;h: Here, jj jj0;h denotes a mesh dependent product norm on Xh Mh jj(v; )jj20;h := jjvjj20; + h3 jjjj20;S :

Note that assumptions of this form can be satisfied for other types of symmetric saddle point problems, e.g. see B RENNER [18]. In case that the bilinear form a(; ) is given by (2), this type of approximation result can be found in [35]. 3. Multigrid methods for saddle point problems We analyze multigrid methods for saddle point problems in the standard setting for W-cycle convergence, and we refer to H ACKBUSCH [24] for the definition of the algorithms and for basic properties. The W-cycle convergence is derived from two-grid convergence, the stability of the smoothing operation and the interpolation. The two essential tools in the two-grid analysis are smoothing and approximation properties. Smoothing property. The variational problem (5) has the following algebraic formulation

Kz :=

A BT B 0

x y

=

f 0

=: b;

where K is a symmetric indefinite matrix. Here, the positive semi-definite matrix A has a block structure associated with the decomposition of into subdomains. The matrix B has full rank


5

and BB T is spectrally equivalent a mass matrix on the interfaces (scaled depending on the mesh size). Then, the j th smoothing step is given by

z j +1 := z j + K~ 1 (b Kz j ); j 0; ~ is a symmetric operator. Assuming that A is positive definite, then K can be decomwhere K posed in 1 T A 0 A 0 A B K = B Id 0 Id : 0 BA 1 B T

(7)

~ . In This decomposition is the starting point for the construction of the smoothing operator K case of the Stokes problem, a smoother having the same algebraic structure as K is investigated in [14]. This approach is applied to the mortar situation in [11, 12, 33]. It requires the exact solution of a modified Schur complement system within each smoothing step

~ 1 A A~ B T = A~ 0 B 0 B Id 0

0 ~ BA 1BT

A~ B T : 0 Id

This observation motivates the introduction of a modified smoother involving an inner iteration scheme. In particular, the modified Schur complement S := B A~ 1 B T will be replaced by a symmetric matrix S~, and we define

~ A

~ 0 A~ 1 0 A~ B T A = B Id 0 Id : B S S~ 0 S~ Here, we assume that S~ 1 can be easily applied and that S~ approximates S well enough. The

K~ :=

BT

following lemma which can be found in [36] guarantees the smoothing property of the operator K~ under some conditions on S~. L EMMA 1. Assume that 0 A A~ and that

(8) is satisfied. Then, for

0 < S~ S (1 + )S~

< 1=3, the smoothing property jjK (1 K~ 1 K ) jj ( ) jjK~ holds with ( ) ! 0 for ! 1.

K jj

We refer to [36] for the proof, and note that the assumption on can be weakened if a damping strategy is used. To construct an adequate S~ satisfying (8) with < 1=3, we use an inner iteration scheme. The starting point for the definition of S~ is the choice of a symmetric positive definite S^ satisfying S < 2S^. Then, there exists for any given " > 0 an integer k (") such that the spectral radius satisfies ((Id S^ 1 S )k ) < " for k k ("). Let S~(k; ) be given by

Id S~(k; ) 1 S = (Id

Then, a straightforward computation shows that

S^ 1 S )k :

1 S~(k; ) = S 1=2 Id (Id S 1=2 S^ 1 S 1=2 )k S 1=2 is symmetric and positive definite for each integer k 0 and > 0. Now, the idea is to find an integer k and an such that S~(k; ) satisfies (8) with < 1=3. It is easy to see that 1 " ~ 1+" ~ S (k; ) S S (k; ) for k k ("). Setting := (1 ") yields 2" ~ 1+" ~ S (k; ) = (1 + ) S (k; ): S~(k; ) S 1 " 1 "

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The last inequality shows that the assumptions of Lemma 1 are satisfied for 7" < 1. We define ~ = t can S~ := S~(k("); 1 ") and set " = 0:1 in the numerical experiments. The solution of Sy be formally obtained by

y=

1

1 "

S

1=2

Id (Id S 1=2 S^ 1 S 1=2 )k(") S

1=2 t:

In the multigrid algorithm, each smoothing step (7) requires the solution of

f K~ = g ; which is done in the following way (depending on A~, S^ and "): s := A~ 1 f; t0 := Bs g; y0 := 0 for i = 1; 2; 3; ::: do (9) yi := yi 1 + S^ 1 ti 1 ; ti := t0 B A~ 1 B T yi until jjti jj " jjt0 jj; 1 y y := 1 " i x := s A~ 1 B T y:

x y

The linear iteration (9) in the implementation is accelerated by a cg-method. If the condition number of the modified Schur complement B A~ 1 B T is level independent, it is easy to construct a suitable preconditioner S~. The multigrid analysis. We consider a sequence of discretizations Xl Ml , l = 0; :::; lmax on quasi-uniform meshes corresponding to a mesh size parameter hl such that hl 1 = 2hl . For l simplicity of the analysis, we assume that a scaled nodal basis fi gni=1 of Xl , nl := dimXl and ml f igi=1 of Ml , ml := dimMl with

jjijj0; C; 1 i nl ; h3l =2 jj

jj0;S C; 1 i ml is given. Using such a basis has the advantage that the mesh dependent norm jj jj0;hl of an element in Xl Ml is equivalent to the Euclidean norm of its vector representation. For convenience, we use the same symbol for a finite element function in Xl Ml and its vector n +m i

R

l l . Restriction and prolongation operators are given in terms of the grid representation in nl 1 +ml 1 nl +ml transfer matrix Il 2 IR . In case of conforming finite elements, Il is defined by the natural embedding of Xl 1 Ml 1 in Xl Ml . If nonconforming Crouzeix-Raviart elements are used, the sequence of spaces Xl is non-nested. In this case, we use a L2 -stable operator from Xl 1 Ml 1 on Xl Ml to define Il [15, 17].

L EMMA 2. Under the assumptions of Lemma 1 and

hl 2 A~l Chl 2 ; the two-grid algorithm has a convergence rate < 1 independent of the mesh size provided that the number of smoothing steps is large enough. P ROOF. The two-grid iteration matrix in case of pre-smoothing steps is given by

K~ l 1 Kl ) :

(Id Il Kl 11 IlT Kl ) (Id

Then, the spectral radius of the two-grid iteration is bounded by

:= [(Id Il Kl 11 IlT Kl ) (Id K~ l Kl ) ℄ jjKl

1

Il Kl 11 IlT jj jjKl (Id K~ l 1 Kl ) jj:


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In case of nested spaces Xl 1 Ml 1 Xl Ml , the approximation property (6) in the analytic formulation yields immediately the approximation property

jjKl 1 Il Kl 11IlT jj Ch2l

(10)

in the algebraic setting. To get the algebraic approximation property in case of non-nested spaces, we refer to [17]. The assumption on A~l and the choice of the basis functions guarantee that the modified Schur complement Bl A~l 1 BlT is spectrally equivalent to hl 2 Id. Thus, the norm of the positive semi~ l Kl is bounded by a constant times hl 2 . Using this obserdefinite block diagonal matrix K vation and applying Lemma 1, we obtain the smoothing property (11)

jjKl (Id K~ l 1Kl ) jj C( )hl 2:

Combining (10) and (11) yields < 1 for sufficiently large.

R EMARK 3. In Lemma 2, it is assumed that the condition number of A~ is bounded independently of the mesh size hl . This is satisfied for Jacobi type smoothers, but not for ILU-type smoothers. In this case, a convergence result independent of the mesh size can be still obtained. However, a different mesh dependent norm for the Lagrange multiplier involving the Schur complement B A~l 1 B T has to be used. Following the approach of H ACKBUSCH [25, Ch. 10.6] the W-cycle convergence can be established from the two-grid convergence. In addition to the approximation and smoothing property, stability of the smoothing operation independent of the number of smoothing steps and refinement levels is required. The iteration matrix in case of smoothing steps is

(Id

K~ l 1 Kl ) = K~ l 1 (K~ l = K~ l 1 (K~ l

1 1 ~ ~ Kl ) Kl (Kl Kl ) 1 =2 Kl ) (K~ l Kl )1=2 K~ l 1 (K~ l

Under the assumptions of Lemma 1, the spectral radius of bounded by one. Eventually, we find (12)

(K~ l

K )1=2

l

1

(K~ l

Kl )1=2 K~ l 1 (K~ l

Kl )1=2 : Kl )1=2 is

jj(Id K~ l 1Kl ) jj jjK~ l 1jj jjK~ l Kl jj Chl 2 h2l C:

In the last inequality, we have used that A~l is spectrally equivalent to hl 2 Id. Together with the smoothing and approximation properties and the stability of Il , the stability of the smoothing algorithm (12) gives a W-cycle convergence rate independent of the refinement level, provided that is large enough. 4. Numerical experiments The analysis in the previous sections gives only asymptotic results for the W-cycle. However, no explicit bounds for the constants are given. In addition, the assumptions on the regularity and the smoothness of the data and the global triangulation are quite restrictive, and for many interesting applications they do not hold. Thus, our analysis is supplemented by numerical experiments. Here, the performance of the method is demonstrated for different types of problem settings. The mortar finite elements are included in the software toolbox UG [5, 6] and its finite element library [31]. Within this code, it is easy to test various solvers and smoothers, and for the development of the method we used the experimental environment extensively to learn about the features of different approaches. We want to point out that the interaction of analysis and experiment plays a crucial role in the understanding of the algorithm.

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A model problem with highly discontinuous coefficients. In the first experiment we consider problem (1) with the bilinear form given by (2) on := ( 1; 1)2 . The domain is decomposed into four squares according to Figure 1. The coefficients are highly discontinuous and have large jumps at the interfaces.

106

1

1

106

Figure 1: Decomposition of and coefficients (left), initial triangulation (right). In each subdomain, we use bilinear conforming elements on a uniform mesh (see Figure 1). The right hand side is given by f := 1 and homogeneous Dirichlet boundary conditions are used. This problem is a classical test problem in multilevel theory (e. g. D ENDY [20]) which is investigated by many authors. It is well-known that for some constellations of diffusion coefficients the solution has a strong singularity at the crosspoint. We recall that no continuity of the mortar solution at the crosspoint is enforced. Here, numerical results for the V- and W-cycle using a smoother given by (9) are presented. The parameter " for the inner iteration is chosen to be " := 0:1. We consider two different possibilities for A~. The first one is a Jacobi method with damping factor 0:7, i. e. A~ := 0:7diagA. In this case, we use a W-cycle with two pre- and post-smoothing steps. In a second test, we take a V-cycle with one preand post-smoothing step. Here, a symmetric Gauß-Seidel smoother is applied. We present the asymptotic convergence rates for the W(2,2)-cycle and the V(1,1)-cycle, see Table 1. The number of required inner iteration steps is bounded independently of the refinement level by 4 for the Jacobi smoother and by 8 for the Gauß-Seidel smoother. This reflects the fact that the condition number of the approximated Schur complement is worse for the Gauß-Seidel smoother. However for both types of smoothers, the condition number of the modified Schur complement is bounded independent of the refinement level. level

number of elements

4 5 6 7 8

1024 4096 16384 65536 262144

W(2,2)-cycle damped Jacobi smoother 0.084 0.128 0.137 0.144 0.146

V(1,1)-cycle symmetric Gauß-Seidel smoother 0.080 0.091 0.095 0.098 0.102

Table 1: Asymptotic convergence rates for highly discontinuous coefficients The asymptotic convergence rate is measured by the reduction factor of the defect within one multigrid cycle, which is stable after approximately 30 iteration steps. This approximates the spectral radius of the iteration matrix. We recall that in case of a nested iteration, the reduction factor is much better at the beginning. Only a few iterations are necessary to obtain an iteration error of the same order as the discretization error. Further experiments show that the multigrid method for the V(1,1)-cycle with Jacobi smoother converges. However, no bound for the convergence rate independent of the number of levels can be observed. In contrast to standard conforming discretization methods [16], the choice of the coefficients k does not disturb the level independent convergence rate. In practical applications, we use the V-cycle as a preconditioner for Krylov methods. If " in (9) is small, the iterates are almost in Vh , and we can use a cg-method for accelerating the multigrid


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method. Here, we use the bi-cg-stab method. Since the resulting schemes are nonlinear and the convergence depends on the condition number of the iteration matrix, we will not consider the asymptotic convergence rate for the Krylov methods. Instead, we present in the following the average convergence rates for a defect reduction by a factor of 10 10 . A model for drug permeation through the skin. In the next example, a geometrical more complex situation is considered. The domain is decomposed into subdomains with bad aspect ratios and we obtain eight crosspoints. We use strongly different mesh sizes on the different subdomains and have highly discontinuous coefficients. This example is motivated by a model for drug permeation through the skin (cf. [26]). Here, large cells (corneocytes) are separated by a very thin lipid layer (with a ratio 1:300), see Figure 2 for the geometric model.

Figure 2: Skin geometry Using nonmatching grids, it is possible to avoid elements with a large aspect ratio for this problem. Furthermore, the robustness of the mortar method for large skips in the diffusion is an important feature required for this application. Here, we consider a linear problem corresponding to a single time step of the given parabolic problem. For this test, we prescribe a drug concentration u on the left and right boundary, and we consider nontrivial Neumann boundary condition on the lower boundary. The permeability in the cells and in the lipid layers differ by a factor 10 6 , so that the medicine is transported mainly in the channels. Here, we obtain stable convergence rates for the cg-method preconditioned with a V(2,2)-cycle, symmetric Gauß-Seidel smoother A~, and " = 0:1 and three inner iterations in (9) for the approximated Schur complement (Table 2). The performance is much better, if the mortar side is on the side with the smaller diffusion constant. However, in this case the discretization error is worse.

level elements conv. rate 1 5128 0.05 2 20512 0.07 3 82048 0.07 4 328192 0.08 Table 2: Drug permeation through stratum corneum – zoom of the grid (left), average convergence rate (middle), and flow aru (right). A rotating geometry. A typical time-dependent problem for nonmatching grids arises if parts of the geometry rotate, cf. Figure 3. This can be modeled easily within the mortar finite element framework on curved interfaces. Here, test this situation for the Laplace problem, Dirichlet boundary conditions at the outer boundary, and Neumann boundary conditions at the inner boundary. In particular, A is singular in this application. For the multigrid solver, this problem is easy to solve. We use a preconditioned bi-cg-stab method. As preconditioner a V(1,1)cycle with an ILU-smoother is applied. On all refinement levels, an error reduction of 10 10 is obtained within 3 steps. On the highest level, 327680 elements are used.

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Figure 3: The gradient of the solution on a geometry which rotates in time. The coupling of conforming and nonconforming finite elements. Coupled problems with different underlying physical models are a strong motivation for the development of mortar techniques. As a first step in this direction we couple different discretizations. Here, we present an example for problem (1) where the bilinear form is given by (2). The first subdomain represents a polygonal channel. It is discretized by Crouzeix-Raviart elements, and a finer triangulation is used as on the rest of the domain. The other subdomains are discretized with linear conforming elements. We consider a simple flow problem in heterogeneous media, modeled with Darcy’s law for a pressure potential u and the flow aru. We choose the diffusion parameters a1 := 1, a2 = 0:001. Inflow and outflow boundary conditions in the channel region are used and homogeneous boundary conditions elsewhere. We recall that piecewise constant Lagrange multipliers are used. Here, it is not advisable to use a Jacobi smoother. Tests with this type of smoother show that the damping factor has to be decreased and the number of smoothing steps has to be increased to obtain a robust method. Stable convergence rates are obtained for a preconditioned cg-method. As preconditioner serves a V(2,2)-cycle with ILU smoother and up to three inner iterations. Table 3 shows the performance of the preconditioned cg-method.

level elements conv. rate 1 4028 0.05 2 16112 0.08 3 64448 0.10 4 257792 0.11

Table 3: Initial mesh for the channel domain (left), multigrid convergence for the coupling P1 /CR (center), and resulting velocity v = aru (right). The coupling of elastic materials with different material parameters. Finally, we apply the mortar technique to linear elasticity problem. The bilinear form is given by (3). We consider a composite material constructed of large bricks of hard material, joined with a thin layer of a softer material (see Fig. 4). Here, the multigrid method is much more sensible with respect to large aspect ratios of the mesh size and the material parameters. Our experiments show that the smoothing property still


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Figure 4: Deformation of the composite (left) and zoom to the thin layer (right). The deformation is scaled by a factor of 1000. holds, whereas the approximation property deteriorates in some cases. Nevertheless, we could obtain results for this problem, but the multigrid performance is very bad compared with the previous examples. The parameters are given by 1 = 110743, 1 = 80193, 2 = 135671 and 2 = 67837. The average convergence rate of the preconditioned bi-cg-stab is 0:5. A V(3,3)cycle with symmetric Gauß-Seidel smoother serves as preconditioner. Up to 360448 elements are used. For better results, in particular for nearly incompressible materials, the mortar method must be combined with a robust method with respect to the Poisson ratio [18, 29, 32]. Conclusion. Our numerical experiments indicate that V-cycle convergence can be obtained for Gauß-Seidel smoother as part of the saddle point smoother with local regularity, local quasiuniform meshes, and curved interfaces. Here, further analysis is required to enhance the Wcycle theory for full regularity and quasi-uniform meshes. To obtain efficient iterative schemes it is necessary to have a modified Schur complement system B A~ 1 B T with a moderate condition number or to have at least a good preconditioner S^ for B A~ 1 B T . References [1] Y. ACHDOU AND Y. K UZNETSOV, Substructuring preconditioners for finite element methods on nonmatching grids, East-West J. Numer. Math., 3 (1995), pp. 1–28. [2] Y. ACHDOU , Y. K UZNETSOV, AND O. P IRONNEAU, Substructuring preconditioners for the Q1 mortar element method, Numer. Math., 71 (1995), pp. 419–449. [3] Y. ACHDOU , Y. M ADAY, AND O. W IDLUND, Iterative substructuring preconditioners for mortar element methods in two dimensions, SIAM J. Num. Anal., 36 (1999), pp. 551–580. [4] T. A RBOGAST, L. C. C OWSAR , M. F. W HEELER , AND I. YOTOV, Mixed finite element methods on nonmatching multiblock grids, Tech. Rep. TICAM 96-50, Texas Inst. Comp. Appl. Math., University of Texas at Austin, 1996. Submitted to SIAM J. Num. Anal. [5] P. BASTIAN, Parallele adaptive Mehrgitterverfahren, Teubner Skripten zur Numerik, Teubner-Verlag, 1996. [6] P. BASTIAN , K. BIRKEN, K. JOHANNSEN , S. L ANG , N. N EUSS , H. R ENTZ -R EICHERT, AND C. W IENERS, UG – a flexible software toolbox for solving partial differential equations, Computing and Visualization in Science, 1 (1997), pp. 27–40. [7] F. B EN B ELGACEM, The mortar finite element method with Lagrange multipliers, tech. rep., Laboratoire d’Analyse Numérique, Univ. Pierre et Marie Curie, Paris, 1995. To appear in Numer. Math. [8] F. B EN B ELGACEM AND Y. M ADAY, The mortar element method for three dimensional finite elements, M 2 AN , 31 (1997), pp. 289–302. [9] C. B ERNARDI , Y. M ADAY, AND A. PATERA, Domain decomposition by the mortar element method, in In: Asymptotic and numerical methods for partial differential equations with critical parameters, H. K. et al., ed., Reidel, Dordrecht, 1993, pp. 269–286. , A new nonconforming approach to domain decomposition: the mortar element method, in In: Nonlin[10] ear partial differential equations and their applications, H. B. et al., ed., Paris, 1994, pp. 13–51. [11] D. B RAESS AND W. DAHMEN, Stability estimates of the mortar finite element method for 3-dimensional problems, (1998). to appear in East-West J. Numer. Math.

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