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1 Universität Stuttgart Multilevel Additive Schwarz Preconditioner For Nonconforming Mortar Finite Element Methods Masymilian Dryja, Andreas Gantner, Olof B. Widlund, Barbara I. Wohlmuth Berichte aus dem Institut für Angewandte Analysis und Numerische Simulation Preprint 2003/002

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3 Universität Stuttgart Multilevel Additive Schwarz Preconditioner For Nonconforming Mortar Finite Element Methods Masymilian Dryja, Andreas Gantner, Olof B. Widlund, Barbara I. Wohlmuth Berichte aus dem Institut für Angewandte Analysis und Numerische Simulation Preprint 2003/002

4 Institut für Angewandte Analysis und Numerische Simulation IANS) Faultät Mathemati und Physi Fachbereich Mathemati Pfaffenwaldring 57 D Stuttgart WWW: ISSN c Alle Rechte vorbehalten. Nachdruc nur mit Genehmigung des Autors. IANS-Logo: Andreas Klime. L A T E X-Style: Winfried Geis, Thomas Merle.

5 MULTILEVEL ADDITIVE SCHWARZ PRECONDITIONER FOR NONCONFORMING MORTAR FINITE ELEMENT METHODS MAKSYMILIAN DRYJA, ANDREAS GANTNER, OLOF B. WIDLUND, AND BARBARA I. WOHLMUTH Abstract. Mortar elements form a family of special non-overlapping domain decomposition methods which allows the coupling of different triangulations across subdomain boundaries. We discuss and analyze a multilevel preconditioner for mortar finite elements on nonmatching triangulations. The analysis is carried out within the abstract framewor of additive Schwarz methods. Numerical results show a performance of our preconditioner as predicted by the theory. Our condition number estimate depends quadratically on the number of refinement levels. Key words. domain decomposition, elliptic mortar finite element method, non-matching triangulations, preconditioned conjugate gradients, additive Schwarz methods. 1. Introduction. Domain decomposition techniques for the numerical solution of partial differential equations have been analyzed extensively and used successfully. Large problems are decomposed into a set of smaller ones by breaing up the given domain into subdomains. Then, with parallel computation in mind, flexible methods are established by using characteristic properties of the given partial differential equation. In this paper, we consider two different aspects of domain decomposition. Within the framewor of mortar finite element methods, we use a discretization scheme based on the coupling of nonmatching triangulations along the interfaces. Then in order to construct an efficient solver for the resulting linear system of equations, we introduce and analyze a special multilevel Schwarz method. We will use the mortar finite elements introduced by Bernardi, et al. in [3, 4]. A characteristic feature of mortar methods is that subdomain meshes may be constructed separately in each subdomain and are, in general, nonmatching along the interfaces. This is in contrast to traditional domain decomposition methods, where a globally conforming triangulation is used. Mortar finite elements therefore provide a more flexible approach than standard conforming formulations. Often, mortar methods are recommended when the splitting into subdomains is somehow prescribed for physical or geometrical reasons. Then, for each subdomain an optimal approximation scheme can be chosen, involving the choice of the triangulation as well as the discretization. The strong condition of pointwise continuity across the interfaces is replaced by a weaer one resulting in a nonconforming approximation scheme. In spite thereof, mortar finite elements provide the same accuracy as standard conforming finite elements. In Section 2, we briefly recall the standard mortar formulation. An introduction of our multilevel preconditioner can be found in Section 3. In Section 4, we establish our main result, which is an upper bound for the condition number of the additive Schwarz operator in terms of the refinement level. In Section 5, we discuss some aspects of the implementation of the method. Finally in Section 6, we present some numerical results illustrating the performance of the method. 2. Mortar spaces. Let the polygonal domain Ω be decomposed into K nonoverlapping polygonal subdomains Ω, 1 K. We assume that the diameter of Ω is of order one, and that the diameter of the subdomains is of order H. For simplicity, we restrict ourselves to the case of geometrically conforming decompositions. For each interface of the decomposition, we introduce a master subdomain Ω m) and a slave subdomain Ω s) with respect to such that Department of Mathematics, Warsaw University, Banacha 2, Warsaw, Poland. dryja@mimuw.edu.pl. This wor was supported in part by the National Science Foundation under Grant NSF- CCR and in part by the Polish Science Foundation under Grant 2P03A Math. Institut, Universität Augsburg, Universitätsstr. 14, Augsburg, Germany. gantner@math.uni-augsburg.de, Courant Institute of Mathematical Sciences, New Yor University, 251 Mercer Street, New Yor, NY 10012, USA. widlund@cs.nyu.edu. This wor was supported in part by the US Department of Energy under Contract DE-FG02-92ER Department of Mathematics, Universität Stuttgart, Pfaffenwaldring 57, Stuttgart, Germany. wohlmuth@mathemati.uni-stuttgart.de. 5

6 = Ω s) Ω m). The choice which subdomain is the master subdomain is arbitrary but should be fixed. We consider the following selfadjoint second order model problem: Find u H0 1 Ω) such that where au, v) = fv), v H0 1 Ω), 2.1) K K au, v) := a u, v) := u, v) L 2 Ω ), =1 =1 fv) := f, v) L 2 Ω). Each subdomain Ω is associated with an initial shape regular triangulation T 0). The meshsize is denoted by h 0) and the triangulations are, in general, nonmatching across the common interfaces. For simplicity we assume that h 0) is on the order of H, the diameter of Ω which itself is supposed to be on the order of H. Using uniform refinement, we obtain a sequence of nested triangulations in each subdomain Ω : T 0) T 1) T L), with the meshsizes at level l given by h l) = 2 l h 0) l). We denote by N the set of nodal points in Ω \ Ω of the triangulation T l), and we set N := N L) and n := N. Consequently, the corresponding finite element spaces X l) of continuous and piecewise linear functions satisfying homogeneous boundary conditions on Ω Ω are nested on each subdomain: X 0) X 1) X L). We can now define the unconstrained product space on level l, l = 0,..., L, by X l) h := K =1 X l), X h := X L) h. One of the main ideas of the mortar finite element method is to replace the pointwise continuity across the interfaces by a weaer one. To specify this interface condition, we introduce some trace spaces: { W l) h ) := } v l) v X s), W h ) := W L) h ), W l) l) h;0 ) := W h ) H1 0 ), W h;0) := W L) h;0 ). We say that a function v X h satisfies the mortar condition if its restrictions v m) and v s) to the master and the slave side, respectively, satisfy v m) η dσ = v s) η dσ, η M h ), 2.2) where M h ) is an appropriate Lagrange multiplier space. The Lagrange multiplier space is associated with the mesh on the interface inherited from the triangulation T L) s) on the slave subdomain. We denote this one dimensional mesh on by Σ and its elements by e. Optimal a priori estimates can be obtained for several different Lagrange multiplier spaces, see, e.g. [9]. Here, we restrict ourselves to the standard Lagrange multiplier space which is a subspace of the trace space W h ): M h ) := { η W h ) η e = const. if the element e Σ touches }. 2.3) 6

7 If we denote by N s) the interior nodes of Σ, a nodal basis of M h ) is provided by {ψ p } p Ns), where ψ p q) = δ pq p, q N s). We introduce the mortar projection Π : L 2 ) W 0;h ) Π vη dσ := vη dσ, η M h ). We note that dim M h ) = dim W 0;h ) and that Π is H 1/2 00 -stable, i.e., Π v 1/2 H 00 ) C v H 1/2 00 ). 2.4) The constrained mortar space V h is defined in terms of the mortar condition 2.2) V h := {v X h Π [v] = 0, for all interfaces }, where the jump across an interface is set to [v] := v m) v s). Now, the discrete problem for 2.1) in V h reads as follows: Find u h V h such that au h, v h ) = fv h ), v h V h. 2.5) The discrete problem has a unique solution, and the rate of convergence is comparable to that of conforming discretizations; see, e.g., [1, 2, 4]. In contrast to the unconstrained product space X h, the mortar finite element space V h cannot be written as the product of spaces locally defined on the Ω. According to 2.2), each basis function φ p of V h is associated with one of the following sets of degrees of freedom: all nodes interior to the subdomains, all nodes interior to the master sides of the interfaces, all subdomain vertices multiple values) except those on Ω. The third set of the nodal points is denoted N V. We note that the nodal basis functions of V h can be obtained in terms of the nodal basis functions of X h by applying the mortar projection. Figure 2.1 shows a basis function associated with a node on the interior of a master side and the resulting basis function in V h. master Z master slave slave 2 Figure 2.1. Construction of a basis function of V h In the following, we will denote functions and operators using italics, e.g., v, A, and their discrete algebraic representations, i.e., vectors and matrices, using boldface, e.g., v, A. Each function v X can be represented uniquely as a vector v := v p)) p N R n in terms of the nodal basis {ϕ p } p N. Then, an element v X h is given by v = v ) K =1 Rn, n := K =1 n. Matrices A R n n are now associated with the bilinear forms a, ) : a v, w) = w T A v v, w X. These A are symmetric and positive semidefinite. The matrix representation of a, ) on X h X h is then given by A =diaga 1, A 2,... A K ). To give a matrix representation of a, ) on V h V h, we 7

8 use interface matrices. We denote by s 0, s 1,... s ns+1 the nodes on the slave side and by m 0, m 1,... m nm+1 the nodes on the master side of the interface. Both sets of nodes are assumed to be ordered lexicographically. For a function v X h, its vector v can now be written, after a permutation, as v = ν s, ν c, ν m, ν) T, with entries given by ν s = vs i )) ns i=1, ν c = vs 0 ), vs ns+1)) T on the slave side, ν m = vm i )) nm+1 i=0 on the master side, and where ν represents the values of v at the nodes that do not lie on the interface. To compute ν s for v V h, we introduce a master matrix ψsi ) M :=, ϕ mj L 2 ) ) ns n m+1 i=1;j=0 R ns nm+2), 2.6) a slave matrix ψsi ) S :=, ϕ sj L 2 ) ) ns n s i=1;j=1 Rns ns, 2.7) and a corner slave matrix ψsi ) C :=, ϕ sj L 2 ) ) ns i=1;j=0,n s+1 Rns 2. The slave matrix S can be computed easily since suppψ si = suppϕ si. We note that in the case of standard Lagrange multipliers the matrix S is tridiagonal, and in the case of dual Lagrange multipliers it is diagonal, see, e.g., [9]. In both cases, the matrix C is almost empty; it has n s 2) zero rows and only two non-zero entries. The computation of the master matrix M is more difficult because the structure of suppψ si suppϕ mj depends on the triangulation. The mortar condition 2.2) can now be written in terms of these interface matrices ν s = S 1 M ν m C ν c ). 2.8) By applying this condition, we are able to eliminate the values ν s from the vector v. Each v V h can now be represented by a shorter vector v V, and the corresponding long vector can be obtained in terms of a global switching matrix Q: v = Qv V. Defining A V := Q T AQ and f V := Q T f, the mortar matrix formulation on the constrained space is given by A V u V = f V. 2.9) 3. A multilevel mortar preconditioner. Additive Schwarz methods provide a new operator equation which can be much better conditioned than the original problem. They are defined in terms of a suitable decomposition of V h and bilinear forms. Before we introduce our multilevel decomposition of V h, we define an extension operator Z : L 2 ) X s) on each interface Here, E l) subdomain, and P l) We note that E l) Z v := l=0 E l) P l) ) P l 1) Π v. : W l) l) h;0 ) X s), is the zero extension operator on level l associated with the slave is the L 2 -projection P l) w, w W l) h;0 : W h;0 ) W l) 1) h;0 ), where we set P := 0. ), is non-zero only in a strip of width hl) 8 s).

9 In a next step, we decompose the boundary Ω \ Ω into two disjoint sets Γ s, Γm, of interfaces such that Ω \ Ω =, Γ s Γ m where Γ s if Ω is the slave subdomain of the interface, and Γ m if Ω is the master subdomain of the interface. We now define our operator Z : X V h in terms of Z and the two interface sets Γ s and Γm v Γ Z s v, on Ω Ω, Z v := Z v Ω, on Ω s), Γ m, 3.1) 0, elsewhere. We note that Z v is, in general, a fine level function even if v is a coarse level function, and that it is supported in Ω and its adjacent subdomains. Moreover, E l), restricted to the interface, is the identity and thus Z v) = Π v. Now, it is easy to verify that Z v V h. We point out that the mortar condition with respect to the finest level holds. Figure 3.1 illustrates the mapping Z. Ω Figure 3.1. Action of Z indicated by the arrows leading from the mortar the dar thic lines) to the slave subdomain. Lemma 3.1 The spaces V l) V h, i.e., := Z X l) V h, 1 K, 0 l L, provide a decomposition of V h = K l=0 =1 Z X l). Proof. It is sufficient to show that Z := K =1 Z restricted to V h is the identity. We restrict ourselves to a subdomain Ω j Zv) Ω = v j Ω Z v j + Ω Z j v = v Ω m) + Z Ω [v] = v j Ω. j Γ s j Γ s j Γ s j Here, we have used the fact that the mortar projection applied to [v], v V h, is zero. The second main ingredient in the construction of additive Schwarz methods, in the nonnested setting, are bilinear forms b l) l) l), ) : X X R, see Section 5.1 in [8]. We use inexact solvers and define for x l), yl) X l) b l) xl), yl) ) := p N l) 9 p)yl) p). 3.2) x l)

10 The corresponding projection-lie operators T l) : V h X l), are then given by b l) T l) v, xl) ) := av, Z x l) ), xl) X l). We now define the operators T l) T l) R NV n R NV NV of T l) of Z. Denoting by R l) l) := Z T V h. The algebraic representation can easily be obtained in terms of the algebraic representation Z : V h V l) l) the prolongation matrix from X T l) = Z R l) Z R l) onto X L), we find ) T AV. 3.3) According to the additive Schwarz framewor, we can now define a preconditioner C V linear system A V u V = f V by for the C V := K =1 l=0 ) T Z R l) Z R l). 3.4) 4. Upper bound for the condition number. In this section, we establish an upper bound for the condition number of C V A V. Before we start to state the estimates, we briefly recall some important results for standard multilevel methods, which we will need in the sequel. Let P l) be the L 2 -projection from X into X l) and use the weighted norm x l) := P l) and let P 1) P l 1) u 2 H 1 Ω ) := u 2 H 1 Ω ) + 1 H 2 u 2 L 2 Ω ). Then, the following well-nown result holds, see, e.g., [7][Theorem 15]: Lemma 4.1 l=0 x l) 2 L 2 Ω ) h l) )2 C x 2 H 1 Ω ). := 0. For x = {x } K =1 X h, we set ) x, 4.1) We get a similar result if we consider functions on the interfaces: Let P l) L 2 -projection from W h;0 ) into W l) 1) ) and let P w l) := P l) Lemma 4.2 P l 1), as before, be the ) h;0 := 0. Then, for w W h;0 ) we set w and get, see [7][Theorem 15]: l=0 w l) 2 L 2 ) h l) s) C w 2 H 1/2 00 ). Now, following the abstract Schwarz framewor, see [8], we show that the spaces V l) provide a stable splitting of V h : Lemma 4.3 There exists a decomposition of v V h with v = K a constant C = 1/c 0 independent of H and L such that K =1 l=0 =1 b l) xl), xl) ) C2 H 2 av, v). 10 L l=0 Z x l), xl) X l) and

11 Let v V h X h be decomposed as K L =1 l=0 vl) Proof., where v := v Ω X and := P l) P l 1) )v X l). In view of Lemma 3.1, the operator Z := K =1 Z restricted to V h is the identity, and thus we get v = K L =1 l=0 Z v l) l). Using the definition of V and a discrete norm equivalence, we find by using Lemma 4.1 v l) K =1 l=0 b l) vl) K, vl) ) = =1 C l=0 K =1 l=0 p N l) v l) v l) p) ) 2 2 L 2 Ω ) h l) )2 C K v 2 H 1 Ω ). The use of the ellipticity of a, ) : V h V h R, and the scaling of the H 1 -norm completes this proof. We note, that for x l) X l), the function Z x l) is nonzero only in Ω and its adjacent subdomains. We apply a standard coloring argument to obtain a strengthened Cauchy-Schwarz inequality, see [10]. Lemma 4.4 There exists a constant C independent of L and H such that ρe) C1+L), where ρe) is the spectral radius of E = ɛ ;l),i;j) ) 1,i K,0 l,j L with av l), vj) i ) ɛ ;l),i;j) av l), vl) ) 1 j) 2 av i, v j) i ) 1 l) 2, v =1 Vl), vj) i V j) i. In a next step, we give a ind of one-sided measure of the approximation properties of the bilinear forms b l), ). Lemma 4.5 There exists a constant c 1 independent of H and L such that Proof. For x l) az x l), Z x l) ) c 11 + L)b l) X l) az x l), Z x l), we find that ) = a x l) + Γ m Γ s xl) Z x l), xl) ),, xl) a s) Z x l), Z x l) xl) X l). Z x l) + Γ s 4.2) ). A standard inverse estimate shows that the first term is bounded in terms of the bilinear forms, ) a b l) x l), xl) ) Cbl) xl), xl) ). Moreover, by a strengthened Cauchy-Schwarz inequality, an inverse inequality and Lemma 4.2, we get for any interface Γ s a Z x l), Z x l) ) C L C C i=0 E i) P i) 1 i=0 h i) Ei) )2 i=0 1 h i) P i) C Π x l) 2 H 1/2 ) ) P i 1) Π x l) 2 H 1 Ω ) P i) ) P i 1) Π x l) 2 L 2 Ω ) ) P i 1) Π x l) 2 L 2 ) 4.3)

12 To obtain an upper bound for the H 1/2 00 -norm of Π x l), we decompose xl) into two parts. With p s and p e denoting the two endpoints of and ϕ s l) and ϕ l) e the nodal basis functions associated with p s and p e at the l-level, we introduce a function χ by Now, x l) can be written on as where χ i H 1/2 00 ), and thus χ := x l) p s)ϕ l) s + x l) p e)ϕ l) e. x l) = χ + χ i, Π x l) H 1/2 00 ) Π χ 1/2 H 00 ) + Π χ i 1/2 H 00 ). 4.4) The H 1/2 00 -stability of Π, 2.4), in combination with an inverse inequality yields Π χ i 2 H 1/2 00 ) C χ i 2 H 1/2 00 ) C p N l) x l) p) ) 2. To give an estimate of the second term in 4.4), we assume that is the segment 0, H ). By the definition of the H 1/2 00 ) norm, Using [6][Lemma 5], we get Π χ 2 = Π H χ 1/2 00 ) 2 H 1/2 ) + + H 0 Π χ 2 dx + x H { ) 2 ) 2 Π χ 2 H 1/2 ) C x l) p s) Π ϕ l) s 2 H 1/2 ) + x l) p e) Π ϕ l) e { ) 2 ) } 2 C x l) p s) + x l) p e). 0 Π χ 2 4.5) H x dx 2 H 1/2 ) Since the last two terms of 4.5) are very similar, we concentrate on the first. We split the integral into two, over 0, h L) ) and hl), H ), respectively. A standard analysis yields )) H Π χ 2 H dx C 1 + log Π h L) x h L) χ 2 L ), and h L) Again using [6][Lemma 5], we obtain 0 Π χ 2 dx C Π χ 2 L x ). { ) 2 ) } 2 Π χ 2 L ) C x l) p s) + x l) p e). } Recalling that h l) = 2 l h 0) and using h 0) H, we find Π x l) H 1/2 00 ) C1 + L) 12 p N l) x l) p) ) 2,

13 and thus a Z x l), Z x l) ) C1 + L)bl) xl), xl) ). Proceeding as before, we get for Γ m a s) Z x l), Z x l) ) C1 + L)bl) xl), xl) ). Finally, substituting the above results into 4.2) gives az x l), Z x l) ) C1 + L)bl) xl), xl) ). Remar 4.1 In the case of two subregions the factor L disappears from the estimate in Lemma 4.5. This follows immediately from the proof since χ = 0 in this case. Relying on the abstract Schwarz framewor, we have shown the following estimate for our preconditioned system: Theorem 4.1 The operator T := K L =1 l=0 T l) satisfies, for any v V h, c 0 H 2 av, v) at v, v) c L) 2 av, v), where the constants c 0 and c 1 are independent of L and H. Remar 4.2 We remar that by introducing a coarse space based on a continuous vertex basis function for each subdomain vertex, we could eliminate the dependence of H in Theorem 4.1. Remar 4.3 It can be proved that in the case of two subregions, the factor 1 + L) 2 disappears in the estimate of Theorem 4.1; see also the numerical results in Section Implementation. Our aim is to give a simple matrix representation of our preconditioner K ) T K L ) ) T C V = Z R l) Z R l) = Z R l) R l) Z T. =1 l=0 In most iterative methods, the preconditioner has to be applied to the residual r V R NV. The vector r := Z T r V is in R n, and we have to compute C r := l=0 =1 R l) R l) l=0 ) T r. But this is a standard mapping since C is of the same type as a BPX-preconditioner for the subdomain Ω, c.f. [8]. We now consider Z in more detail. A matrix representation of the operator Π can easily be given in terms of the mass matrices M, S and C. We introduce the prolongation matrix I l) from W l) L) h;0 onto W h;0 and the prolongation matrices il) from W l) l+1) h;0 onto W h;0 associated with the natural embedding operators and the mass matrices G l) := ϕ l) i ), ϕ l) j L 2 ) ) Nl N l i=1;j=1 R N l N l, 0 l L, 5.1) where N l denotes the number of interior nodes of the interface triangulation on inherited from l). We obtain a matrix representation of P by T l) s) P l) := G l) ) 1 13 I l) ) T G L). 5.2)

14 We are now able to give a matrix representation of the operator Z Here, E l) Z v = l=0 R l) s) El) G l) I l) ) 1 i l 1) ) T G L) S 1 G l 1) ) 1 i l 1) S C M 0) ) T ) is a matrix representation of the operator E l), the trivial extension from to Ω s) at the l-level. We note, that most of the wor in applying the mapping Z can be done in parallel for two adjacent subdomains. To see this, let be an interface and let v m) and v s) be the vectors on the master and slave subdomain, respectively. On the master subdomain there is no change of the vector v m), while on the slave subdomain we have to calculate an update for the interface of the form Z v m) Z v s). This can be done by first calculating see 2.8), and then l=0 R l) s) El) G l) ) νs x := M ν m S C ), ν c ) 1 i l 1) G l 1) ) 1 i l 1) ) T ) I l) ν s ν c ν m ν i. ) T G L) S 1 x. The cost of applying Z is of order O n ). It is well nown [5] that applying the BPX preconditioner C is of order On ). Altogether we get a computational cost of order O N ) for the preconditioner C V. Remar 5.1 We remar that the computational cost can be further reduced if we use dual Lagrange multipliers. Then S is a diagonal matrix. Moreover, we can replace P l) by a quasiprojection operator which gives rise to a diagonal matrix. 6. Numerical results. In this section, we present some numerical test examples illustrating the performance of our mortar multilevel additive Schwarz method. We use standard conforming P 1 -Lagrange finite elements in each subdomain Ω and nonmatching simplicial triangulations. Starting with an initial triangulation, we decompose each element into four subelements in each refinement step. To solve the algebraic system A V u V = f V, we apply a preconditioned conjugate gradient method where the preconditioner is defined in Section 3. If the decomposition into subdomains is fixed, Theorem 4.1 yields a condition number of C V A V which is bounded by CL 2, and thus the number of iterations which are necessary to reach a given tolerance is OL). We start with the following test example: u = f, in Ω := 1, 1) 2, u = 0 and the right-hand side f is chosen to be on Ω, f = 2π 2 sinπx) sinπy). Then, the wea solution is given by ux, y) = sinπx) sinπy). In a first step Example 1), we decompose the unit square into two subdomains. The initial nonmatching triangulation and the decomposition into two subdomains are shown in the left picture of Figure

15 a) Example 1 b) Example 2 Figure 6.1. Decomposition into subdomains and initial triangulation The number of iteration steps to obtain the given tolerance 10 8 is given in the left part of Figure 6.2. Asymptotically, we obtain level independent convergence rates for our preconditioner. The numerical results for this test setting are better than predicted by Theorem 4.1. This fact is due to our special decomposition; we have no crosspoints, c.f. Remar 4.3. The condition number estimates for this setting are in the following table: Level No Condition No Number of iteraions Level a) Example 1 Number of iterations Level b) Example 2 Figure 6.2. Number of iterations For our second example, we use the same boundary value problem but decompose the unit square into nine subdomains. Here, we have four interior crosspoints, see the right part of Figure 6.1. The numerical results are shown in the right picture of Figure 6.2. For this example, we do not obtain level independent convergence rates. The number of iteration steps increase with the number of levels, and a linear growth can be observed. Using a continuous coarse space in our second example improves considerably the condition numbers. Without a coarse space, we observe a quadratic growth of the condition of the precondi- 15

16 tioned system. The numerical results confirm our theoretical results. Adding a coarse space yields qualitative better numerical results. Level No Condition No without coarse space Condition No with coarse space REFERENCES [1] F. Ben Belgacem, The mortar finite element method with Lagrange multipliers, Numer. Math, ), pp [2] C. Bernardi and Y. Maday, Raffinement de maillage en elements finis par la methode des joints, C. R. Acad. Sci., 1995). This paper appeared also as a preprint, Laboratoire d Analyse Numérique, Univ. Pierre et Marie Curie, Paris, R94029, including more details. [3] C. Bernardi, Y. Maday, and A. Patera, Domain decomposition by the mortar element method., in Kaper, Hans G. ed.) et al., Asymptotic and numerical methods for partial differential equations with critical parameters, Reidel, Dordrecht, 1993, pp [4] C. Bernardi, Y. Maday, and T. Patera, A new nonconforming approach to domain decomposition: The mortar element method, in Nonlinear Partial Differential Equations and Their Applications, H. Brezis and J. L. Lions, eds., Pitman, Paris, 1994, pp [5] J. H. Bramble, J. Pascia, and J. Xu, Parallel multilevel preconditioners, Math. Comp., ), pp [6] M. Casarin and O. Widlund, A hierarchical preconditioner for the mortar finite element method, ETNA, Electron. Trans. Numer. Anal., ), pp [7] P. Oswald, Multilevel Finite Element Approximation: Theory and Applications, Teubner Sripten zur Numeri, B. G. Teubner, Stuttgart, [8] B. F. Smith, P. E. Bjørstad, and W. D. Gropp, Domain Decomposition: Parallel Multilevel Methods for Elliptic Partial Differential Equations, Cambridge University Press, [9] B. I. Wohlmuth, Discretization Methods and Iterative Solvers Based on Domain Decomposition, Springer, Berlin, [10] X. Zhang, Multilevel Schwarz methods, Numer. Math., ), pp

17 Masymilian Dryja Department of Mathematics, Warsaw University, Banacha 2, Warsaw, Poland. Andreas Gantner Math. Institut, Universität Augsburg, Universitätsstr. 14, Augsburg, Germany. Olof B. Widlund Courant Institute of Mathematical Sciences, New Yor University, 251 Mercer Street, New Yor, NY 10012, USA. Barbara I. Wohlmuth Department of Mathematics, Universität Stuttgart, Pfaffenwaldring 57, Stuttgart, Germany. 17

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19 Erschienene Preprints ab Nummer 2003/001 Komplette Liste: /001 Lamichhane, B. P., Wohlmuth, B. I.: Mortar Finite Elements for Interface Problems 2003/002 Dryja, M., Gantner, A., Widlund, O. B., Wohlmuth, B. I.: Multilevel Additive Schwarz Preconditioner For Nonconforming Mortar Finite Element Methods.