The mortar element method for quasilinear elliptic boundary value problems

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1 The mortar element method for quasilinear elliptic boundary value problems Leszek Marcinkowski 1 Abstract We consider a discretization of quasilinear elliptic boundary value problems by the mortar version of finite element method. We show that the error estimate is of the same order as in the standard conforming finite element method. We propose a method of solving the discrete problem and prove its convergence. The method combines the Schwarz preconditioning technique with the Richardson-Newton method. Keywords. Nonlinear elliptic problem, mortar finite element, additive Schwarz method AMS(MOS) subject classifications: 65N30,65N15,65N55 1. INTRODUCTION Recently variational conforming and nonconforming decomposition methods are analyzed and used to approximate differential equations. A large problem is split into some smaller ones that can, for example, be solved independently. It was natural to consider a method that use locally in each subdomain an independent discretization adapted to the local properties of the solution. One of the invented methods was the mortar element method which ensures a good transmission of information between adjacent subdomains. This transmission (in some sense) is optimal. We refer for a general presentation of the mortar method for linear problems to [7], [5] and [6] and for a presentation of the matching constraints in terms of Lagrange multipliers to [4]. Recently, there is a development of parallel algorithms devoted to solve systems of linear equations arising from the mortar version of finite element discretization for linear elliptic problems, see [2, 3, 11, 13]. The goal of this paper is to give an analysis of the error of the mortar version of finite element discretization applied to some nonlinear problems and discuss a domain decomposition method for solving the discrete problem in geometrically conforming case. For our knowledge there is no results devoted to such topics for nonlinear problems. Namely, in this paper we discuss the application of the mortar element method to a second-order nonlinear elliptic boundary value problem with the strongly monotone and Lipschitz continuous operator in a polygonal region Ω with the Lipschitz boundary. We consider first the geometrically conforming case of the mortar element method, i.e. the intersection of the closures of two subdomains can be the empty set, an edge or a vertex, and a more general case: the geometrically nonconforming one, when we do not impose this condition. In the first case, we see that the estimate of the error is of the same order as in the standard conforming piecewise linear finite element discretization provided that the solution of the differential problem is in H 2 (Ω). In the latter one we have to strengthen our regularity assumption. Namely, we shall assume that the solution of the differential problem belongs to the space H 5/2 (Ω) in order to derive the same error estimate. The technique that we use to obtain our estimates is a generalization of that used for linear problems, cf. [7] and [5]. We propose an algorithm for solving discrete problem in case of the geometrically conforming version of the mortar element method, combining Schwarz methods with Newton s one. Namely we construct a preconditioner in terms of Additive Schwarz 1 Institute of Applied Mathematics and Mechanics, Warsaw University, Banacha 2, Warsaw, Poland. Electronic mail address lmarcin@mimuw.edu.pl. This work has been supported in part by Polish Scientific Grant 102/P03/95/09 1

2 Method for a linear problem and apply it to the Richardson iteration for the discretized nonlinear problems considered in this paper. The implementation of our preconditioner can be done in parallel and consists of solving one global linear problem on coarse grid and two kinds of local problems. The global problem is the standard conforming finite element one defined for Poisson s equation. The first kind of local problems consists of ones that are one dimensional and are associated with the vertices of subdomains. The second kind includes the ones that are defined on two substructures with a common edge. We use only nodal basis functions of mortar method corresponding to nodes, such that values of functions at these nodes are degrees of freedom of the method. The outline of the paper is as follows. In Section 2, we formulate the differential problem. Section 3 is devoted to presenting the mortar element method and studying the error estimate in the geometrically conforming case. In Section 4, we present the geometrically nonconforming case of the mortar method. In Section 5, we discuss an application of Richardson-Newton method to the system of nonlinear equations which arise from discretization of the boundary value problem by the geometrically conforming mortar element method. We show that this method is almost optimal convergent. 2. DIFFERENTIAL PROBLEM In this section we formulate the differential problem. We consider the following differential equation x i a i (x, u, u ) + a 0 (x, u, u ) = f(x) in Ω (2.1) with the homogeneous Dirichlet boundary condition, where Ω R 2 is Lipschitz continuous bounded polygonal region. The weak formulation is of the form: Find u H 1 0 (Ω) such that a(u, v) = f(v) v H 1 0 (Ω), (2.2) where a(u, v) = ( a i (x, u, u ) D i v + a 0 (x, u, u) v) dx, f(v) = Ω Ω fv dx (2.3) Here D i = x i, f L 2 (Ω), u = ( x 1 u, x 2 u) T. Let a i (x, p 0, p 1, p 2 ) = a i (x, u, u x 1, u x 2 ) and p = (p 0, p 1, p 2 ). We assume that the functions a i : Ω R 3 R, i = 0, 1, 2 satisfy the following conditions: For some positive constants M, µ 0, a i C 1 (Ω R 3 ), i = 0, 1, 2 (2.4) max{ a i (x, 0, 0, 0), a i x k (x, p), a i p j (x, p) } M, for i, j = 0, 1, 2; k = 1, 2; (2.5) i,j=0 p j a i (x, p)ξ i ξ j µ 0 ξi 2 (2.6) for any ξ = (ξ 0, ξ 1, ξ 2 ) R 3, ξ 0. As a direct consequence of the above assumptions we obtain that for all p, q R 3 there is a positive constant L such that in Ω p, q R 3 a k (x, p) a k (x, q) L( 2 p i q i 2 ) 1/2 k = 0, 1, 2 (2.7) i=0

3 Under the above assumptions it can be proven that the form a(, ) is strongly monotone and Lipschitz continuous and that the problem (2.2) has the unique solution, see [16] or [18]. 3. THE GEOMETRICALLY CONFORMING CASE We now define a discrete space V h L 2 (Ω) that is not a subspace of H 1 0 (Ω). In that sense our method is nonconforming. In this section, we consider a geometrically conforming version of the mortar method. In the next section, we consider the geometrically nonconforming one. We consider a partition of Ω into polygonal subdomains i.e. N Ω = Ω i with Ω i Ω j = if i j that are arrange in such a way that the intersection of Ω k Ω l for k l is either the empty set, an edge or a vertex. We call this partition geometrically conforming. A more general case is considered in the next section. The mortar element method first deals with the union of all edges (interfaces). Γ = N Ω k \ Ω (3.1) and consists of choosing one of the decomposition of Γ, that is made of disjoint open segments (that are edges of subdomains) called mortars, denoted by γ k, 1 k M i.e. Γ = M γ k γ k γ l = if k l. We denote an common, open edge to Ω i and Ω j by γ ij. By γ k(i) we denote an edge of Ω i that is a mortar (master) and by δ k(j) an edge of Ω j that occupies geometrically the same place, called nonmortar (slave). There is no rule in selecting γ k(i) as a mortar. Let us introduce some notation. We define a local space and a global one H 1 D(Ω i ) = {v H 1 (Ω i ) : v Ω Ωi = 0} X = N HD(Ω 1 i ) with a seminorm v X = ( N v i 2 H 1 (Ω i) )1/2 and a norm v X = ( v 2 L 2 (Ω) + v 2 X )1/2. With each Ω k we associate a quasiuniform triangulation T h (Ω k ) made of elements that are triangles, cf. [12]. By h k we denote a parameter of this triangulation, that is the maximum diameter of the triangles. As the triangulation T h (Ω k ) is chosen over each Ω k, so we can give the definition of the finite element functions. Let us assume that we work with the simple generic case of linear finite elements. We first define the finite element functions locally and introduce the space X h (Ω k ) = {v k,h C(Ω k ) : v k,h Ω = 0, t T h (Ω k ), v k,h t P 1 (t)} where P 1 (t) is the set of all linear polynomials over the t triangle in T h (Ω k ). Let W hj (γ ij ) be the restriction of X h (Ω j ) to γ ij. We also introduce the global space X h as X h (Ω) = N X h (Ω k ) 3

4 that can be considered as a subspace of X consisting of functions whose restriction over each Ω k belongs to X h (Ω k ). Note that, since the triangulations on two adjacent subdomains are independent, the interface γ ij = γ k(i) = δ k(j) is provided with two different and independent (1D) triangulations the h i and h j ones and two different spaces W hi (γ k(i) ) and W hj (δ k(j) ). Additionally we define an auxiliary test space M hj (δ k(j) ) being a subspace of the nonmortar space W hj (δ k(j) ) such that its functions are constant on elements which intersect the ends of δ k(j). The dimension of M hj (δ k(j) ) is equal to dimension of W hj (δ k(j) ) minus two. In what follows we express the matching condition that is sufficient to ensure the optimality of the global approximation and define our discrete space V h : V h = {u h X h (Ω) : δ m(j) Γ, ψ M hj (δ m(j) ) (u i,h u j,h ) δm(j) ψ ds = 0 } (3.2) δ m(j) where u i,h, u j,h in the integral are the traces of u h onto δ m(j) = γ ij, the common edge to Ω i and Ω j. The condition (3.2) is called mortar. Our discrete problem is to find u h V h such that where and b(u, v) = ( Ω k b(u h, v h ) = f(v h ) v h V h (3.3) a i (x, u k, u k ) D i v k + a 0 (x, u k, u k ) v k ) dx f(v) = Remark 3.1 If u, v H0 1 (Ω) then a(u, v) = b(u, v). Ω k f v k dx With the help of the Lemma 3.2 and Lemma 3.3, which are proven below in this section, we can prove that under our assumptions (2.4)- (2.6) there exists the unique solution of (3.3), cf [18]. Now we state the main theorem of this section. Theorem 3.1 Let u and u h be the solutions of (2.2) and (3.3), respectively and let u H 2 (Ω). Then we have u u h X c h where c is a constant independent of h k and h = max k {h k }. To prove this theorem we first state and prove some auxiliary lemmas. First we define an auxiliary operator associated with the form (2.3), which is a generalization of normal derivative in case of the linear operators. Definition 3.1 Let γ Ω i be an segment. Then let l i : H 3/2 (Ω i ) L 2 (γ) be defined as (l i u)(x) = a i (x, u(x), u(x))n i on γ (3.4) where n = (n 1, n 2 ) is the normal vector to γ (in Ω i ). It is easy to show that under assumptions (2.4) - (2.5) l i u is well defined in L 2 (γ), cf. [18]. Next lemma states the continuity of l i across γ, the part of the interface. 4

5 Lemma 3.1 Let γ Ω i Ω j be an segment and Ω ij = Ω i Ω j. l i : H 3/2 (Ω i ) L 2 (γ) and l j : H 3/2 (Ω j ) L 2 (γ) be defined by (3.4). Then for u H 3/2 (Ω ij ) we have l j u j = l i u i a.e. on γ i.e. a k (x, u i (x), u i (x))n k = a k (x, u j (x), u j (x))n k a.e. on γ where n = (n 1, n 2 ) is the normal vector to γ and u i, u j are the restrictions of u to Ω i and Ω j respectively. Proof. First we prove that l i is Lipschitz continuous. We may assume that γ is parallel to the x axis and then we have l i u(x) = a 2 (x, u(x), u(x)). We use x 0 u(x) = u(x) to simplify the notation. Let now u, v H 3/2 (Ω ij ), then we conclude that γ a 2 (x, u, u) a 2 (x, v, v) 2 dx L 2 (u v) 2 dx C u v 2 H x 3/2 (Ω ij) i γ i=0 We have used the standard trace theorem and (2.7). As the statement of the lemma obviously is satisfied for u C (Ω ij ), we finish the proof using the density of C (Ω ij ) in H 3/2 (Ω ij ) and Lipschitz continuity of l i and l j. The next corollary can be proven using the same ideas as in [18]. Corollary 3.1 Let γ Ω i be an segment, l i : H 3/2 (Ω i ) L 2 (γ) be defined by (3.4) and u H 2 (Ω i ) then under our assumptions (2.4)- (2.5) we have l i u H 1/2 (γ) with l i u H 1/2 (γ) c ( Ω i + u 2 H 2 (Ω i) )1/2. Now we define the restriction of the form a(, ) to H 1 (Ω i ). Definition 3.2 Let a bilinear form b i (, ) : H 1 (Ω i ) H 1 (Ω i ) R be defined as u, v H 1 (Ω i ) b i (u, v) = ( Ω i a i (x, u, u ) D i v + a 0 (x, u, u) v) dx Remark 3.2 Note that for all u, v X b(u, v) = N b i(u i, v i ). Using the assumptions (2.4) - (2.6) and the results of [6] we can prove that the form b(u h, v h ) is strongly monotone and Lipschitz continuous in V h. We state that in two lemmas Lemma 3.2 The form b(, ) is strongly monotone in V h, i.e. for all u h, v h V h we have where c is a positive constant independent of h k. b(u h, u h v h ) b(v h, u h v h ) c u h v h 2 X Proof. Let u h, v h V h. As V h X we get that restrictions of u h, v h to Ω k denoted by u k,h, v k,h are in H 1 D (Ω k). In [18] was proven that So we can deduce that u, v H 1 (Ω i ) b i (u, u v) b i (v, u v) c u v 2 H 1 (Ω i) b(u h, u h v h ) b(v h, u h v h ) = b k (u k,h, u k,h v k,h ) b k (v k,h, u k,h v k,h ) 5

6 c u k,h v k,h 2 H 1 (Ω k ) = c u h v h 2 X c u h v h 2 X The last estimate follows from the fact that for u h V h u h X is equivalent to u h X, with a constant independent of all h k, what was proved in [6]. Lemma 3.3 The form b(, ) is Lipschitz continuous in X (and thus in V h ), i.e. for all u, v, w X we have b(u, w) b(v, w) M u v X w X where M is a positive constant. Proof. In [18] was proven that u, v, w H 1 (Ω i ) b i (u, w) b i (u, v) M u v 2 H 1 (Ω i) w 2 H 1 (Ω i) Summing over all subdomains, using the above result for u, v, w X and Schwarz inequality ( for the standard inner product in R N ) we end the proof. We now formulate and prove the lemma which is a generalization of the second Strang lemma for the boundary value problem considered in this paper, cf. [12] for the proof in the linear case. Lemma 3.4 Let u and u h be the solutions of (2.2) and (3.3), respectively. Let u H 2 (Ω). Under assumptions (2.4) - (2.6) we have u inf u h X C v h V h u sup γ v h X + m l m u [w h ]ds w h V h (3.5) w h X γ m Γ where [w h ] is a jump of w h across γ m, l m u γ m = l i(m) u is defined in (3.4), the sum is taken over all mortars γ m and C is a constant independent of h i. Proof. We have for all v h V h u u h X u v h X + u h v h X (3.6) Let denote w h = u h v h, then Lemma 3.2 and (3.3) yield that (1/c) u h v h 2 X b(u h, w h ) b(v h, w h ) = f(w h ) b(v h, w h ) (3.7) We now can deduce that ( ) f(w h ) = ( a i (x, u, u )) + a 0 (x, u, u ) x i + Ω k Ω k ( a i (s, u, u )n i w k,h ) ds = b(u, w h ) + γ m Γ w k,h dx = b(u, w h )+ γ m l m u [w h ]ds The last equality follows from Lemma 3.1. From this, Lemma 3.3 and (3.7) it follows that (1/c) w h 2 X b(u, w h ) b(v h, w h ) + l m u [w h ]ds γ m M u v h X w h X + γ m Γ γ m Γ γ m l m u [w h ]ds 6

7 Dividing by w h X and substituting in (3.6) we complete the proof. The first term in (3.5) is known as the approximation error and the second one we call the consistency error which is a consequence of the discontinuities of the elements of V h through the interface. We can now turn to the proof of Theorem 3.1. Proof of Theorem 3.1. Let us first consider the approximation error. In [7] or in [5] for 3D case it was proven that if v H0 1(Ω) with v Ω k H 2 (Ω k ) then there exists constant c independent of h k such that inf v h V h v v h 2 X c h 2 k v Ωk 2 H 2 (Ω k ) (3.8) Let us turn to the consistency term. We now prove that δ m δ m l m u [w h ]ds c 1/2 h k { Ω k + u 2 H 2 (Ω k )} wk,h H 1 (Ω k ) (3.9) where c is a constant independent of h k. It follows the same lines as in [5]. Let us consider one interface γ ij common to Ω i and Ω j. Assume that γ m(i) is a mortar, then δ m(j) is a nonmortar. From (3.2) we have ψ M hj (δ m(j) ) l m u [w h ] ds = (l m u ψ) (w j,h w i,h ) ds δ m(j) δ m(j) Hence l m u [w h ]ds δ m(j) Using the trace theorem, cf. [1], we have l m u [w h ]ds c δ m(j) It can be proven, e.g. see [5], that inf ψ M hj (δ m(j) ) l mu ψ [H 1/2 (δ m(j) )] w j,h w i,h H 1/2 (δ m(j) ) inf ψ M hj l m u ψ [H 1/2 (δ m(j) )] ( w i,h H 1 (Ω i) + w j,h H 1 (Ω j )) inf ψ M hj (δ m(j) ) l mu ψ [H 1/2 (δ m(j) )] c h j l m u H 1/2 (δ m(j) ) (3.10) Now we sum over all nonmortars δ m(j) = γ ij and use Corollary 3.1 what proves (3.9). Combining (3.9) and (3.8) completes the proof of the theorem. 4. THE GEOMETRICALLY NONCONFORMING CASE In this section we consider the geometrically nonconforming version of the mortar finite element method. As in the previous section we assume that Ω is divided into disjoint, polygonal subregions Ω k. As in the third section we introduce in each subdomain a quasi-uniform triangulation. Let γ ij = Ω i Ω j. The local spaces X h (Ω k ) and the global spaces X, X h (Ω) be defined as in the previous section. The mortar element method consists of choosing one of the decomposition of Γ defined in (3.1), made of mortars γ m that are disjoint i.e. Γ = M m=1 γ m, γ m γ n =, m n 7

8 and that satisfy the assumption that each mortar is an edge of one subdomain i.e. γ m = γ m(i) is an edge of Ω i. We assume that there is at least one such decomposition. The mortar sides of Ω i we denote by γ m(i) and the slave sides (the edges that are not mortars) we denote by δ k(i). For each edge that is not a mortar we define a space of traces W hj (δ k(j) ) and a test space M hj (δ k(j) ) in the same way as in the previous section, i.e. W hj (δ k(j) ) is the restriction of X h (Ω j ) to δ k(j) and M hj (δ k(j) ) is a subspace of W hj (δ k(j) ) such that its functions are constant on elements which intersect the ends of δ k(j). Now we define our discrete space as V h = { v h X h (Ω) : δ k(j) not a mortar ψ M hj (δ k(j) ) δ k(j) (v j,h m(i) where the sum is taken over m such that γ m(i) δ k(j). The discrete problem is to find u h V h such that v i,h δk(j) ) ψ ds } (4.1) b(u h, v h ) = f(v h ) v h V h (4.2) where the form b(, ) is defined as in the third section. We have as in the previous section that norm X and seminorm X are equivalent and that analogous lemmas to Lemma 3.2 and 3.3, are also valid in this case, cf. [7]. Thus there exists the unique solution of (4.2), see [18]. Now we state the main theorem of this section. Theorem 4.2 Let u and u h be the solutions of (2.2) and (4.2), respectively. Let u H 5/2 (Ω). Then under the hypotheses of (2.4) - (2.6) we have u u h X c h where c is a positive constant independent of h k and h = max k (h k ). To prove Theorem 4.2 we state an analogue of Lemma 3.4 for the geometrically nonconforming case. The proof is similar to that of Lemma 3.4. Lemma 4.5 Let u and u h be the solutions of (2.2) and (4.2), respectively. Let u H 2 (Ω). Under assumptions (2.4) - (2.6) we have u inf u h X C v h V h u sup δ v h X + m l m u [w h ]ds w h V h (4.3) w h X where [w h ] is a jump of w h across δ m, l m u δ m = l i(m) u that is defined in (3.4). The sum is taken over all δ m, sides of substructures that are not mortars. Proof of Theorem 4.2. The first term, the approximation error, can be estimated from (3.8) whose statement is also true in this case, cf. [7] or [5]. It remains to estimate the second term, the consistency error. First we prove that l m u [w h ]ds c h j l m u H1 (δ m(j) ) ( w j,h H1 (Ω j ) + w k,h H1 (Ω k )) (4.4) δ m(j) k δ m Γ 8

9 that is an analogue to (3.9). The sum is taken over all Ω k such that Ω k δ m(j). Proceeding as in the proof of (3.9) we see that l m u [w h ]ds = l m u (w j,h w k,h δm ) ds δ m(j) δ m(j) From (4.1) we can deduce that for all ψ M hj (δ m(j) ) l m u [w h ]ds = (l m u ψ) (w j,h δ m(j) δ m(j) Ω k δ m Ω k δ m Using Schwarz inequality we have l m u inf [w h ]ds δ m(j) ψ M hj (δ m(j) ) l mu ψ L2 (δ m(j) ) w j,h w k,h δm ) ds Ω k δ m w k,h δm L2 (δ m(j) ) Combining the trace theorem and the standard approximation result for l m u H 1 (δ m(j) ) we prove (4.4). As u H 5/2 (Ω), we can obtain a result similar to that of Corollary 3.1. Combining this with (4.4) and summing over all δ m(j) we finish the proof of Theorem THE RICHARDSON-NEWTON METHOD In this section we propose a method for solving the problem (3.3) arising from discretization of the boundary value problem (2.2) by the geometrically conforming mortar finite element method. We do not know, how to generalize this method to the geometrically nonconforming case. For simplicity of presentation we describe the method with one additional assumption that the subdomains Ω i are triangles and form a quasiuniform triangulation with a parameter H, cf. [12]. To define our method, we first have to introduce some special functions and subspaces of V h. We will primarily work with nodal basis of the mortar finite element space associated with the following sets of nodes: all nodes interior to the substructures all nodes interior to the mortars all nodes of vertices of subregions except those on Ω We associate a basis function with each node of these sets. The functions corresponding to nodes in the interiors of the substructures are standard nodal basis functions as in the conforming finite element discretization. A function associated with a node x k interior to the mortar γ m(i), we define as follows. It is one at x k and zero at the remaining nodes defined above, i.e. nodes interior to all substructures, vertices of all substructures and all nodes in the interiors of the mortars except x k. The values of this function at the interior nodes of nonmortars δ m(j) = γ m(i) are determined by the mortar condition (3.2), with zero values at the ends of δ m(j). We now define basis functions associated with the vertices of the substructures. We first denote by ν the set of vertices of the substructures that are associated with degrees of freedom of V h, i.e. those which are not on Ω. Each crosspoint of Γ belongs to numbers of subdomains and therefore corresponds to several nodes of ν, to one degree of freedom for each of subregions that meet at that point. These nodes are in the same geometrically position, but are assigned to different subdomains. Let v n ν be a vertex of Ω i. Then a basis function associated with v n ν we denote as φ vn. It is defined as one at v n and zero at all other vertices of ν and at all interior nodes of all substructures. We now define this function 9

10 on Γ, i.e. on all mortars and nonmortars. There are three possible situations: the vertex v n can be a common end of two mortars γ n(i) and γ m(i), a common end of two nonmortars δ l(i) and δ k(i) or a common end of a nonmortar δ s(i) and a mortar γ p(i). In the first case φ vn restricted to γ n(i) and γ m(i) is a standard nodal function corresponding to v n, i.e. is one at v n and zero at the remaining nodes of the both mortars. On nonmortars δ n = γ n(i) and δ m = γ m(i) this function is determined by the mortar condition (3.2) with zero values at the ends of δ n and δ m, respectively. In the second case φ vn restricted to the mortars γ l = δ l(i) and γ k = δ k(i) is zero and on δ l(i) and δ k(i) is determined by the mortar condition with one at v n and zero at the other ends of δ l(i) and δ k(i). In the last case φ vn is defined on the mortar γ p(i) (and δ p = γ p(i) ) as in the first case while on the nonmortar δ s(i) (and γ s = δ s(i) ) analogously to the second case. In all cases φ vn is defined as zero on the remaining mortars and nonmortars. Using these basis functions we have V h = span{φ 1,..., φ n }. Let the solution of (3.3) be represented as u h = n u iφ i and introduce n k i (u 1,..., u n ) = b u j φ j, φ i, f i = (f, φ i ) L 2 (Ω) j=1 Let B = (k 1,..., k n ) T, u = (u 1,..., u n ) T and f = (f 1,..., f n ) T. With these notations we rewrite the problem (3.3) as the system of nonlinear algebraic equations B(u) = f (5.1) Here and below we use u either as a function in V h or as a vector representation in terms of the nodal basis i.e. u = (u 1,..., u n ) T or u = n u iφ i. Additionally we introduce a bilinear form on V h V h as a (u, v) = Ω i u i v i dx Let D be its matrix representation. We should point out that a (u, u) 1/2 = (Du, u) 1/2 R = u X, n therefore a (, ) is positive definite in V h. We solve (5.1) by a method that combines the additive Schwarz preconditioning technique with Newton s method. The Schwarz method is determined by subspaces of V h, and bilinear forms defined on these subspaces, cf. [15]. In our case those forms are equal to a (, ). We now define subspaces that form the decomposition of V h. Let V 0 be the coarse space of continuous, piecewise linear functions on the coarse triangulation with zero on Ω. Next we define one dimensional vertex spaces V vn, that are associated with v n ν: V vn = span{φ vn }. Finally we introduce V ij spaces which are associated with all pairs of two subdomains Ω i and Ω j that have a common edge γ ij which is the mortar γ m(i) Ω i and the nonmortar δ m(j) Ω j. We define V ij as a subspace of V h such that its functions can be nonzero at the interior nodes of Ω i and Ω j and at the interior nodes of γ m(i) and δ m(j). It is easy to see that V h = V 0 + V vn + v n ν γ ij Γ V ij We then define operators T 0 : V h V 0, T vn : V h V vn and T ij : V h V ij, by a (T 0 (u), v) = b(u, v) v V 0 (5.2) a (T vn (u), v) = b(u, v) v V vn (5.3) 10

11 and a (T ij (u), v) = b(u, v) v V ij (5.4) The matrix representation of these operators are denoted by the same symbols. They have the following form: T 0 = R0 T D 1 0 R 0B(u), T vn = Rv T n Dv 1 n R vn B(u) and T ij = Rij T D 1 ij R ijb(u) where D 0, D vn and D ij are the matrix representations of a (, ) in the corresponding subspaces and R 0 : V h V 0, R vn : V h V vn and R ij : V h V ij are the restrictions operators defined as in [8]. We note that T 0, T vn and T ij are nonlinear in general. To define an additive Schwarz method, let T = T 0 + T vn + T ij (5.5) v n ν γ ij Γ We replace the problem (5.1) by the problem of finding u V h such that T (u) = g (5.6) where g = g 0 + v g n ν v n + γ g ij Γ ij with g 0 = T 0 (u), g vn = T vn (u) and g ij = T ij (u). Here u is the solution of (5.1). We show that problems (5.6) and (5.1) have the same unique solution. These g i can be pre-computed without knowing the exact solution u. Introducing M 1 = R0 T D0 1 R 0 + Rv T n Dv 1 n R vn + RijD T 1 ij R ij (5.7) we have v n ν T (u) = M 1 B(u) For solving (5.6) we use the following algorithm: γ ij Γ Algorithm 5.1 For τ defined in Theorem 5.4, see below, iterate for n = 0, 1,... until convergence. u n+1 = u n τ(t (u n ) g) To prove the convergence of the algorithm we need the following auxiliary result. Theorem 5.3 For any u V h c(1 + log(h/h)) 2 (Du, u) R n (DM 1 Du, u) R n C(Du, u) R n where c, C are positive constants independent of H, h i and h = inf i h i. This theorem is proved in the last part of this section. Theorem 5.3 yields that (5.6) has the unique solution equal to the solution of (5.1) and that M 1 is invertible. The next corollary plays an important role in the proof of convergence of the algorithm. Corollary 5.2 There exists constants δ 0 and δ 1 such that and (B(u) B(v), u v) R n δ 0 u v 2 M B(u) B(v) M 1 δ 1 u v M where M was defined in (5.7), δ 0 = C (1 + log(h/h)) 2, and C, δ 1 are constants independent of H, h i. The proof of the corollary follows from Theorem 5.3, Lemma 3.2 and Lemma 3.3. We now state the main theorem of this section that can be proven in the standard way using Corollary 5.2, cf. [10] or [17]. 11

12 Theorem 5.4 If we choose 0 < τ < 2δ 0 /δ 1, where δ 0 and δ 1 are defined in Corollary 5.2, then Algorithm 5.1 is convergent in the sense that u n u M ρ(τ) n u 0 u M where ρ 2 = 1 τ(2δ 0 /δ 1 τ) < 1. The optimal τ opt = δ 0 /δ 1 and ρ 2 opt = 1 δ2 0 /δ 1. We now prove Theorem 5.3 using the general theory of ASM, cf. [15]. It reduces to check three key assumptions. Proof of Theorem 5.3. Assumption (i) We want to prove that there is a positive constant c independent of h i and H such that for all u V h there exist functions u 0 V 0, u vn V vn and u ij V ij such that u = u 0 + v u n ν v n + γ u ij Γ ij and a (u 0, u 0 ) + a (u vn, u vn ) + a (u ij, u ij ) c(1 + log(h/h)) 2 a (u, u) (5.8) v n ν γ ij Γ We first select u 0 V 0 = V H by making u 0 (c r ) = u cr, where c r Γ is a crosspoint and u cr is the average value of u at the vertices of ν that coincide geometrically with c r. We further denote ν(c r ) as the set of these vertices and ν(i) as the set of vertices of Ω i. Let N(c r ) be the number of vertices in ν(c r ). Thus we have that u cr = (1/N(c r )) v u(v n ν(c r) n). Let now define u vn V vn by the pointwise interpolation of u u 0 at v n, i.e. u vn = (u u 0 )(v n ) φ vn Note that w defined as vanishes at all vertices v n ν. We now decompose w in Ω i as w = u u 0 v n ν u vn w i = w Ωi = P i w i + H i w i where H i w i is the discrete harmonic part of w i and P i w i is the H 1 0 (Ω i ) projection on X h (Ω i ) H 1 0 (Ω i ) of w i, i.e. H i w i = w i on Ω i and ψ X h (Ω i ) H 1 0 (Ω i ) a (H i w i, ψ) = 0, a (P i w i, ψ) = a (w i, ψ) (5.9) Both P i w i and H i w i we extend as zero off Ω i. With each space V ij we associate the common edge γ ij = Ω i Ω j which is the mortar γ m(i) Ω i and the nonmortar δ m(j) Ω j. Then let w ij V ij be equal to w on γ m(i) and δ m(j), be zero on Ω i \ γ m(i), Ω j \ δ m(j) and be extended as the discrete harmonic function in Ω i and Ω j. Note that w ij is zero off Ω i Ω j because w ij V ij. We finish the decomposition of u by setting u ij = w ij + (1/N e (i))p i w i + (1/N e (j))p j w j where N e (k) is a number of edges γ kl Γ Ω k and N e (k) is equal to 3 if Ω k Ω = and 2 or 1 otherwise. Note that u = u 0 + u vn + v n ν γ ij Γ u ij 12

13 We first estimate a (u 0, u 0 ). Let u i be the average value of u over Ω i. Using the inverse inequality we have a (u 0, u 0 ) = u 0 2 H 1 (Ω = N i) u 0 u i 2 H 1 (Ω c N i) v n ν(i) u 0 (v n ) u i 2 We consider one vertex v n0 of Ω i, which geometrically coincides with a crosspoint c r and we have that u 0 (v n0 ) = u 0 (c r ). Hence u 0 (v n0 ) u i 2 1 = N(c r ) v n ν(c r) u(v n ) u i 2 c v n ν(c r) u(v n ) u i 2 Note that average values of u over a mortar γ m(i) and a nonmortar δ m(j) that occupies geometrically the same place, are equal to each other. Using this, the standard Sobolev-like inequality for finite elements, see e.g. [14], and the Poincare inequality, we obtain u 0 (v n0 ) u i 2 c (1 + log(h/h n )) u n 2 H 1 (Ω n) (5.10) v n ν(c r) The sum is taken over all subdomains with the common vertex c r. Summing over all subregions and all their vertices gives the estimate a (u 0, u 0 ) c (1 + log(h/h i )) u i 2 H 1 (Ω i) (5.11) We now estimate v n ν a (u vn, u vn ). We deduce that a (u vn, u vn ) = u(v n ) u 0 (v n ) 2 φ vn 2 X c (1 + log(h/h)) u(v n) u 0 (v n ) 2 We have used the fact that φ vn 2 X c (1 + log(h/h)), cf. [11]. We further use the same arguments as for the estimate of a (u 0, u 0 ) to get a (u vn, u vn ) c v i ν(c r) (1 + log(h/h)) 2 u i 2 H 1 (Ω i) (5.12) where c r is a crosspoint geometrically coinciding with v n. Summing over all subdomains and their vertices we obtain a (u vn, u vn ) c (1 + log(h/h)) 2 u i 2 H 1 (Ω i) (5.13) v n ν We now estimate γ ij Γ a (u ij, u ij ). We deduce that a (u ij, u ij ) c { } w ij 2 H 1 (Ω + w j) ij 2 H 1 (Ω + P i) iw i 2 H 1 (Ω + P i) jw j 2 H 1 (Ω j) We first estimate P i w i 2 H 1 (Ω ( P i) jw j 2 H 1 (Ω j) can be estimated in the same way). Using (5.12) we get P i w i 2 H 1 (Ω i) c ( u i 2 H 1 (Ω i) + v n ν(i) u vn 2 H 1 (Ω n) ) c ( u i 2 H 1 (Ω i) + n (1 + log(h/h)) 2 u n 2 H 1 (Ω n) ) where the last sum is taken over all substructures that have a common vertex with Ω i. 13

14 We now estimate the norms of w ij the discrete harmonic part of u ij. We first note that w ij 2 X = H i w i 2 H 1 (Ω i) + H jw j 2 H 1 (Ω j) c { w i 2 H 1/2 00 (γ m(i)) + w j 2 H 1/2 00 (δ m(j)) } c w i 2 H 1/2 00 (γ m(i)) The first inequality follows from extension property of discrete harmonic functions, cf. [9], and the second one from H 1/2 00 stability of functions in V h over each edge γ ij Γ, cf. [4]. Thus it remains to prove the estimate of w i 2. H 1/2 00 (γ m(i)) Let denote by v n1, v n2 the ends of γ m(i). Then w i γm(i) = z 2 z(v n i )φ vni z z 0(v ni )φ where z = u vni i u i and z 0 = u 0 u i. From this we obtain w i 2 H 1/2 00 (γ m(i)) c { z z(v ni )φ vni 2 H 1/2 00 (γ m(i)) + z 0 z 0 (v ni )φ vni 2 H 1/2 00 (γ m(i)) The first term we can estimate by c (1 + log(h/h i )) 2 u i 2 H 1 (Ω i), cf. [15] while the second one easily by z 0 z 0 (v ni ) 2 c (1 + log(h/h H 1/2 00 (γ i )) u 0 (v ni ) u i 2 m(i)) cf. [13]. Now using (5.10) we deduce that } w i 2 H 1/2 00 (γ m(i)) c (1 + log(h/h))2 n u n 2 H 1 (Ω n) where the sum is taken over all indices of subdomains that has a vertex that geometrically coincides with one of the ends of γ m(i). Summing over all subspaces V ij we have a (u ij, u ij ) c (1 + log(h/h)) 2 u 2 X = c (1 + log(h/h)) 2 a (u, u) γ ij Γ Combining this, (5.11) and (5.13), we get (5.8) what ends the proof of Assumption (i). Assumptions (ii) It is obviously satisfied with ω = 1 as all local forms equals a (, ). Assumption (iii) It is satisfied with ρ(ɛ) C. ACKNOWLEDGMENTS The author is greatly indebted to prof. Maksymilian Dryja for his encouragement and his invaluable advises. References [1] R. A, Adams, Sobolev spaces. Academic Press, [2] Y. Achdou, Yu. A. Kuznetsov, and O. Pironneau, Substructuring preconditioners for the Q 1 mortar element method. Numer. Math. (1995), 71, [3] Y. Achdou, Y. Maday, and O. Widlund, Substructuring preconditioners for the mortar method in 2-D. In: Proceeding of the Eight International Conference on the Domain Decomposition Method, Beijing, May

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