An additive average Schwarz method for the plate bending problem

Transcription

1 J. Numer. Math., Vol. 10, No. 2, pp (2002) c VSP 2002 Prepared using jnm.sty [Version: v1.2] An additive average Schwarz method for the plate bending problem X. Feng and T. Rahman Abstract The original additive average Schwarz method and its variants were introduced for solving second order elliptic boundary value problems. In this paper, we extend the original idea to fourth order problems by designing a variant of the additive average Schwarz method for the plate bending problem using the nonconforming Morley finite element for the discretization. Like the original average method this method uses only nonoverlapping subdomains, and its coarse space does not need any explicit coarse mesh. An analysis of the method, and some numerical experiments to validate the theory are presented. Keywords: Domain decomposition, additive Schwarz, plate bending, Morley finite element 1. INTRODUCTION The additive Schwarz method refers to a general methodology, based on the idea of domain decomposition, for solving system of linear algebraic equations resulting from the discretization of partial differential equations which arise in various application problems from science and engineering. An abstract theoretical framework for the design and analysis of additive Schwarz methods developed by Dryja- Widlund can be found in their paper [10]. Independent work on additive Schwarz methods can be found in the literature by Matsokin, Nepomnyaschikh and several others, see [11,14] and the references therein for a detailed exposition. In this paper, we follow the abstract framework and develop an additive Schwarz method for solving system of equations resulting from the discretization of the biharmonic equation in the plane, using the Morley finite element. The method is based on the idea of using nonoverlapping subdomains and the fact that the interactions between the subdomains take place only through a coarse space. The coarse space is constructed without explicitly introducing any coarse mesh, instead, it is defined as the range of some interpolation like operator. The action of this operator is local, and it is cheap to compute. The basic construction of the method is based Department of Mathematics, University of Tennessee, Knoxville, TN 37996, U.S.A.; xfeng@math.utk.edu. Institute for Mathematics, University of Augsburg, Universitätsstr. 14, D Augsburg, Germany; talal.rahman@math.uni-augsburg.de

2 110 X. Feng and T. Rahman on the idea of the original additive average Schwarz method of Bjørstad-Dryja- Vainikko [2], which was primarily developed for second order elliptic problems with discontinuous coefficients. The construction and the analysis of our additive Schwarz method for fourth order problems is, however, more complicated and delicate. Additive Schwarz methods using nonconforming plate elements have been studied extensively by Brenner [3,4], see also [9]. We refer to [18,19] for earlier work on additive Schwarz methods using conforming plate elements. All these additive methods were designed using overlapping subdomains. It is known, cf. [3,4,9], that the condition number bound for the two level additive Schwarz method with overlap is of the form (H/δ) 4, where H is the subdomain size and δ is the overlap. It is also shown in [3] that for small overlap h δ H, and under some shape regularity on the subdomains, this bound can be improved to (H/h) 3. Here h is the mesh size. This result was, however, presented as a theoretical result and no numerical result, to our knowledge, has ever been reported. Brenner-Sung [7] later showed that this bound is sharp. The average method presented in this paper, as we shall see in the next sections, possesses this improved upper bound. The method is quite simple. An important feature of it is that, in each iteration, the communication needed between the processors (virtual processors) which are responsible for the subdomains is minimal, since only nonoverlapping subdomains are used. The method is therefore very well suited for parallel computations. In each iteration, however, the method requires to solve a special coarse problem whose unknowns are associated with the nodal variables on all subdomain interfaces. The idea of the average method can further be extended to three dimensional biharmonic problems. For works on nonoverlapping methods for the plate problem, in the class of substructuring type algorithms, we refer to [6,13,15,16]. The paper is organized as follows. In Section 2, we recall the plate bending problem and its nonconforming Morley finite element approximation. In Section 3, we introduce and analyze the additive average Schwarz method for the Morley finite element. We construct the coarse space as the range of an interpolation like operator, present the stability and the approximation property of the operator, and give an estimate of the condition number of the iteration operator using the abstract Schwarz framework. We discuss some implementation issues in Section 4, and in Section 5, we present some numerical experiments to verify our theory. 2. PRELIMINARIES We consider the biharmonic problem 2 u = f, in Ω, (2.1) u = u = 0, on Ω. (2.2) Where Ω R 2 is a bounded domain and f L 2 (Ω). This boundary value problem describes the bending of a clamped plate Ω under the distributed load f (cf. [8]).

3 Additive Schwarz for the plate bending problem 111 The weak formulation of (2.1) (2.2) is defined by seeking u H0 2 (Ω) such that where F(v) = Ω f v dx, and a Ω (u,v) = Ω a Ω (u,v) = F(v), v H 2 0 (Ω), (2.3) [ u v + (1 σ) { 2 2 u x 1 x 2 2 v 2 u 2 v x 1 x 2 x1 2 x2 2 2 u x v x 2 1 } ] dx. (2.4) Here 0 < σ < 1 2 is Poisson s ratio for plate. Let Ω be a polygonal domain and T h be a quasi uniform triangulation of Ω with the mesh size h. Let V h be the Morley finite element space associated with T h, which is defined as follows. Any v V h will satisfy: v L 2 (Ω) with v K P 2 (K) for each triangle K T h. v is continuous at the vertices of T h and vanishes at the vertices along Ω. v is continuous at the midpoints of the edges of T h and vanishes at the midpoints of the edges along Ω. Clearly, V h C 0 (Ω), so the Morley element is a strongly nonconforming plate element. The finite element solution u V h is defined as the solution of where and a K (u,v) is defined by (2.4). Define for any v V h a h (u,v) = F(v), v V h, (2.5) a h (u,v) = K T h a K (u,v) (2.6) v h,m,ω = ( 12 v 2 H (K)) m. (2.7) K T h,k Ω It is well known (see [8]) that v h,2,ω is a norm on V h and that there exists two positive constants c 1 and c 2 such that c 1 v 2 h,2,ω a h (v,v) c 2 v 2 h,2,ω, v V h. (2.8)

4 112 X. Feng and T. Rahman Using the basis functions of the space V h the finite element problem (2.5) reduces to the following linear system of equations, Au = b. (2.9) The stiffness matrix A whose components are k i j = a h (ϕ i,ϕ j ), where ϕ i and ϕ j are the standard Morley basis functions, is both symmetric and positive definite. u is the solution vector containing the nodal values and b is the load vector with elements F(ϕ i ). Iterative methods like the Conjugate Gradient (CG) method can be used to solve (2.9), see [17]. The convergence of such iterative methods in general depend on the condition number of the coefficient matrix, the smaller the number the better the convergence. It is well known that the (matrix 2 norm) condition number of A (κ(a)) is of the order O(h 4 ) (cf. [3,4,9]). The linear system thus becomes illconditioned for small mesh size h. The idea of an additive Schwarz method is to solve instead, an equivalent system, Tu = g, (2.10) where κ(t) will be much smaller. The analysis of such methods, therefore, revolves around finding an estimate for the number often in the form of an upper bound. For background knowledge and a general theory on additive Schwarz methods, we refer to [11,14]. 3. THE ADDITIVE SCHWARZ METHOD Let {Ω j } N j=1 be a nonoverlapping partition of Ω, i.e Ω = N j=1 Ω j, and for i j, Ω i Ω j = /0. We consider each Ω j to be a polygonal domain consisting of a cluster of finite elements (triangles) of T h. Let H j be the diameter of Ω j, and h j the mesh size associated with the triangulation of Ω j. We assume that the subdomains and the triangles are shape regular, and H = max Ω j ΩH j is the largest diameter. We use Ω jh and Ω jh to denote the set of interior and the set of boundary vertices of the subdomain Ω j, respectively. Correspondingly, Ω j and Ω jh will denote the set of interior and the set of boundary edge midpoints of Ω j, respectively. We decompose the space V h into smaller subspaces, Vj h V h, j = 0,,N, so that their sum equals the whole space, i.e., V h = V h 0 +V h 1 + +V h N. (3.1) For j = 1,,N, the subspace Vj h as below corresponds to the subdomain Ω j and is defined V h j = {v V h : v(x) = 0, x Ω jh ; v (x) = 0, x Ω jh; v = 0, outside of Ω j }. (3.2)

5 Additive Schwarz for the plate bending problem 113 The coarse space V h 0 is constructed using the idea given in [2], i.e., by defining V h 0 as the range of some interpolation like operator I a, in other words, V h 0 = {I a v : v V h }. (3.3) We need that the operator I a should preserve all linear functions rather than just constants. It is known for fourth order problems that the coarse space should include all linear functions in order to get a scalable algorithm. The linear operator I a : V h V h is defined below. Let ṽ j be a linear function on Ω j, calculated as the P 1 -least squares approximation that fits the boundary data {v(p) : p Ω jh } of v at the boundary vertices of Ω j (cf. Lemma 3.1). I a v V h on Ω j, for any v V h, is then defined by its nodal values as { v(x), x Ω (I a v)(x) = jh, (3.4) ṽ j (x), x Ω jh. and (I a v) (x) = v (x), x Ω jh, ṽ j (x), x Ω jh. (3.5) η is the unit normal at x. Now, using this coarse space and the subspaces Vj h (3.2), it is straight forward to verify that (3.1) holds. from Remark 3.1. ṽ j can also be seen as the L 2 projection of v Ω j onto P 1 (Ω j ) with respect to the discrete inner product (u,v) = u(x) v(x). x Ω jh We formulate the additive Schwarz method using this coarse space and present an analysis of the method. Let the spaces Vj h, j = 1 N be defined as in (3.2). We then define the linear projection operators T j : V h Vj h, j = 0,1,...,N in the standard way as a j (T j u,v) = a h (u,v), v V h j, (3.6) where the bilinear form a j (, ) : V h j V h j R is given by a j (u,v) = a h (u,v). (3.7) Let T = N j=0 T j be the additive operator, then the discrete problem (2.5) can be replaced by the equation Tu = g, (3.8)

6 114 X. Feng and T. Rahman where g = N j=0 g j, and g j = T j u is defined as the solution of a j (g j,v) = F(v), v V h j. (3.9) Theorem 3.1. The operator T is self adjoint and positive definite with respect to the inner product a h (, ), and for any u V h c 1 β 1 a h (u,u) a h (Tu,u) c 2 a h (u,u), (3.10) where c 1 and c 2 are positive constants independent of the parameters h j, H j and β = max Ω j Ω(H j /h j ) 3. Proof. We apply the general Schwarz framework, cf. Chapter 5 of [14], to prove the above theorem. According to the framework three key assumptions need to be verified. Taking V0 h as the coarse space, and referring to the notations used in [14] we have for Assumption 2 that ρ(e ) = 1. Similarly, for Assumption 3, we have ω = 1 since only exact bilinear form a h (u,v) is used everywhere. So, it remains to check the stability assumption, i.e., N a j (u j,u j ) cβa h (u,u), (3.11) j=0 for any u V h such that u = N j=0 u j with u j Vj h and c is a positive constant. Throughout this paper c is used to represent a generic positive constant which is independent of the mesh sizes h j, H j and β. For any u V h, let u 0 = I a u and u j = (u u 0 ) Ω j, then clearly u j Vj h and u = N j=0 u j. Now using this decomposition we get N a j (u j,u j ) = j=0 N a j (u u 0,u u 0 ) + a 0 (u 0,u 0 ) j=1 = a h (u u 0,u u 0 ) + a h (u 0,u 0 ) 2a h (u,u) + 3a h (u 0,u 0 ). (3.12) Hence, for (3.11) to be true, the following estimate should be true, i.e., a h (u 0,u 0 ) cβa h (u,u), (3.13) which, in fact, is an immediate consequence of the stability estimate (3.26) of the operator I a (cf. Lemma 3.5), and the inequality (2.8). In order to establish the desired stability estimate for I a, needed for (3.13), we need several auxiliary lemmas. The first lemma gives an explicit calculation of the linear polynomial ṽ j. The proof of the lemma is not given since it is a simple consequence of least squares fitting.

7 Additive Schwarz for the plate bending problem 115 Lemma 3.1. Let v be any function in L 2 (Ω j ) whose function values at Ω jh are well defined. Let ṽ j P 1 (Ω j ) be the function defined as above. If z = (z 1,z 2 ) t denotes an arbitrary point in R 2 then ṽ j can be written as ṽ j (z) = c 0 + c 1 z 1 + c 2 z 2, (3.14) where the coefficients c 0,c 1,c 2 are given by the following formulas. c 1 = (x 1v x 1 v)(x 2 2 x2 2 ) (x 2v x 2 v)(x 1 x 2 x 1 x 2 ) (x 1 x 2 x 1 x 2 ) 2 (x 2 1 x2 1 )(x2 2 x2 2 ), c 2 = (x 1v x 1 v)(x 1 x 2 x 1 x 2 ) (x 2 v x 2 v)(x 2 1 x2 1 ) (x 1 x 2 x 1 x 2 ) 2 (x 2 1 x2 1 )(x2 2 x2 2 ), c 0 = v c 1 x 1 c 2 x 2, where v = 1 n j x Ω jh v(x) and x i = 1 n j x Ωjh x i, for i = 1,2, with n j being the number of vertices on Ω j. Note that x 2 i x 2 i and x 1 x 2 x 1 x 2. In addition, ψ P 1 (Ω j ) [v(p) ṽ j (p)] 2 [v(p) ψ(p)] 2. (3.15) The second lemma states two useful estimates which we need for the analysis. These two estimates relate the Morley element to its conforming relative, the fifth degree Argyris element, see [3]. Recall that the Argyris element has a fifth degree polynomial as its shape function which is uniquely determined by the function value, the two first partial derivatives and the three second order partial derivatives at each vertex of a triangle, and the first normal derivatives at the midpoints of the edges of that triangle. Argyris element is a conforming C 1 element. Clearly, the nodal variables of a Morley element is a subset of the nodal variables of an Argyris element. Let U h denote the Argyris finite element space associated with the triangulation T h. Let E h be the extension operator, as defined in [3], which maps the Morley space V h into the Argyris space U h. In this mapping the function values at the vertices and the normal derivatives at the edge midpoints are preserved. The following estimates were established in [3] by Brenner. Lemma 3.2. For any v V h the following inequalities hold. E h v H 2 (Ω) c v h,2,ω, (3.16) E h v v L 2 (Ω) ch 2 v h,2,ω. (3.17)

8 116 X. Feng and T. Rahman q3 PSfrag replacements q j1 Tj q1 q j2 Oj q j3 Ω j Hj T j H j q2 Figure 1. The three vertices, q j1,q j2 and q j3, of the triangle T j are three vertices on the boundary of the subdomain Ω j. T j has the size of order H j. In the following lemma, we state some estimates associated with the averaged Taylor polynomials (cf. Chapter 4 of [5]). We use the averaged Taylor polynomials in place of the classical Taylor polynomials due to the reason that the functions we deal here with have only low differentiability. Lemma 3.3. For any w U h, let Q 2 w be the averaged Taylor polynomial of degree 2 of w as defined in Chapter 4 of [5]. Let R 2 w = w Q 2 w denote the 2nd order remainder term. Then for j = 1,2,,N, R 2 w L 2 (Ω j ) ch 2 j w H 2 (Ω j ), (3.18) R 2 w W r, (Ω j ) ch j h r j w H 2 (Ω j ), r = 0,1. (3.19) Proof. We first remark that Q 2 w(x) is only a polynomial of degree 1 in x (see Proposition of [5]), and that w H 2 (Ω j ) makes sense since U h H 2 (Ω). Now the proof of inequality (3.18) and inequality (3.19) with r = 0 are given in [5] (cf. Lemma and Proposition of [5] respectively). With r = 1, (3.19) follows from Lemma of [5] (with p = ) and the inverse inequality as follows. R 2 w W 1, (Ω j ) ch j w W 2, (Ω j ) ch j h 1 j w H 2 (Ω j ), where we have used the fact that w W 2, (Ω j ) for every w U h. We introduce an auxiliary function, a linear interpolant of v in Ω j, and state some of its properties in the following lemma. For each Ω j, Let q j1,q j2 and q j3 be any three vertices on Ω jh such that the big triangle T j with three vertices q j1,q j2 and q j3, is shape regular and has the size of order H j, see Figure 1. Note that this implies that the subdomain itself should be shape regular. Let ϕ j1,ϕ j2,ϕ j3 denote the nodal basis of P 1 (T j ) and be extended naturally to the whole subdomain Ω j. We define the linear interpolant π H j v of v and the corresponding interpolation operator as the following. π H j π H j v(x) = v(q j1 )ϕ j1 (x) + v(q j2 )ϕ j2 (x) + v(q j3 )ϕ j3 (x), x Ω j. (3.20)

9 Additive Schwarz for the plate bending problem 117 Lemma 3.4. For the interpolation operator π H j, the following inequalities hold. π H j w L (Ω j ) c w L (Ω j ), w C 0 (Ω j ). (3.21) w π H j w L 2 (Ω j ) ch 2 j w H 2 (Ω j ), w U h. (3.22) (w π H j w) L (Ω j ) ch j h 1 j w H 2 (Ω j ), w U h. (3.23) Proof. Note that ϕ jk (x) L (T j ) 1 for k = 1,2,3 (cf. [5]), where T j is the triangle defined above. The first inequality (3.21) thus follows directly from the definition of π H j and the fact that diam(ω j ) = O(H j ). Let w U h. For any x Ω j, since w(x) = Q 2 w(x) + R 2 w(x) (cf. Lemma 3.3) and that π H j preserves any P 1 (Ω j ) polynomial, we have w(x) π H j w(x) = [Q 2 w(x) π H j w(x)] + R 2 w(x) = π H j (Q 2 w w)(x) + R 2 w(x) = (I π H j )R 2 w(x), (3.24) where I stands for the identity operator. Hence, by using (3.24), the triangle inequality, (3.18), (3.21), and (3.19) with r = 0 we get the second inequality (3.22), i.e., w π H j w L 2 (Ω j ) = (I π H j )R 2 w L 2 (Ω j ) R 2 w L 2 (Ω j ) + π H j R 2 w L 2 (Ω j ) { } c H 2 j w H 2 (Ω j ) + H j R 2 w L (Ω j ) ch 2 j w H 2 (Ω j ). Finally to show (3.23), note that π H j R2 w(x) is linear everywhere in Ω j, giving (π H j R 2 w) L (Ω j ) = (π H j R 2 w) L (T j ) ch 1 j π H j R 2 w L (T j ). (3.25) As before, we use (3.24), the triangle inequality, (3.25), (3.21), and (3.19) (first with r = 1 and then with r = 0) to get (w π H j w) L (Ω j ) = (I π H j )R 2 w L (Ω j ) R 2 w L (Ω j ) + (π H j R 2 w) L (Ω j ) { } c H j h 1 j w H 2 (Ω j ) + H 1 j R 2 w L (Ω j ) { } c H j h 1 j w H 2 (Ω j ) + w H 2 (Ω j ), and the inequality (3.23) then follows.

10 118 X. Feng and T. Rahman We are now ready to establish the desired stability estimate for the operator I a, which we present in the following lemma. Lemma 3.5. There exists a positive constant c such that for any v V h, I a v 2 h,2,ω c β v 2 h,2,ω. (3.26) Proof. For any Ω j, let ṽ j and π H j v P 1(Ω j ) be defined as in Lemma 3.1 and (3.20), respectively. Clearly, I a v 2 h,2,ω = N j=1 I a v ṽ j 2 h,2,ω j. (3.27) By the definition of the operator I a, I a v equals ṽ j inside all interior triangles of Ω j. Apply the inverse inequality, the discrete equivalent of the L 2 norm in the Morley space (cf. [4]), the triangle inequality and (3.15) to get I a v ṽ j 2 h,2,ω j ch 4 j ch 4 j ch 2 j I a v ṽ j 2 L 2 (K) K Ω j h2 j [v(p) ṽ j (p)] 2 + h 4 j [v(p) π H j v(p)] 2 + h 2 j m Ω jh m Ω jh [ v (m) (πh j v) (m) + h 2 j m Ω jh [ (π H j v) [ v (m) ṽ ] 2 j (m) ] 2 ] 2 (m) ṽ j (m). (3.28) Again, π H j v ṽ j is linear over Ω j, and the following inequality holds. h 2 j [ ] (π H 2 j v) (m) ṽ j (m) c [π H j v(p) ṽ j (p)] 2. (3.29) m Ω jh

11 Additive Schwarz for the plate bending problem 119 Combining the above two inequalities and using (3.15), we get I a v ṽ j 2 h,2,ω j c = c h 2 j h 2 j [v(p) π H j v(p)] 2 + m Ω jh [ v (m) (πh j v) ] 2 (m) [w(p) π H j w(p)] 2 [ w + (m) (πh j w) ] 2 (m), (3.30) m Ω jh where w = E h v, the conforming relative (in the Argyris space) of the Morley function v (see in paragraph before Lemma 3.2), and π H j w denotes the linear interpolant of w as defined in (3.20). The last equality follows from the fact that the function values at the vertices and the normal derivatives at the edge midpoints of v are the same as those of w following the construction of E h. In addition π H j w is exactly equal to π H j v which follows immediately from the definitions of E h and π H j. For any fixed x Ω j, let w(x) = Q 2 w(x) + R 2 w(x) (cf. Lemma 3.3), then by using (3.24), (3.21) and (3.19) with r = 0 it follows that w π H j w L (Ω j ) = (I π H j )R 2 w L (Ω j ) From (3.30), (3.31) and (3.23) we obtain I a v ṽ j 2 h,2,ω j c c R 2 w L (Ω j ) ch j w H 2 (Ω j ). (3.31) h 2 j H 2 j w 2 H 2 (Ω j ) + H 2 j h 2 m Ω jh j w 2 H 2 (Ω j ) ch 3 j h 3 j w 2 H 2 (Ω j ). (3.32) Finally, the proof of Lemma 3.5 is completed by replacing (3.32) in (3.27) and then using (3.16). As a corollary of Lemma 3.5 we point out that the following approximation property for the average operator I a holds.

12 120 X. Feng and T. Rahman Corollary 3.1. There exists a positive constant c such that for any v V h, v I a v L 2 (Ω) c H 2 v h,2,ω. (3.33) Proof. Let the functions ṽ j and π H j v be defined as before, then by using the triangle inequality and the fact that I a v equals ṽ j inside all interior triangles of Ω j we get v I a v 2 L 2 (Ω j ) 2 v ṽ j 2 L 2 (Ω j ) + 2 ṽ j I a v 2 L 2 (K) K Ω j = T 1 + T 2. (3.34) From (3.32) we immediately get an upper bound for T 2, which is, T 2 ch 3 j h j w 2 H 2 (Ω j ). We estimate T 1 in the following. At first, we use the triangle inequality to get T 1 4 v π H j v 2 L 2 (Ω j ) + 4 πh j v ṽ j 2 L 2 (Ω j ). (3.35) Recall from the previous proof that π H j w πh j v, where w = E hv. For the first term on the right hand side of the above inequality we use the triangle inequality and the estimate (3.22) to get v π H j v 2 L 2 (Ω j ) 2 v w 2 L 2 (Ω j ) + 2 w πh j w 2 L 2 (Ω j ) 2 v w 2 L 2 (Ω j ) + ch4 j w 2 H 2 (Ω j ). (3.36) It follows from the definitions of ṽ j and w j that ṽ j w j. Hence, by using (3.15) and the estimate (3.31), for the second term on the right hand side of (3.35), we get π H j v ṽ j 2 L 2 (Ω j ) ch j h j π H j v(p) ṽ j (p) 2 = ch j h j π H j w(p) w j (p) 2 ch j h j w(p) π H j w(p) 2 ch j h j H 2 j w 2 H 2 (Ω j ) ch 4 j w 2 H 2 (Ω j ). (3.37) Now by using (3.37) and (3.36) in (3.35) we get an upper bound for T 1 as follows, { } T 1 c v w 2 L 2 (Ω j ) + H4 j w 2 H 2 (Ω j ).

13 Additive Schwarz for the plate bending problem 121 Finally, the proof is completed by replacing T 1 and T 2 in (3.34) by their bounds, summing over all subdomains and then using the (3.17) and (3.16). We conclude this section by the following remark on an extension of the additive average Schwarz method of this section to other plate elements. Remark 3.2. The results of this section can be extended to the incomplete biquadratic element (cf. [12]), a rectangular version of the Morley finite element. 4. IMPLEMENTATION ISSUES The subproblems associated with the subspaces Vj h, j = 1,,N, are the standard local subproblems, it is well known how to implement their solvers. In this section, we discuss only the coarse problem. The basis functions {ϕ k } of the Morley finite space V h, are associated with the vertices Ω h and the edge midpoints Ω h. Let u be the vector representation of u V h with respect to the basis of V h, containing the nodal values of u. The matrix A of the discrete problem can be split as follows. Let λ j be the union of Ω jh and Ω jh except those vertices and edge midpoints on Ω. Let R j be the restriction matrix containing only ones and zeros, which, if multiplied with u from the left, will return a vector of length λ j. This vector will contain the nodal values of u only at the vertices and the edge midpoints of λ j. Let R T j be the corresponding extension operator. Now, let A j be the local stiffness matrix associated with the subdomain Ω j, whose rows and columns correspond to the vertices and the edge midpoints of the set λ j. The (l,i)-th element of A j equals to K Ω j a K (ϕ i,ϕ l ), where x i,x l λ j. Using the above three types of matrices we can split the global stiffness matrix as A = N j=1 RT j A j R j. Note that A j, j = 1,,N can be constructed locally by the processor (virtual processor) responsible for Ω j. We use these local matrices to construct our coarse stiffness matrix. The coarse space V0 h can be defined as the span of some basis functions {Ψ l : x l S h Sh }, where S h and Sh are respectively the sets of all vertices and edge midpoints on all subdomain boundaries, except those on Ω. We use the Morley basis of V h to represent each basis function Ψ l as Ψ l Ψ l = Ψ l (x k )ϕ k + x k Ω h x m Ω (x m)ϕ m h We define the basis functions as follows: associate each function Ψ l with a node x l from the set S h S h by setting the nodal value to 1 at x l and 0 at all other nodes in S h S h, and then use the definition of I a to decide the nodal values at other vertices and edge midpoints. Let ϑ l be a Morley function associated with a vertex x l S h Ω h, which has the value ϑ l (x l ) = 1 at x l and 0 at all other vertices of Ω h, and the normal derivatives of

14 122 X. Feng and T. Rahman ϑ l are 0 at all edge midpoints of Ω h. Let ϑ l j P 1 (Ω j ) be the corresponding P 1 -least squares approximation as defined in the previous section. We use (3.4)-(3.5) to get 1, x k = x l,x l S h, 0, x k (S h \ {x l }), Ψ l (x k ) = ϑ l j(x k ), x k Ω jh,x l Ω jh, 0, at all other vertices of Ω h, and Ψ l (x m) = ϑ l j 1, x m = x l,x l Sh, 0, x m (Sh \ {x l}), (x m), x m Ω jh,x l Ω jh, 0, at all other edge midpoints of Ω h. Let v l be the vector representation of Ψ l with respect to the basis of V h. Let A (0) be the corresponding stiffness matrix of size n n, where n is the total number of vertices and edge midpoints in S h S h. The (p,q)-th element of A(0) equals to a h (Ψ q,ψ p ) = v T q Av p = N j=1 (R jv q ) T A j (R j v p ). Note that the term inside the summation is nonzero only when both x p and x q lie on Ω j. The matrix A (0) can thus be assembled in the standard way, from its element matrices A (0) j which contain elements from the set {(R j v q ) T A j (R j v p )} q,p, where x p,x q Ω j. Finally, to solve the coarse problem, a careful ordering of the unknowns, like the nested disection ordering or the minimum degree ordering etc., will result in reasonably sparse factors for the coarse stiffness matrix. 5. NUMERICAL EXPERIMENTS We present here some numerical results showing the performance of the additive average Schwarz method of this paper for the plate bending problem using the Morley finite element for discretization. For our experiment, we consider the domain Ω = [0,1] 2, and we choose the force function f so that the exact solution is u = x 2 1 x2 2 (x 1 1) 2 (x 2 1) 2. The domain Ω is triangulated by first generating a rectangular grid using equidistant grid lines parallel to the sides and then by dividing each rectangular block into a pair of triangles. The overall domain is then partitioned into nonoverlapping rectangular subdomains each containing equal number of blocks. We use the CG method to solve (3.8), and the iteration stops whenever r 2 < 10 6 b 2 is reached, where r is the computed residual and b is the right hand side. We compare the average method of this paper with its slightly modified version. We get the modified version as follows. Replace the term ṽ j in (3.4) with the term v j = 1 n j x Ωjh v(x), where n j is the number of vertices on Ω j, and set the value of (I av) (x) in (3.5) to zero at all x Ω jh, i.e., we use only constant extension inside

15 Additive Schwarz for the plate bending problem 123 Table 1. H/h is fixed. The number of iterations required to reduce the residual norm by 10 6 and a condition number estimate (in parentheses) for each test are shown. Additive average method Subdomains Blocks weaker ver. standard ver ( ) 55 (47.70) ( ) 59 (49.34) ( ) 59 (50.05) Table 2. In the first three tests H is fixed, while in the last three tests h is fixed. The number of iterations required to reduce the residual norm by 10 6 and a condition number estimate (in parentheses) for each test are shown. Additive average method Subdomains Blocks weaker ver. standard ver ( ) 55 ( ) ( ) 152 ( ) ( ) 423 ( ) ( ) 229 ( ) ( ) 89 ( ) ( ) 43 ( ) Ω jh instead of the linear extension as in the previous section. Note that this modified operator I a will preserve only the constant functions. This results in a method whose convergence depends on the number of subdomains. We call the resulting method the weaker version of the average method. Results from different tests using the average method of this paper (denoted by standard ver. ) and its weaker version (denoted by weaker ver. ), are presented in Table 1 and Table 2; a condition number estimate (in parentheses) and the iteration count for each test are reported. Table 1 shows that as the ratio H/h is kept constant, the condition number estimates for the average method remain constant. The same estimates for the weaker version show a growth proportional to the factor 1/h 2. The results here reflect also the well known fact that the interpolation operator I a should preserve linear functions in order to get a scalable algorithm. In Table 2 we see how the condition number estimates depend on the parameters h and H separately. In the first three tests of the table, H is kept fixed, while h varies with a factor of 2. For both methods, as we move downwards along the table, the condition number estimates grow by a factor of about 8, or in other words proportional to the factor 1/h 3. In the last three tests, h remains fixed while H varies, also with a factor of 2. The condition number estimates for the weaker version show a linear dependence on H whereas for the average method the estimates vary as H 3. The experimental results thus support the theory for the average method presented

16 124 X. Feng and T. Rahman in this paper. 6. CONCLUDING REMARKS An additive Schwarz method, based on the idea of the additive average Schwarz method of Bjørstad-Dryja-Vainikko for second order elliptic problems [2], is developed for the plate bending problem. The method is easy to implement, it requires no subdomain overlap, and imposes only mild geometric condition on the subdomains (they are required to be shape regular in the proofs of Lemmas 3.4 and 3.5). Due to the nonoverlapping feature, the method is quite suitable for parallel implementation. Our theory and experiments show that the condition number of its iteration operator grows like (H/h) 3. The computational complexity of the coarse problem may however be quite large. This situation can be improved by following the idea of Bjørstad-Dryja in [1], i.e., by replacing the coarse space with two suitable coarse spaces, the sum of which contains the original coarse space, and then exploiting the structure of the new stiffness matrices. This is a subject of an ongoing work. Acknowledgments. We are indebted to both Prof. P. Bjørstad and Prof. M. Dryja for their valuable discussions and suggestions concerning the work. REFERENCES 1. P. E. Bjørstad and M. Dryja. A coarse space formulation with good parallel properties for an additive schwarz domain decomposition algorithm. Submitted to Numer. Math., P. E. Bjørstad, M. Dryja, and E. Vainikko. Additive Schwarz methods without subdomain overlap and with new coarse spaces. In R. Glowinski, J. Périaux, Z. Shi, and O. B. Widlund, editors, Domain Decomposition Methods in Sciences and Engineering. John Wiley & Sons, Proceedings from the Eight International Conference on Domain Decomposition Metods, May 1995, Beijing. 3. S. C. Brenner. A two-level additive Schwarz preconditioner for nonconforming plate elements. Numer. Math., 72(4): , S. C. Brenner. Two level additive schwarz preconditioners for nonconforming finite element methods. Math. Comp., 65: , S. C. Brenner and L. R. Scott. The Mathematical Theory of Finite Element Methods, volume 15 of Texts in Applied Mathematics. Springer-Verlag, 2nd edition, S. C. Brenner and Li-Y. Sung. Balancing domain decomposition for nonconforming plate elements. Numer. Math., 83:25 52, S. C. Brenner and Li-Y. Sung. Lower bounds for two-level additive schwarz preconditioners for nonconforming finite elements. In Lecture Notes in Pure and Appl. Math., 202, pages Dekker, New York, P. G. Ciarlet. The Finite Element Method for Elliptic Problems. North-Holland Publishing Company, 2nd edition, Q. Deng and X. Feng. Two level Schwarz methods for conforming and nonconforming finite element approximations of plate bending problems. Mathematical Report 94-01, Dept. of Mathematics, Univ. of Tennessee, 1994.

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