A FETI-DP Method for Mortar Finite Element Discretization of a Fourth Order Problem

Transcription

1 A FETI-DP Method for Mortar Finite Element Discretization of a Fourth Order Problem Leszek Marcinkowski 1 and Nina Dokeva 2 1 Department of Mathematics, Warsaw University, Banacha 2, Warszawa, Poland, lmarcin@mimuw.edu.pl 2 Department of Mathematics and Computer Science, Clarkson University, PO Box 5815, Potsdam, NY , USA, ndokeva@clarkson.edu Summary. In this paper we present a FETI-DP type algorithm for solving the system of algebraic equations arising from the mortar finite element discretization of a fourth order problem on a nonconforming mesh. A conforming reduced Hsieh- Clough-Tocher macro element is used locally in the subdomains. We present new FETI-DP discrete problems and later introduce new parallel preconditioners for two cases: where there are no crosspoints in the coarse division of subdomains and in the general case. 1 Introduction The mortar methods are effective methods for constructing approximations of PDE problems on nonconforming meshes. They impose weak integral coupling conditions across the interfaces on the discrete solutions, cf. [1]. In this paper we present a FETI-DP method (dual primal Finite Element Tearing and Interconnecting, see [6, 9, 8] for solving discrete problems arising from a mortar discretization of a fourth order model problem. The original domain is divided into subdomains and a local conforming reduced HCT (Hsieh-Clough-Tocher macro element discretization is introduced in each subdomain. The discrete space is constructed using mortar discretization, see [10]. Then the degrees of freedom corresponding to the interior nodal points are eliminated as usually in all substructuring methods. The remaining system of unknowns is solved by a FETI-DP method. Many variants of FETI-DP methods for solving systems arising from the discretizations on a single conforming mesh of second and fourth order problems are fully analyzed, cf. [9, 8]. Recently there have been a few FETI-DP type algorithms for mortar discretization of second order problems, cf. [11, 5, 4, 3], and [7]. To our knowledge there are no FETI type algorithms for solving systems of equations arising from a mortar discretization of a fourth order problem in the literature. This work was partially supported by Polish Scientific Grant 2/P03A/005/24.

2 584 L. Marcinkowski, N. Dokeva The remainder of the paper is organized as follows. In Section 2 we introduce our differential and discrete problems. When there are no crosspoints in the coarse division of the domain, the FETI operator takes a much simpler form and therefore this case is presented separately together with a parallel preconditioner in Section 3, while Section 4 is dedicated to a short description of the FETI-DP operator and a respective preconditioner in the general case. 2 Differential and Discrete Problems Let Ω be a polygonal domain in R 2. Then our model problem is to find u H 2 0(Ω such that a(u, v = f(v v H 2 0(Ω, (1 where u is the displacement, f L 2 (Ω is the body force, a(u, v = [ u v + (1 ν (2u x1 x 2 v x1 x 2 u x1 x 1 v x2 x 2 u x2 x 2 v x1 x 1 ] dx. Here Ω H 2 0(Ω = {v H 2 (Ω : v = nv = 0 on Ω}, n is the normal unit derivative outward to Ω, and u xi x j := 2 u x i x j for i, j = 1, 2. We assume that the Poisson ratio ν satisfies 0 < ν < 1/2. From the Lax-Milgram theorem and the continuity and ellipticity of the bilinear form a(, it follows that there exists a unique solution of this problem. Next we assume that Ω is a union of disjoint polygonal substructures Ω i which form a coarse triangulation of Ω, i.e. the intersection of the boundaries of two different subdomains Ω k Ω l, k l, is either the empty set, a vertex or a common edge. We also assume that this triangulation is shape regular in the sense of Section 2, p. 5 in [2]. An important role is played by the interface Γ, defined as the union of all open edges of substructures, which are not on the boundary of Ω. In each subdomain Ω k we introduce a quasiuniform triangulation T h (Ω k made of triangles. Let h k = max τ Th (Ω k diam τ be the parameter of this triangulation. In each Ω k we introduce a local conforming reduced Hsieh-Clough-Tocher (RHCT macro finite element space X h (Ω k as follows, cf. Figure 1: X h (Ω k = {v C 1 (Ω k : v τ P 3(τ i, for triangles τ i, i = 1, 2, 3, (2 formed by connecting the vertices of τ T h (Ω k to its centroid, nv is linear on each edge of τ, and v = nv = 0 on Ω k Ω}. The degrees of freedom of RHCT macro elements are given by {u(p i, u x1 (p i, u x2 (p i}, i = 1, 2, 3, (3 for the three vertices p i of an element τ T h (Ω k, cf. Figure 1. We introduce next an auxiliary global space X h (Ω = N k=1 X h(ω k, and the so called broken bilinear form:

3 FETI-DP for Mortar FE for 4th Order Problem 585 Fig. 1. Reduced HCT element a h (u, v = N a k (u, v, where a k (u, v = [ u v + (1 ν (2u x1 x 2 v x1 x 2 u x1 x 1 v x2 x 2 u x2 x 2 v x1 x 1 ] dx. Ω k k=1 Then let X(Ω be the subspace of X h (Ω consisting of all functions which have all degrees of freedom (dofs of the RHCT elements continuous at the crosspoints the vertices of the substructures. The interface Γ kl which is a common edge of two neighboring substructures Ω k and Ω l inherits two 1D independent triangulations: T h,k (Γ kl the h k one from T h (Ω k and T h,l (Γ kl the h l one from T h (Ω l. Hence we can distinguish the sides (or meshes of this interface. Let γ m,k be the side of Γ kl associated with Ω k and called master (mortar and let be the side corresponding to Ω l and called slave (nonmortar. Note that both the master and the slave occupy the same geometrical position of Γ kl. The set of vertices of T h,k (γ m,k on γ m,k is denoted by γ m,k,h and the set of nodes of T h,l ( on by,h. In order to obtain our results we need a technical assumption of a uniform bound for the ratio h γm /h δm for any interface Γ kl = γ m,k =. An important role in our algorithm is played by four trace spaces onto the edges of the substructures. For each interface Γ kl = Ω k Ω l let W t,k (Γ kl be the space of C 1 continuous functions piecewise cubic on the 1D triangulation T h,k (Γ kl and let W n,k (Γ kl be the space of continuous piecewise linear functions on T h,k (Γ kl. The spaces W t,l (Γ kl and W n,l (Γ kl are defined analogously, but on the h l triangulation T h,l (Γ kl of Γ kl. Note that these four spaces are the tangential and normal trace spaces onto the interface Γ kl Γ of functions from X h (Ω k and X h (Ω l, respectively. We also need to introduce two test function spaces for each slave = Γ kl. Let M t( be the space of all C 1 continuous piecewise cubic on T h,l ( functions which are linear on the two end elements of T h,l ( and let M n( be the space of all continuous piecewise linear on T h,l ( functions which are constant on the two end elements of T h,l (. We now define the global space M(Γ = Γ Mt( M n( and the bilinear form b(u, ψ defined over X(Ω M(Γ as follows: let u = (u 1,..., u N X(Ω and ψ = (ψ m δm = (ψ m,t, ψ m,n δm M(Γ, then let

4 586 L. Marcinkowski, N. Dokeva b(u, ψ = b m,t(u, ψ m,t + b m,n(u, ψ m,n δ m Γ with b m,t(u, ψ m,t = (u k u l ψ m,y ds (4 δ m b m,n(u, ψ m,n = ( nu k nu l ψ m,n ds. (5 δ m Then our discrete problem is to find the pair (u h, λ X(Ω M(Γ such that a h (u h, v + b(v, λ = f(v v X(Ω (6 b(u h, φ = 0 φ M(Γ. (7 Note that if we introduce the discrete space V h = {u X(Ω : b(u, φ = 0 φ M(Γ} then u h is the unique function in V h that satisfies a h (u h, v = f(v v V h, which is a standard mortar discrete problem formulation, cf. e.g. [10]. Note that we can split the matrix K (l the matrix representation of a l (u, v in the standard nodal basis of X h (Ω l as: K (l := K (l ii K (l ci K (l ic K (l K(l ir cc K (l cr K (l ri K rc (l K rr (l, (8 where in the rows the indices i, c and r refer to the unknowns u (i corresponding to the interior nodes, u (c to the crosspoints, and u (r to the remaining nodes, i.e. those related to the edges. 2.1 Matrix Form of the Mortar Conditions Note that (7 is equivalent to two mortar conditions on each slave = γ m,k = Γ kl : b m,t(u, φ = (u k u l φ ds = 0 φ M t( (9 δ m b m,n(u, ψ = ( nu k nu l ψ ds = 0 ψ M n(. (10 δ m Introducing the following splitting of two vectors representing the tangential and normal traces u δm,l and nu δm,l we get u δm,l = u (r + u (c and nu δm,l = nu (r + nu (c on a slave Ω l, cf. (8. We can now rewrite (9 and (10 in a matrix form as B (r t, u (r + B (c t, u (c = B (r t,γ m,k u (r γ m,k + B (c t,γ m,k u (c γ m,k, (11

5 FETI-DP for Mortar FE for 4th Order Problem 587 B (r n, nu (r + B (c n, nu (c = B (r n,γ m,k nu (r γ m,k + B (c n,γ m,k nu (c γ m,k, where the matrices B t,δm,l = (B (r t,, B (c t, and B n,δm,l = (B (r t,γ m,k, B (c t,γ m,k are mass matrices obtained by substituting the standard nodal basis functions of W t,l (, W n,l ( and M t(, M n( into (9 and (10, respectively i.e. B t,δm,l = {(φ x,s, ψ y,r} x,y δm,l,h φ x,s W t(, ψ y,r M t(, (12 s,r=0,1 B n,δm,l = {(φ x, ψ y} x,y δm,l,h φ x W n(, ψ y M n(, (13 where φ x,s, (ψ x,s is a nodal basis function of W t(, (M t( associated with a vertex x of T h,l ( and is either a value if s = 0 or a derivative if s = 1, and φ x W n,l ( and ψ x, M n( are nodal basis function of these respective spaces equal to one at the node x and zero at all remaining nodal points on. The matrices B t,γm,k = (B (r n,, B (c n,, and B n,γm,k = (B n,γ (r m,k, B n,γ (c m,k are defined analogously. Note that B (r t,, B (r n, are positive definite square matrices, see e.g. [10], but the other matrices in (11 are in general rectangular. We also need the block-diagonal matrices ( ( Bt,δm,l 0 Bt,γk,l 0 B δm,l = B 0 B γk,l = n,δm,l 0 B n,γk,l. (14 3 FETI-DP Problem No Crosspoints Case In this section we present a FETI-DP formulation for the case with no crosspoints, i.e. two subdomains are either disjoint or have a common edge, cf. Figure 2. In this case both the FETI-DP problem and the preconditioner are fully parallel and simple to describe and implement. Ω 1 Ω 1 Ω 2 Ω 3 Ω 4 Ω 2 Ω 3 Fig. 2. Decompositions of Ω into subdomains with no crosspoints 3.1 Definition of the FETI Method We now reformulate the system (6 (7 as follows K ii K ir 0 u (i f i K := K ri K rr Br T u (r = f r, (15 0 B r 0 λ 0

6 588 L. Marcinkowski, N. Dokeva where B r = diag{b r,δm,l } δm with B r,δm,l = and K ii are block diagonal matrices of K (l rr ( I δm,l, (B (r 1 B γ (r m,k. Here K rr and K (l ii, respectively, cf. (8, and λ = {(B (r T }λ. Next the unknowns related to interior nodes and crosspoints, i.e. u (i in (15, are eliminated, which yields a new system Su (r + B T r λ = g r, B ru (r = 0, (16 where S = K rr K ri (K ii 1 K ir and g r = f r K ri (K ii 1 f i. We now eliminate u (r and we end up with the following FETI-DP problem find λ M(Γ such that F( λ = d, (17 where d = B rs 1 g r and F = B rs 1 B T r. Note that both S and B are block diagonal matrices due to the assumption that there are no crosspoints. Next we introduce the following parallel preconditioner M 1 = B rsb T r. ( Convergence Estimates We say that the coarse triangulation is in Neumann-Dirichlet ordering if every subdomain has either all edges as slaves or all as mortars. In the case of no crosspoints it is always possible to choose the master-slave sides so as to obtain an N-D ordering of subdomains. We have the following theorem in which a condition bound is established: Theorem 1. For any λ M(Γ it holds that c (1 + log(h/h p Mλ, λ Fλ, λ C Mλ, λ, where c and C are positive constants independent of any mesh parameters, H = max k H k and h = min k h k, p = 0 in the case of Neumann-Dirichlet ordering and p = 2 in general case. 4 General Case Here we present briefly the case with crosspoints: the matrix formulation of (6 (7 is as follows: K ii K ic K ir 0 u (i f i K := K ci Kcc K cr Bc T u (c K ri K rc K rr Br T u (r = f c f r, (19 0 B c B r 0 λ 0 where the global block matrices B c = diag{b c,δm,l } and B r = diag{b r,δm,l } are split into local ones defined over the vector representation spaces of traces on the interface Γ kl = γ m,k = :

7 B c,δm,l = FETI-DP for Mortar FE for 4th Order Problem 589 ( (B (r 1 B (c, (B (r 1 B γ (c m,k, (20 and B r,δm,l is defined in (15. Here K cc is a block built of K (l cc taking into account the continuity of dofs at crosspoints, λ = {(B (r T }λ, and K rr and K ii are block diagonal matrices as in (15. Next we eliminate the unknowns related to the interior nodes and crosspoints i.e. u (i, u (c in (19 and we get Ŝu (r + ˆB T λ = ˆf r, ˆBu (r + Ŝcc λ = ˆf c, ( 1 Kir where the matrices are defined as follows: Ŝ = K rr (K ri K rc K i&c K ( ( cr Kir 0 1 ˆB = B r (0 B c K i&c K i&c = ( Kii K ic such that K ci Kcc 1, and Ŝcc = (0 Bc K i&c (21 K cr Bc T with. We now eliminate u (r and we end up with finding λ M(Γ F( λ = d, (22 where d = f c ˆBŜ 1 f r and F = Ŝcc ˆBŜ 1 ˆBT. Next we introduce the following parallel preconditioner: M 1 = B rs rrbr T where S rr = diag{s rr (l } N l=1 with S rr (l = (K rr (l K (l ri (K(l ii 1 K (l ir, i.e. S(l rr is the respective submatrix of the Schur matrix S (l over Ω l. Then in the case of Neumann-Dirichlet ordering we have that the condition number κ(m 1 F is bounded by (1 + log(h/h 2 and in the general case by (1 + log(h/h 4., References [1] C. Bernardi, Y. Maday, and A.T. Patera. A new nonconforming approach to domain decomposition: the mortar element method. In Nonlinear Partial Differential Equations and Their Applications. Collège de France Seminar, Vol. XI (Paris, , volume 299 of Pitman Res. Notes Math. Ser., pages Longman Sci. Tech., Harlow, [2] S.C. Brenner. The condition number of the Schur complement in domain decomposition. Numer. Math., 83(2: , [3] N. Dokeva, M. Dryja, and W. Proskurowski. A FETI-DP preconditioner with a special scaling for mortar discretization of elliptic problems with discontinuous coefficients. SIAM J. Numer. Anal., 44(1: , [4] M. Dryja and W. Proskurowski. On preconditioners for mortar discretization of elliptic problems. Numer. Linear Algebra Appl., 10(1-2:65 82, [5] M. Dryja and O.B. Widlund. A FETI-DP method for a mortar discretization of elliptic problems. In Recent developments in domain decomposition methods (Zürich, 2001, volume 23 of Lect. Notes Comput. Sci. Eng., pages Springer, Berlin, [6] C. Farhat, M. Lesoinne, and K. Pierson. A scalable dual-primal domain decomposition method. Numer. Linear Algebra Appl., 7(7-8: , 2000.

8 590 L. Marcinkowski, N. Dokeva [7] H.H. Kim and C.-O. Lee. A preconditioner for the FETI-DP formulation with mortar methods in two dimensions. SIAM J. Numer. Anal., 42(5: , [8] A. Klawonn, O.B. Widlund, and M. Dryja. Dual-primal FETI methods for three-dimensional elliptic problems with heterogeneous coefficients. SIAM J. Numer. Anal., 40(1: , [9] J. Mandel and R. Tezaur. On the convergence of a dual-primal substructuring method. Numer. Math., 88(3: , [10] L. Marcinkowski. A mortar element method for some discretizations of a plate problem. Numer. Math., 93(2: , [11] D. Stefanica and A. Klawonn. The FETI method for mortar finite elements. In Eleventh International Conference on Domain Decomposition Methods (London, 1998, pages DDM.org, Augsburg, 1999.