A two-level Schwarz preconditioner for heterogeneous problems

Transcription

1 A two-eve Schwarz preconditioner for heterogeneous probems V. Doean 1, F. Nataf 2, R. Scheich 3 and N. Spiane 2 1 Introduction Coarse space correction is essentia to achieve agorithmic scaabiity in domain decomposition methods. Our goa here is to buid a robust coarse space for Schwarz type preconditioners for eiptic probems with highy heterogeneous coefficients when the discontinuities are not just across but aso aong subdomain interfaces, where cassica resuts break down [3, 6, 15, 9]. In previous work, [7], we proposed the construction of a coarse subspace based on the ow-frequency modes associated with the Dirichet-to-Neumann (DtN) map on each subdomain. A rigorous anaysis was recenty provided in [2]. Simiar ideas to buid stabe coarse spaces, based on the soution of oca eigenvaue probems on entire subdomains, can be found in [4], and even traced back to simiar ideas for agebraic mutigrid methods in [1]. However, we wi argue beow that the DtN coarse space presented here is better designed to dea with coefficient variations that are stricty interior to the subdomain, being as robust as, but eading to a smaer dimension than the coarse space anaysed in [4]. The robustness resut that we obtain, generaizes the cassica estimates for overapping Schwarz methods to the case where the coarse space is richer than just the constant mode per domain [8], or other cassica coarse spaces (cf. [15]). The anaysis is inspired by that in [4, 13] and cruciay uses the framework of weighted Poincaré inequaities, introduced in [12, 10] and successfuy appied aso to other methods in [11, 14]. Laboratoire J.A. Dieudonné, CNRS UMR 6621, Nice Cedex 02, France. doean@unice.fr Laboratoire J.L. Lions, CNRS UMR 7598, Université Pierre et Marie Curie, Paris, France. nataf@ann.jussieu.fr,spiane@ann.jussieu.fr Department of Mathematica Sciences, University of Bath, Bath BA2 7AY, United Kingdom, R.Scheich@maths.bath.ac.uk 1

2 2 V. Doean, F. Nataf, R. Scheich and N. Spiane 2 Two-eve Schwarz method with DtN coarse space We consider the variationa formuation of a second order, eiptic boundary vaue probem with Dirichet boundary conditions: Find u H0 1 (Ω), for a given domain Ω R d (d = 2 or 3) and a source term f L 2 (Ω), such that a(u, v) α(x) u v = fv (f, v), v H0 1 (Ω), (1) Ω and the diffusion coefficient α = α(x) is a positive piecewise constant function that may have arge variations within Ω. We consider a discretization of the variationa probem (1) with continuous, piecewise inear finite eements (FE). For a shape reguar, simpicia trianguation T h of Ω, the standard space of continuous and piecewise inear functions (w.r.t T h ) is then denoted by V h. The subspace of functions from V h that vanish on the boundary of Ω is denoted by V h,0. The discrete FE probem that we want to sove is: Find u h V h,0 such that Ω a(u h, v h ) = (f, v h ), v h V h,0. (2) Given the usua noda basis {φ i } n i=1 for V h,0 consisting of hat functions with n := dim(v h,0 ), (2) can be compacty written as Au = f, with A ij := a(φ j, φ i ) and f i = (f, φ i ), i, j = 1,...,n, (3) where u and f are respectivey the vector of coefficients corresponding to the unknown FE function u h in (2) and to the r.h.s function f. Two-eve Schwarz type methods for (2) are now constructed by choosing an overapping decomposition {Ω j } J of Ω with a subordinate partition of unity {χ j } J, as we as a suitabe coarse subspace V H V h,0. In practice the overapping subdomains Ω j can be constructed automaticay given the system matrix A by using a graph partitioner, such as METIS, and adding on a number of ayers of fine grid eements to the resuting nonoverapping subdomains. A suitabe partition of unity can be constructed from the geometric information of the fine grid. For more detais see e.g. [15] or [2]. We assume that each point x Ω is contained in at most N 0 subdomains Ω j. The crucia ingredient to obtain robust two-eve methods for probems with heterogeneous coefficients is the choice of coarse space V H V h,0. Let us assume for the moment that we have such a space V H and a restriction operator R 0 from V h,0 to V H and define restriction operators R j from functions in V h,0 to functions in V h,0 (Ω j ), or from vectors in R n to vectors in R dim V h,0(ω j), by setting (R j u)(x i ) = u(x i ) for every grid point x i Ω j. The two-eve overapping additive Schwarz preconditioner for (3) is then simpy M 1 AS,2 = J j=0 RT j A 1 j R j where A j := R j ARj T, j = 0,...,J. (4)

3 Two-eve Schwarz for heterogeneous probems 3 In the cassica agorithm V H consists simpy of FEs on a coarser trianguation T H of Ω and R H is the canonica restriction from V h,0 to V H, eading to a fuy scaabe iterative method with respect to mesh/probem size (provided the overap size is proportiona to the coarse mesh size H). However, unfortunatey this preconditioner is not robust to strong variations in the coefficient α. We wi now present a new, competey oca approach to construct a robust coarse space, as we as an associated restriction operator using eigenvectors of oca Dirichet-to-Neumann maps, proposed in [7]. We start by constructing suitabe oca functions on each subdomain Ω j that wi then be used to construct a basis for V H. To this end, et us fix j {1,...,J} and first consider at the continuous eve the Dirichet-to-Neumann map DtN j on the boundary of Ω j. Let Γ j := Ω j and et v Γ : Γ j R be a given function, such that v Γ Ω = 0 if Γ j Ω. We define DtN j (v Γ ) := α v ν j, Γj where ν j is the unit outward norma to Ω j on Γ j, and v satisfies div(α v) = 0 in Ω j, v = v Γ on Γ. (5) The function v is the α harmonic extension of the boundary data v Γ to the interior of Ω j. To construct the (oca) coarse basis functions, we now find the ow frequency modes of the Dirichet-to-Neumann operator DtN j with respect to the weighted L 2 norm on Γ j, i.e. the smaest eigenvaues of DtN j (v (j) Γ ) = λ(j) αv (j) Γ. (6) Then we extend each of these modes v (j) Γ α harmonicay to the whoe domain and et v (j) be its extension. This is equivaent to the Stekov eigenvaue probem of ooking for the pair (v (j), λ (j) ) which satisfies: div(α v (j) ) = 0 in Ω j and α v(j) ν j = λαv (j) on Γ j. (7) The variationa formuation of (7) is to find (v (j), λ (j) ) H 1 (Ω j ) R such that α v (j) w = λ (j) tr j α v (j) w, w H 1 (Ω j ), (8) Ω j Γ j where tr j α(x) :=im y Ωj x α(y). To discretize this generaized eigenvaue probem, we consider for a v, w H 1 (Ω j ) the biinear forms a j (v, w) := α v w and m j (v, w) := tr j αvw Ω j Γ j

4 4 V. Doean, F. Nataf, R. Scheich and N. Spiane and restrict (8) to the FE space V h (Ω j ). The coefficient matrices associated with the variationa forms a j and m j are A (j) k := α φ k φ Ω j and M (j) k := tr j αφ k φ, Γ j where φ k and φ are any two noda basis functions for V h (Ω j ) associated with vertices of T h contained in Ω j. Then the FE approximation to (8) in matrix notation is A (j) v (j) = λ (j) M (j) v (j) (9) where v (j) R nj, n j := dimv h (Ω j ), denotes the degrees of freedom of the FE approximation to v (j) in V h (Ω j ). Let the n j eigenpairs (λ (j),v ) nj =1 corresponding to (9) be numbered in increasing order of λ (j). Since M (j) k 0 ony if φ k and φ are associated with the n Γ vertices of T h that ie on Γ j, it is easy to see that at most n Γ of the eigenvaues λ (j) are finite. Moreover, the smaest eigenvaue λ (j) 1 = 0 with constant eigenvector and the set of eigenvectors {v } nj =1 can be chosen so that they are A (j) orthonorma. The oca coarse space is now defined as the span of the FE functions v (j) V h (Ω j ), m j n Γ, corresponding to the first m j eigenpairs of (9). For each subdomain Ω j, we choose the vaue of m j such that λ (j) < diam(ω j ) 1, for a m j, and λ (j) m diam(ω j+1 j) 1. We wi see in the anaysis in the next section why this is a sensibe choice. Using the partition of unity {χ j } J, we now combine the oca basis functions constructed in the previous section to obtain a conforming coarse space V H V h,0 on a of Ω. The new coarse space is defined as ( ) } V H := span {I h χ j v (j) : 1 j J and 1 m j, (10) where I h is the standard noda interpoant onto V h,0 (Ω). The dimension of V H is J m ( j. By construction each of the functions I h χj v (j) ) Vh0, so that as required V H V h,0. The transfer operator R 0 from V h0 to V H is defined in a canonica way by setting R0 Tu H(x i ) = u H (x i ), for a u H V H and for a vertices x i of T h. We wi see in the next section that under some mid assumptions on the variabiity of α this choice of coarse space eads to a scaabe and coefficientrobust domain decomposition method with supporting theory. 3 Conditioning anaysis To anayse this method et us first define the boundary ayer Ω j for each Ω j that is overapped by neighbouring domains, i.e.

5 Two-eve Schwarz for heterogeneous probems 5 Ω j := {x Ω j : χ j (x) < 1}. We assume that this ayer is uniformy of width δ j, in the sense that it can be subdivided into shape reguar regions of diameter δ j, and that the trianguation T h resoves it. This aso guarantees that it is possibe to find a partition of unity such that χ j = O(1) and χ j = O(δ 1 j ). We now state the key assumption on the coefficient distribution α(x). Assumption 1 We assume that, for each j = 1,...,J, there exists a set α(x) X j Γ j (not necessariy connected) such that (i) max x,y Xk α(y) = O(1) and (ii) there exists a path P y from each y Ω j to X j, such that α(x) is an increasing function aong P y (from y to X j ). Lemma 1 (weighted Poincaré inequaity [10]). Let Assumption 1 hod. α v v Xj 2 C P δ j α v 2, for a v V h (Ω j ), Ω j where v Xj := 1 X j X j v. Ω j Remark 1. Note that Assumption 1 is reated to the cassica notion of quasimonotonicity coined in [3]. It ensures that the constant C P in the Poincarétype inequaity in Lemma 1, as we as a the other (hidden) constants beow are independent of the vaues of the coefficient function α(x). The constants may however depend ogarithmicay or ineary on δ j /h. This depends on the geometry and shape of the paths P y and on the size and shape of the set X j. For more detais see [2] and [12, 10]. The foowing proposition is the centra resut in our anaysis. It proves the stabiity and a weak approximation property for a oca projection onto the span of the first m j eigenvectors. Proposition 1. Let Assumption 1 hod, and for any u V h (Ω j ), define the projection Π j u := m j =1 a j(v (j), u)v(j). Then Π j u a,ωj u a,ωj and (11) u Π j u 0,α,Ω j c j (m j )δ j u a,ωj. (12) where c j (m j ) := C 2 P + ( δ j λ (j) m j+1) 1. Proof. Theorem 3.2 in [2]. As usua (cf. [15]), the foowing condition number bound can then be obtained via abstract Schwarz theory by constructing a stabe spitting. Theorem 2. Let Assumption 1 be satisfied. Then the condition number of the two-eve Schwarz agorithm with the coarse space V H based on oca DtN maps and defined in (10) can be bounded by

6 6 V. Doean, F. Nataf, R. Scheich and N. Spiane κ(m 1 AS,2A) J max {c j(m j )} CP 2 + J max ( δ j λ (j) m j+1) 1. The hidden constant is independent of h, δ j, diam(ω j ), and α. Proof. This is Theorem 3.5 in [2]. The stabe spitting for a function u V h,0 is constructed using the projections Π j, j = 1,..., J, in Proposition 1 to define the coarse quasi-interpoant u 0 := I h ( J χ j Π j u Ωj ) V H. (13) If we now choose u j := I h (χ j (u Π j u)) V h,0 (Ω j ), then u = J j=0 u j and J α u j 2 j=0 Ω max J {c j(m j )} α u 2 Ω For detais see [2]. Remark 2. Note that by choosing the number m j of modes per subdomain such that λ (j) m j+1 diam(ω j) 1, as stated in Section 2, we have ( κ(m 1 AS,1 A) CP 2 + max J ) diam(ω j ). δ j Hence, provided the constant C P is uniformy bounded, independenty of any jumps in the coefficients, we retrieve the cassica estimate for the twoeve additive Schwarz method independenty of any variations of coefficients across or aong subdomain boundaries. 4 Numerica resuts We choose Ω = (0, 1) 2 and discretize (1) on reguar grid with m m eements. We appy a homogeneous Dirichet boundary condition u = 0 on the eft hand side boundary and homoegenous Neumann boundary conditions u ν = 0 on the remainder. We use the METIS partitioner to spit the domain into 16 irreguar subdomains as shown in Figure 1. Then we construct the overapping partition using Freefem++ [5] by extending each subdomain by one ayer of eements on a or the boundary. As the coarse space we use the DtN coarse space described in Section 2 with m j chosen such that λ m (j) j < diam(ω j ) 1 λ (j) m j+1, for a j = 1,...,16 (abeed D2N beow). We compare this preconditioner with the one-eve additive Schwarz method (abeed NONE beow) and the two-eve method with

7 Two-eve Schwarz for heterogeneous probems 7 partition of unity coarse space, i.e. choosing m j = 1, for a j = 1,...,16 (abeed POU beow). To confirm in some sense the optimaity of our choice for m j, we aso incude resuts with the DtN coarse space choosing m j +1 and max{1, m j 1} basis functions per subdomain (abeed D2N+ and D2N-, respectivey). We use the preconditioners within a conjugate gradient iteration and terminate when the residua has been reduced by a factor In the first test case, we choose m = 80 and α to be a reaization of a ognorma distribution with exponentia covariance function (variance σ 2 = 4 and correation ength λ = 4/m) and mean(og α) = 3 (cf. Figure??). In Figure 3 we pot u ū (where ū is the soution of (3) obtained via a direct sover) against the iteration count. We compare three methods: AS : the one eve preconditioner M 1 AS,1 = J RT j A 1 j R j. AS + Z NICO : the two eve preconditioner defined by (4) with a coarse grid which consists simpy of constant functions on each subdomain weighted by a partition of unity. AS + Z D2N : the two eve preconditioner defined by (4) with the new coarse grid we have introduced. The AS and AS + Z NICO methods require roughy the same number of iterations (89 versus 92 iterations) whereas the new AS + Z D2N stands out reducing the number of iterations to 38. Finay, in Figure 4 we show that the criterion for the number m j of eigenmodes that we seect in each subdomain is somewhat optima since adding one has hardy any impact on performance whie removing one has strong negative impact. References [1] T. Chartier, R. D. Fagout, V. E. Henson, J. Jones, T. Manteuffe, S. Mc- Cormick, J. Ruge, and P. S. Vassievski. Spectra AMGe (ρamge). SIAM J. Sci. Comput., 25(1):1 26, [2] V. Doean, F. Nataf, Scheich R., and N. Spiane. Anaysis of a twoeve Schwarz method with coarse spaces based on oca Dirichet to Neumann maps [3] M. Dryja, M. V. Sarkis, and O. B. Widund. Mutieve Schwarz methods for eiptic probems with discontinuous coefficients in three dimensions. Numer. Math., 72(3): , [4] J. Gavis and Y. Efendiev. Domain decomposition preconditioners for mutiscae fows in high contrast media: Reduced dimension coarse spaces. Mutiscae Modeing & Simuation, 8(5): , [5] Frédéric Hecht. FreeFem++. Numerica Mathematics and Scientific Computation. Laboratoire J.L. Lions, Université Pierre et Marie Curie, edition, 2010.

8 8 V. Doean, F. Nataf, R. Scheich and N. Spiane IsoVaue Fig subdomains AS AS + Z Nico AS + Z D2N Fig. 2 og(α) (0 < α < ) Error AS AS + AS + Z NICO Z D2N max(m j 1, 1) 50 m j m j Iteration count Fig. 3 Convergence Fig. 4 Optimaity of the criterion Number of iterations needed when we add or remove one mode per subdomain as compared to the number m j given by the automatic method [6] J. Mande and M. Brezina. Baancing domain decomposition for probems with arge jumps in coefficients. Math. Comp., 65: , [7] F. Nataf, H. Xiang, and V. Doean. A two eve domain decomposition preconditioner based on oca Dirichet-to-Neumann maps. C. R. Mathématique, 348(21-22): , [8] R. A. Nicoaides. Defation of conjugate gradients with appications to boundary vaue probems. SIAM J. Numer. Ana., 24(2): , [9] C. Pechstein and R. Scheich. Scaing up through domain decomposition. App. Ana., 88(10-11): , [10] C. Pechstein and R. Scheich. Weighted Poincaré inequaities. Technica Report NuMa-Report , Institute of Computationa Mathematics, Johannes Keper University, Linz, December submitted. [11] C. Pechstein and R. Scheich. Anaysis of FETI methods for mutiscae PDEs - Part II: Interface variation. Numer. Math., Pubished onine 21 February [12] C. Pechstein and R. Scheich. Weighted Poincaré inequaities and appications in domain decomposition. In Y. Huang, R. Kornhuber, O. Widund, and J. Xu, editors, Domain Decomposition Methods in Science and Engineering XIX, voume 78 of LNCSE, pages Springer, 2011.

9 Two-eve Schwarz for heterogeneous probems 9 [13] R. Scheich, P. S. Vassievski, and L. T. Zikatanov. Weak approximation properties of eiptic projections with functiona constraints. Technica Report LLNL-JRNL , Lawrence Livermore Nationa Lab, [14] R. Scheich, P.S. Vassievski, and L.T. Zikatanov. Mutieve methods for eiptic probems with highy varying coefficients on non-aigned coarse grids. SIAM J Numer Ana, Accepted subject to minor corrections. [15] A. Tosei and O. B. Widund. Domain decomposition methods agorithms and theory. Springer, Berin, 2005.