UNCORRECTED PROOF. Domain Decomposition Methods for the Helmholtz 2 Equation: A Numerical Investigation 3. 1 Introduction 7

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1 Domain Decomposition Methods for the Helmholtz 2 Equation: A Numerical Investigation 3 Martin J. Gander and Hui Zhang 4 University of Geneva martin.gander@unige.ch 5 2 hui.zhang@unige.ch 6 Introduction 7 We are interested in solving the Helmholtz equation 8 { u(x,y,z) k 2 (x,y,z) u(x,y,z) =g(x,y,z), (x,y,z) Ω, () n u(x,y,z) ik(x,y,z) u(x,y,z) =0, (x,y,z) Ω, where k := 2π f /c is the wavenumber with frequency f R and c := c(x,y,z) is 9 the velocity of the medium, which varies in space. The geophysical model SEG 0 SALT is used as a benchmark problem on which we will test some existing domain decomposition methods in this paper. In this model, the domain Ω is defined as 2 (0,3,520) (0,3,520) (0,4,200) m 3, the velocity is described as piecewise con- 3 stants on cells and varies from,500 to 4,500 m/s, and the source 4 g is a Dirac function at the point (6,000,6,760,0). 5 To discretize the problem () on a coarser mesh, the velocity is sub-sampled to 6 less number of cells such that every cell has a constant velocity and contains one 7 or more mesh elements. Then the problem () is discretized with Q finite elements 8 (i.e. trilinear local basis functions on brick elements). 9 We first test the direct solver A\b in Matlab; the results are listed in Table where 20 nw is the number of wavelength along the x-direction at the lowest velocity. At f = 2, 2 the direct solver runs out of memory after 6 h on a computer with 64 GB of mem- 22 ory. The inefficiency in both memory and time of the direct solver for large scale 23 problems calls for cheaper iterative methods. For a review of current iterative meth- 24 ods for the Helmholtz equation, we refer to [6]. In this work, we focus on domain 25 decomposition methods which are easily parallelized Overview of Some Existing Methods 27 Due to the indefiniteness of the Helmholtz equation, the classical Schwarz method 28 with Dirichlet transmission conditions fails to converge. As a remedy, [5] introduced 29 R. Bank et al. (eds.), Domain Decomposition Methods in Science and Engineering XX, Lecture Notes in Computational Science and Engineering 9, DOI 0.007/ , Springer-Verlag Berlin Heidelberg 203 Page 223

2 Martin J. Gander and Hui Zhang Table. Test of the direct solver (backslash in Matlab) f /4 /2 2 nw mesh CPU.28s 27.5s 829.9s > 6h first-order absorbing transmission conditions to replace the Dirichlet transmission 30 conditions. This type of interface condition was also adopted in [7] to regularize 3 subdomain problems. More general local transmission conditions of zero or second 32 order were proposed and analyzed in [0, ] with parameters optimized for acceler- 33 ating convergence. More advanced and even non-local transmission conditions can 34 be used, see [3, 2, 8], and also [2, 3] in this volume. In this paper, however, we 35 will restrict ourselves to local transmission conditions. 36 Another remedy is to modify the usual coarse problem, which probably origi- 37 nated from the multigrid context, first suggested by Achi Brandt and presented in 38 [9]. In their paper [7], Farhat et al. used plane waves on the interface as basis of the 39 coarse space. The idea turns out to be very successful and was followed by Farhat 40 et al. [8], Kimn and Sarkis [5], and Li and Tu [7], and will also be used for the 4 optimized Schwarz methods in this paper. Note that, however, the coarse problem 42 does not change the underlying subdomain problems. 43 In the following paragraphs, we will give a brief introduction to these methods at 44 the (almost) continuous level The Non-overlapping Methods 46 We partition the domain into non-overlapping subdomains denoted by Ω := i Ω i, 47 and we call the set of points shared by more than two subdomains (or shared by two 48 subdomains and the outer boundary Ω) corners. In three dimensions, this includes 49 vertices and edges. We call all the points shared by exactly two subdomains the 50 interface Γ, and in particular a connected component of the interface shared by Ω i 5 and Ω j is called interface segment Γ ij. 52 If we know the Neumann, Dirichlet or Robin data (denoted by λ ) of the exact 53 solution on the interface, then we can recover the exact solution from the corre- 54 sponding boundary value problems defined on subdomains (as long as they are well- 55 posed) with continuous constraints at corners. Since on every subdomain there is a 56 recovered solution that gives Dirichlet, Neumann or Robin traces on the interface, 57 we expect for each interface segment Γ ij the traces from Ω i and Ω j to be equal. The 58 above process indeed sets up an equation, denoted by Fλ = d, for the interface data 59 λ of the exact solution. For the Helmholtz equation, an additional coarse problem is 60 introduced such that (I FQ(Q FQ) Q )Fλ =(I FQ(Q FQ) Q )d is solved, 6 where the columns of Q are traces of plane waves on the interface. 62 From the above point of view, we summarize some existing non-overlapping 63 domain decomposition methods in Table 2. The (first-order) absorbing boundary data 64 Page 224

3 DDM for Helmholtz: A Numerical Investigation is defined as λ := n u iku. The lumped preconditioner is the stiffness submatrix 65 A ΓΓ corresponding to the interface. The first three methods share interface data (up 66 to a sign for the normal derivative) on their common interface segments, and are 67 therefore one-field methods. This is in contrast to the last method, since optimized 68 Schwarz methods have two sets of unknowns on each interface segment, and thus 69 belong to the class of two-field methods. Note also that we do not have suitable 70 preconditioners for the last two methods, which can be a subject for future study. 7 Table 2. The non-overlapping methods Algorithms Unknowns Matching Precond. FETI-DPH ([8]) Neumann Dirichlet DtN/lumped BDDC-H ([7]) Dirichlet Neumann NtD FETI-H ([7]) Absorbing Dirichlet (none) Optimized Schwarz ([0]) two-field Robin two-field Robin (none) 2.2 The Overlapping Methods 72 We partition the domain into overlapping subdomains. We will use the substructured 73 form 3 as for the non-overlapping methods in Sect. 2.. Note that in an overlapping 74 setting, subdomains can not share the same interface data, since the interfaces are 75 in different locations, and therefore all overlapping methods are in some sense two 76 field methods, like the non-overlapping optimized Schwarz methods. The interface 77 data used (both as unknowns and matching conditions) and related references are: 78 Dirichlet [6], absorbing [4, 5], Neumann [4], Robin [9]. A coarse problem as in 79 Sect. 2. is adopted but without corner constraints Numerical Experiments 8 All the experiments were done in Matlab with sequential codes. We use GMRES with 82 zero initial guess to solve the substructured systems until the relative residual is less 83 than 0 6 or the maximum iteration number is attained. The domain is partitioned in 84 a Cartesian way. If we vary the mesh size, then the velocity in () is sub-sampled on 85 the coarsest mesh of We introduce the following acronyms: 87 t2. t2.2 t2.3 t2.4 t2.5 FL/FD: FETI-DPH with the lumped/dtn preconditioner 88 FH: FETI-H with corner constraints 89 O0/O2: non-overlapping optimized Schwarz of zero/second order 90 3 Though most of the overlapping methods in the literature are not in this form, we found by numerical experiments it may be cheaper in both time and memory. Page 225

4 Martin J. Gander and Hui Zhang OD/ON/OR: overlapping method with Dirichlet/Neumann/absorbing data 9 OO0/OO2: overlapping optimized Schwarz of zero/second order 92 For the overlapping methods, the overlapping region has a thickness of two mesh 93 elements and the matching conditions are imposed on faces, edges and vertices, re- 94 spectively, without repeats on any degrees of freedom. Due to the absence of relevant 95 results, the parameters for the optimized Schwarz methods are not respecting over- 96 lapping (except OO0), coarse problem and medium heterogeneity. The plane waves 97 used are along six directions that are normal to the x-y, y-z and z-x planes, respec- 98 tively. 99 We found that all the methods outperform the direct solver in CPU time (see 00 Table ) on the mesh. We are interested in how the convergence of these 0 methods depends on the frequency f in (), the mesh size h,thepartition N x N y N z 02 or the subdomain size H and medium heterogeneity. At f = the domain contains 03 nine wavelength along the x-direction, which corresponds to the problem on the unit 04 cube with the wavenumber 8π. 05 In the following tables, the numbers outside/inside parentheses are the iteration 06 numbers with/without plane waves, respectively, and a bar is used instead of when the maximum iteration number is reached. We use e/w to represent the number 08 of elements per wavelength at the lowest velocity. The smallest iteration numbers 09 among the non-overlapping methods and those among the overlapping methods are 0 in bold. Note that for the FETI-DPH method with DtN preconditioner the amount of work per iteration is about.5 times that for the others, and construction of the 2 preconditioner also leads to double LU factorizations in the setup stage. 3 In Tables 3 and 4, we increase the frequency with fh or f 3 h 2 [] kept constant. Table 3. Dependence on the frequency ( fh=constant) f FL FD FH O0 O2 OD OR ON OO0 OO2 partition 3 3 t (5) 4 (8) 9 (5) 5 (2) 8 (4) 8 (20) 8 (2) 9 (20) 7 (5) 6 (4) t (30) 9 (2) 8 (33) 29 (34) 9 (20) 23 (34) 2 (5) 24 (37) 2 (7) (3) 44 (5) 20 (23) 75 (93) 43 (48) 25 (25) 5 (58) 7 (7) 57 (66) 22 (25) 4 (5) t3.4 partition scaling with mesh: H/h = 8(seealsothefirstrowfor f = 4 ) t (46) 5 (30) 0 (73) 7 (7) 0 (50) 4 (73) (33) 2 (03) 8 (55) 8 (5) 9 (83) 7 (-) (-) 2 (-) 2 (-) 27 (-) 5 (74) 52 (-) 6 (-) 5 (-) t3.6 t3. partition scaling with mesh: H/h = 6 (see also the second row for f = 2 ) 39 (27) 32 (03) 74 (-) 59 (3) 27 (39) 76 (7) 26 (38) 4 (-) 26 (53) 22 (32) t3.7 We see that more iterations are usually needed for larger frequency except in the 5 middle of Table 4. 6 In Table 5, the frequency is fixed and the mesh is refined. From the table, the 7 iteration numbers with plane waves almost remain constant. 8 4 Page 226

5 DDM for Helmholtz: A Numerical Investigation Table 4. Dependence on the frequency ( f 3 h 2 =constant) f FL FD FH O0 O2 OD OR ON OO0 OO2 t4. partition 3 3 (see also the first row in Table 3 for f = 0.25) (25) 6 () 4 (25) 30 (33) 8 (2) 8 (29) (4) 9 (32) 9 (5) 9 (3) (4) (5) 33 (49) 37 (42) 25 (26) 38 (46) 6 (7) 39 (50) 5 (20) 3 (4) partition scaling with mesh: H/h = 8(seealsothefirstrowinTable4forf= 0.25) (36) 5 (23) 0 (54) 5 (58) 9 (40) 2 (60) 0 (29) 3 (73) 7 (40) 7 (40) (27) 5 (00) 9 (49) 4 (56) 8 (2) 4 (60) (65) 20 (-) 7 (23) 7 (7) partition scaling with mesh: H/h = 6 (see also the first row for f = 0.40) (89) 8 (53) 8 (9) 43 (25) 8 (75) 33 (3) 6 (35) 36 (2) 3 (75) 3 (75) Table 5. Dependence on the mesh size ( f = 4 ) e/w FL FD FH O0 O2 OD OR ON OO0 OO2 partition 3 3 (seealsothefirstrowintable4fore/w = 0) 20 0 (9) 5 (9) 3 (20) 7 (26) 9 (7) 4 (28) (5) 3 (27) 8 (6) 6 (6) 40 5 (25) 6 (0) 8 (25) 2 (32) (20) 2 (39) 5 (9) 9 (36) 9 (7) 8 (7) partition H/h = 8(seealsothefirstrowinTable4fore/w = 0) 20 7 (2) 5 (2) 0 (32) 4 (47) 8 (32) 0 (46) 9 (25) 0 (44) 7 (29) 6 (30) 40 6 (9) 4 (3) 9 (36) 4 (92) 7 (63) 9 (90) 9 (46) 9 (9) 7 (56) 6 (59) partition H/h = 6 (see also the first row for e/w = 20) 40 (34) 6 (5) 4 (47) 7 (60) 0 (38) 5 (63) 2 (28) 3 (52) 7 (33) 7 (35) Next, we compare the iteration numbers for different partitions with both the 9 frequency and the mesh size fixed in Table 6. One can see that with plane waves H Table 6. Dependence on the partition FL FD FH O0 O2 OD OR ON OO0 OO2 H 0 f = 2, mesh and velocity and H 0 partition (30) 9 (2) 8 (33) 28 (35) 9 (2) 22 (34) 2 (5) 23 (37) (7) (4) 2 8 (47) 5 (30) 0 (73) 6 (72) 9 (5) 4 (75) (34) 2 (05) 8 (62) 7 (57) 4 4 (22) 4 (2) 7 (48) 0 (95) 7 (72) 7 (97) 8 (52) (-) 6 (83) 5 (78) f =, mesh and velocity and H 0 partition (54) 22 (24) 79 (97) 45 (49) 26 (26) 54 (6) 7 (8) 60 (69) 22 (26) 5 (6) 2 43 (33) 35 (09) 82 (-) 63 (7) 28 (40) 82 (76) 27 (39) 36 (-) 28 (56) 24 (34) 4 0 (84) 8 (-) 4 (-) 26 (-) 6 (40) 32 (-) 7 (-) - (-) 25 (-) 22 (-) N x f =, mesh and velocity and partition N x 8 7 (25) 79 (75) 7 (84) 66 (70) 28 (28) 94 (99) 23 (24) 00 (04) 5 (46) 23 (25) 6 84 (-) 92 (99) - (-) 3 (37) 45 (47) - (-) 46 (47) - (-) 72 (8) 43 (45) 32 - (-) - (-) - (-) 72 (73) 87 (90) - (-) 86 (90) 82 (88) 48 (36) 84 (87) t4.2 t4.3 t4.4 t4.5 t4.6 t5. t5.2 t5.3 t5.4 t5.5 t5.6 t6. t6.2 t6.3 t6.4 t6.5 t6.6 t6.7 t6.8 t6.9 t6.0 t6. t6.2 t Page 227

6 Martin J. Gander and Hui Zhang using more subdomains can both increase and decrease the iteration numbers. It is 2 interesting that for the strip-wise partition only the methods based on transmission 22 conditions (O0, O2, OR, OO0 and OO2) work reliably, though with substantial iter- 23 ation numbers, and the plane waves do not help much. 24 Last, we study the influence of the heterogeneity in the velocity. The experiments 25 are carried out on artificial velocity models to have high contrasts. The frequency is 26 fixed as f = 2. The lowest velocity is fixed as c min =,500 and different levels of 27 highest velocity c max = ρc min are considered. It can be seen from Table 7 that the 28 iteration numbers vary only little. Table 7. Influence of medium heterogeneity ρ FL FD FH O0 O2 OD OR ON OO0 OO2 mesh , partition 8 andc = c min,c max on subdomains 58 (76) 37 (46) 83 (94) 60 (64) 28 (29) 70 (8) 27 (26) 69 (79) 37 (44) 24 (24) (36) 42 (58) 30 (37) 37 (55) 26 (3) 37 (53) 27 (29) 63 (75) 5 (26) 3 (22) (36) 49 (58) 33 (37) 45 (55) 26 (3) 43 (53) 29 (30) 7 (75) 9 (26) 7 (22) as above except partition (90) 7 (62) 2 (24) 26 (79) 5 (39) 8 (97) 4 (35) 22 (7) 0 (46) 2 (34) (59) 0 (04) 7 (5) 25 (78) 5 (46) 7 (67) 2 (34) 29 (00) 8 (42) 9 (37) (58) (04) 9 (5) 27 (79) 7 (47) 9 (68) 2 (34) 33 (00) 8 (42) 0 (37) mesh , partition 8 andc = c min,c max on 8 cells 70 (8) 40 (50) 05 (4) 73 (75) 27 (28) 74 (80) 28 (27) 62 (66) 34 (37) 24 (24) (59) 30 (34) 69 (84) 58 (67) 26 (28) 56 (67) 23 (26) 5 (59) 26 (28) 23 (26) (59) 30 (34) 70 (85) 58 (67) 26 (28) 56 (68) 23 (26) 5 (59) 26 (28) 23 (26) mesh , partition 6 6 2andc = c min,c max on cells 2 (05) 8 (65) 6 (44) 34 (96) 9 (4) 24 (2) 7 (37) 25 () 2 (46) 5 (34) (68) 7 (34) 4 (07) 29 (09) 7 (48) 26 () 3 (45) 2 (06) (47) 2 (40) 0 4 (68) 7 (34) 5 (07) 3 (09) 8 (48) 26 (0) 4 (45) 2 (07) (47) 2 (40) mesh , partition 6 6 2andcrandom constants on elements (6) 5 (0) 0 (2) 4 (6) 9 (4) 4 (60) (37) 2 (59) 7 (35) 8 (38) (5) 6 (9) (20) 2 (67) 8 (46) 4 (67) 5 (6) 25 (86) 8 (39) 8 (42) as above except partition (38) 0 (6) 26 (45) 28 (37) 9 (2) 26 (36) 3 (5) 27 (36) 5 (2) 2 (4) 0 2 (7) 6 (8) 5 (20) 8 (33) (2) 6 (35) 5 (23) 6 (42) 7 (7) 8 (9) (7) 6 (8) 6 (2) 5 (39) 9 (24) 8 (40) 6 (3) 7 (52) 8 (20) 9 (22) t7. t7.2 t7.3 t7.4 t7.5 t7.6 t7.7 t7.8 t7.9 t7.0 t7. t7.2 t7.3 t7.4 t7.5 t7.6 t7.7 t Conclusions 30 For the SEG SALT model on the cube domain, we get the following conclusions: 3 among the non-overlapping methods, the FETI-DPH method with DtN precondi- 32 tioner performs best in terms of iteration numbers. Among the overlapping methods, 33 Page 228

7 DDM for Helmholtz: A Numerical Investigation the optimized Schwarz method of second order is usually the best. With a fixed num- 34 ber of plane waves, all the methods can slow down for larger frequencies on properly 35 refined meshes. They also deteriorate for fixed frequency on finer meshes, unless 36 when using plane waves and more subdomains. A smaller subdomain size can both 37 increase and decrease the iteration numbers, and the experiments indicate the exis- 38 tence of some optimal choice. For strip-wise partitions, only the methods based on 39 transmission conditions work well, and plane waves do not help much. We also find 40 the performance of all the method is only little affected by the heterogeneity in the 4 velocity we considered, but other kinds of heterogeneity still need to be investigated. 42 Acknowledgments The authors thank Paul Childs for providing the velocity data of the geo- 43 physical SEG SALT model. This work was partially supported by the University of Geneva. 44 The second author was also partially supported by the NSFC Tianyuan Mathematics Youth 45 Fund Bibliography 47 [] Ivo Babuska, Frank Ihlenburg, Ellen T. Paik, and Stefan A. Sauter. A general- 48 ized finite element method for solving the Helmholtz equation in two dimen- 49 sions with minimal pollution. Comput. Methods Appl. Mech. Engrg., 28(3-4): , [2] Yassine Boubendir, Xavier Antoine, and Christophe Geuzaine. New non- 52 overlapping domain decomposition algorithm for Helmholtz equation. In Twen- 53 tieth International Conference on Domain Decomposition Methods,pageinthis 54 volume, [3] Yassine Boubendir, Xavier Antoine, and Christophe Geuzaine. A quasi-optimal 56 non-overlapping domain decomposition algorithm for the Helmholtz equation. 57 J. Comput. Phys., 23(2): , [4] Xiao-Chuan Cai, Mario A. Casarin, Frank W. Elliott, Jr., and Olof B. Wid- 59 lund. Overlapping Schwarz algorithms for solving Helmholtz s equation. In 60 Domain decomposition methods, 0 (Boulder, CO, 997), volume 28 of Con- 6 temp. Math., pages Amer. Math. Soc., Providence, RI, [5] Bruno Després. Domain decomposition method and the Helmholtz problem. In 63 Mathematical and numerical aspects of wave propagation phenomena (Stras- 64 bourg, 99), pages SIAM, Philadelphia, PA, [6] Olivier G. Ernst and Martin J. Gander. Why it is difficult to solve Helmholtz 66 problems with classical iterative methods. In Numerical Analysis of Multiscale 67 Problems. Durham LMS Symposium 200, Springer Verlag, [7] Charbel Farhat, Antonini Macedo, and Michel Lesoinne. A two-level do- 69 main decomposition method for the iterative solution of high frequency exterior 70 Helmholtz problems. Numer. Math., 85(2): , [8] Charbel Farhat, Philip Avery, Radek Tezaur, and Jing Li. FETI-DPH: a dual- 72 primal domain decomposition method for acoustic scattering. J. Comput. 73 Acoust., 3(3): , Page 229

8 Martin J. Gander and Hui Zhang [9] Martin J. Gander. Optimized Schwarz methods for Helmholtz problems. In Do- 75 main decomposition methods in science and engineering (Lyon, 2000), Theory 76 Eng. Appl. Comput. Methods, pages Internat. Center Numer. Meth- 77 ods Eng. (CIMNE), Barcelona, [0] Martin J. Gander, Frédéric Magoulès, and Frédéric Nataf. Optimized Schwarz 79 methods without overlap for the Helmholtz equation. SIAM J. Sci. Comput.,24 80 ():38 60 (electronic), [] Martin J. Gander, Laurence Halpern, and Frédéric Magoulés. An opti- 82 mized Schwarz method with two-sided Robin transmission conditions for the 83 Helmholtz equation. Int. J. Numer. Meth. Fluids, 55(2):63 75, [2] Souad Ghanemi. A domain decomposition method for Helmholtz scattering 85 problems. In Ninth International Conference on Domain Decomposition Meth- 86 ods, pages 05 2, [3] Murthy N. Guddati and Senganal Thirunavukkarasu. Improving the conver- 88 gence rate of Schwarz methods for Helmholtz equation. In Twentieth Inter- 89 national Conference on Domain Decomposition Methods, page in this volume, [4] Jung-Han Kimn and Blaise Bourdin. Numerical implementation of overlapping 92 balancing domain decomposition methods on unstructured meshes. In Domain 93 decomposition methods in science and engineering XVI, volume 55 of Lect. 94 Notes Comput. Sci. Eng., pages Springer, Berlin, [5] Jung-Han Kimn and Marcus Sarkis. Restricted overlapping balancing domain 96 decomposition methods and restricted coarse problems for the Helmholtz prob- 97 lem. Comput. Methods Appl. Mech. Engrg., 96(8):507 54, [6] An Leong and Olof B. Widlund. Extension of two-level Schwarz precondition- 99 ers to symmetric indefinite problems. Technical report, New York University, 200 New York, NY, USA, [7] Jing Li and Xuemin Tu. Convergence analysis of a balancing domain decom- 202 position method for solving a class of indefinite linear systems. Numer. Linear 203 Algebra Appl., 6(9): , [8] Bruno Stupfel. Improved transmission conditions for a one-dimensional do- 205 main decomposition method applied to the solution of the Helmholtz equation. 206 J. Comput. Phys., 229(3):85 874, [9] Shlomo Ta asan. Multigrid methods for highly oscillatory problems. PhDthe- 208 sis, Weizmann Institute of Science, Rehovot, Israel, Page 230