An Iterative Solver for the HPS Discretization Applied to Three Dimensional Helmholtz Problems
Abstract.
This manuscript presents an efficient solver for the linear system that arises from the hierarchical Poincaré–Steklov (HPS) discretization of three dimensional variable coefficient Helmholtz problems. Previous work on the HPS method has tied it with a direct solver. This work is the first efficient iterative solver for the linear system that results from the HPS discretization. The solution technique utilizes GMRES coupled with a locally homogenized block-Jacobi preconditioner. The local nature of the discretization and preconditioner naturally yield the matrix-free application of the linear system. Numerical results illustrate the performance of the solution technique. This includes an experiment where a problem approximately 100 wavelengths in each direction that requires more than a billion unknowns to achieve approximately 4 digits of accuracy takes less than 20 minutes to solve.
Keywords
MSC codes
Get full access to this article
View all available purchase options and get full access to this article.
Acknowledgment.
The authors wish to thank Total Energies for their permission to publish.
References
1.
I. Babuška, F. Ihlenburg, E. T. Paik, and S. A. Sauter, A generalized finite element method for solving the Helmholtz equation in two dimensions with minimal pollution, Comput. Methods Appl. Mech. Engrg., 128 (1995), pp. 325–359, https://doi.org/10.1016/0045-7825(95)00890-X.
2.
I. M. Babuška and S. A. Sauter, Is the pollution effect of the FEM avoidable for the Helmholtz equation considering high wave numbers?, SIAM J. Numer. Anal., 34 (1997), pp. 2392–2423, https://doi.org/10.1137/S0036142994269186.
3.
S. Balay, S. Abhyankar, M. F. Adams, J. Brown, P. Brune, K. Buschelman, L. Dalcin, A. Dener, V. Eijkhout, W. D. Gropp, D. Karpeyev, D. Kaushik, M. G. Knepley, D. A. May, L. C. McInnes, R. T. Mills, T. Munson, K. Rupp, P. Sanan, B. F. Smith, S. Zampini, H. Zhang, and H. Zhang, PETSc Users Manual, Technical report ANL-95/11—Revision 3.15, Argonne National Laboratory, 2021; also available online from https://www.mcs.anl.gov/petsc.
4.
S. Balay, S. Abhyankar, M. F. Adams, J. Brown, P. Brune, K. Buschelman, L. Dalcin, A. Dener, V. Eijkhout, W. D. Gropp, D. Karpeyev, D. Kaushik, M. G. Knepley, D. A. May, L. C. McInnes, R. T. Mills, T. Munson, K. Rupp, P. Sanan, B. F. Smith, S. Zampini, and H. Zhang, PETSc Web page, https://www.mcs.anl.gov/petsc, 2021.
5.
S. Balay, W. D. Gropp, L. C. McInnes, and B. F. Smith, Efficient management of parallelism in object oriented numerical software libraries, in Modern Software Tools in Scientific Computing, E. Arge, A. M. Bruaset, and H. P. Langtangen, eds., Birkhäuser Press, Basel, Switzerland, 1997, pp. 163–202.
6.
A. Bayliss, C. Goldstein, and E. Turkel, On accuracy conditions for the numerical computation of waves, J. Comput. Phys., 59 (1985), pp. 396–404, https://doi.org/10.1016/0021-9991(85)90119-6.
7.
N. N. Beams, A. Gillman, and R. J. Hewett, A parallel shared-memory implementation of a high-order accurate solution technique for variable coefficient Helmholtz problems, Comput. Math. Appl., 79 (2020), pp. 996–1011, https://doi.org/10.1016/j.camwa.2019.08.019.
8.
T. Beck, Y. Canzani, and J. L. Marzuola, Quantitative Bounds on Impedance-to-Impedance Operators with Applications to Fast Direct Solvers for PDEs, https://arxiv.org/abs/2103.14700, 2021.
9.
J. Boyd, Chebyshev and Fourier Spectral Methods, 2nd ed., Dover, Mineola, NY, 2001.
10.
A. Brandt, Multi-level adaptive solutions to boundary-value problems, Math. Comp., 31 (1977), pp. 333–390.
11.
A. Brandt, Rigorous quantitative analysis of multigrid. I. Constant coefficients two-level cycle with \(L^2\)-norm, SIAM J. Numer. Anal., 31 (1994), pp. 1695–1730, https://doi.org/10.1137/0731087.
12.
S. N. Chandler-Wilde, The impedance boundary value problem for the Helmholtz equation in a half-plane, Math. Methods Appl. Sci., 20 (1997), pp. 813–840, https://doi.org/10.1002/(SICI)1099-1476(19970710)20:10%3C813::AID-MMA883%3E3.0.CO;2-R.
13.
T. Chaumont Frelet, Finite Element Approximation of Helmholtz Problems with Application to Seismic Wave Propagation, Thesis, INSA de Rouen, 2015, https://tel.archives-ouvertes.fr/tel-01246244.
14.
O. G. Ernst and M. J. Gander, Why it is difficult to solve Helmholtz problems with classical iterative methods, in Numerical Analysis of Multiscale Problems, Springer, Berlin, 2012, pp. 325–363, https://doi.org/10.1007/978-3-642-22061-6_10.
15.
M. J. Gander and H. Zhang, A class of iterative solvers for the Helmholtz equation: Factorizations, sweeping preconditioners, source transfer, single layer potentials, polarized traces, and optimized Schwarz methods, SIAM Rev., 61 (2019), pp. 3–76.
16.
P. Geldermans and A. Gillman, An adaptive high order direct solution technique for elliptic boundary value problems, SIAM J. Sci. Comput., 41 (2019), pp. A292–A315, https://doi.org/10.1137/17M1156320.
17.
A. George, Nested dissection of a regular finite element mesh, SIAM J. Numer. Anal., 10 (1973), pp. 345–363, https://doi.org/10.1137/0710032.
18.
A. Gillman, A. H. Barnett, and P.-G. Martinsson, A spectrally accurate direct solution technique for frequency-domain scattering problems with variable media, BIT, 55 (2015), pp. 141–170.
19.
A. Gillman and P. G. Martinsson, A direct solver with O(N) complexity for variable coefficient elliptic PDEs discretized via a high-order composite spectral collocation method, SIAM J. Sci. Comput., 36 (2014), pp. A2023–A2046, https://doi.org/10.1137/130918988.
20.
I. G. Graham and S. A. Sauter, Stability and finite element error analysis for the Helmholtz equation with variable coefficients, Math. Comp., 89 (2020), pp. 105–138.
21.
X. P. H. Calandra, S. Gratton, and X. Vasseur, An improved two-grid preconditioner for the solution of three-dimensional Helmholtz problems in heterogeneous media, Numer. Linear Algebra Appl., 20 (2012), pp. 663–688.
22.
S. Hao and P.-G. Martinsson, A direct solver for elliptic PDEs in three dimensions based on hierarchical merging of Poincaré–Steklov operators, J. Comput. Appl. Math., 308 (2016), pp. 419–434, https://doi.org/10.1016/j.cam.2016.05.013.
23.
P. Hemker, W. Hoffmann, and M. van Raalte, Two-level Fourier analysis of a multigrid approach for discontinuous Galerkin discretization, SIAM J. Sci. Comput., 25 (2003), pp. 1018–1041.
24.
P. W. Hemker, W. Hoffmann, and M. H. van Raalte, Fourier two-level analysis for discontinuous Galerkin discretization with linear elements, Numer. Linear Algebra Appl., 11 (2004), pp. 473–491.
25.
J. Lighthill, Waves in Fluids, Cambridge University Press, Cambridge, 1978.
26.
S. Kim and M. Lee, Artificial damping techniques for scalar waves in the frequency domain, Comput. Math. Appl., 31 (1996), pp. 1–12.
27.
L. E. Kinsler, A. R. Frey, A. B. Coppens, and J. V. Sanders., Fundamentals of Acoustics, Wiley, New York, 2000.
28.
A. V. Knyazev, Toward the optimal preconditioned eigensolver: Locally optimal block preconditioned conjugate gradient method, SIAM J. Sci. Comput., 23 (2001), pp. 517–541, https://doi.org/10.1137/S1064827500366124.
29.
M. Kronbichler and K. Kormann, Fast matrix-free evaluation of discontinuous Galerkin finite element operators, ACM Trans. Math. Software, 45 (2019), https://doi.org/10.1145/3325864.
30.
A. G. Kyurkchan and N. I. Smirnova, Mathematical Modeling in Diffraction Theory, Elsevier, Amsterdam, 2016, https://doi.org/10.1016/B978-0-12-803728-7.09993-0.
31.
J. P. Lucero Lorca and M. J. Gander, Optimization of Two-Level Methods for DG Discretizations of Reaction-Diffusion Equations, https://doi.org/10.48550/ARXIV.2004.14100, 2020.
32.
P. Martinsson, A fast direct solver for a class of elliptic partial differential equations, J. Comput. Phys., 38 (2009), pp. 316–330, https://doi.org/10.1007/s10915-008-9240-6.
33.
P. Martinsson, A direct solver for variable coefficient elliptic PDEs discretized via a composite spectral collocation method, J. Comput. Phys., 242 (2013), pp. 460–479, https://doi.org/10.1016/j.jcp.2013.02.019.
34.
H. Pfeiffer, L. Kidder, M. Scheel, and S. Teukolsky, A multidomain spectral method for solving elliptic equations, Comput. Phys. Commun., 152 (2003), pp. 253–273.
35.
C. D. Riyanti, A. Kononov, Y. A. Erlangga, C. Vuik, C. W. Oosterlee, R.-E. Plessix, and W. A. Mulder, A parallel multigrid-based preconditioner for the 3d heterogeneous high-frequency Helmholtz equation, J. Comput. Phys., 224 (2007), pp. 431–448.
36.
W. E. Roth, On direct product matrices, Bull. Amer. Math. Soc., 40 (1934), pp. 461–468, https://doi.org/10.1090/S0002-9904-1934-05899-3.
37.
B. R. S. Cools and W. Vanroose, A new level-dependent coarse grid correction scheme for indefinite Helmholtz problems, Numer. Linear Algebra Appl., 21 (2014), pp. 513–533.
38.
Y. Saad and M. H. Schultz, GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM J. Sci. Stat. Comput., 7 (1986), pp. 856–869, https://doi.org/10.1137/0907058.
39.
E. Sanchez-Palencia, Non-Homogeneous Media and Vibration Theory, Lecture Notes in Phys., Springer, Berlin, 1980.
40.
B. F. Smith, P. E. Bjørstad, and W. D. Gropp, Domain Decomposition: Parallel Multilevel Methods for Elliptic Partial Differential Equations, Cambridge University Press, Cambridge, 1996.
41.
A. Toselli and O. B. Widlund, Domain Decomposition Methods: Algorithms and Theory, Springer Ser. Comput. Math., Springer, Berlin, 2005.
42.
L. N. Trefethen, Spectral Methods in MATLAB, SIAM, Philadelphia, 2000, https://doi.org/10.1137/1.9780898719598.
43.
Y. A. Erlangga, C. Vuik, and C. W. Oosterlee, On a class of preconditioners for the Helmholtz equation, Appl. Numer. Math., 50 (2004), pp. 409–425.
Information & Authors
Information
Published In
SIAM Journal on Scientific Computing
Pages: A80 - A104
DOI: 10.1137/21M1463380
ISSN (online): 1095-7197
Copyright
© 2024 Society for Industrial and Applied Mathematics.
History
Submitted: 6 December 2021
Accepted: 18 August 2023
Published online: 16 January 2024
Keywords
MSC codes
Authors
Funding Information
National Science Foundation (NSF): DMS-2110886
Total Energies
Funding: The work of the fourth author is supported by the National Science Foundation (DMS-2110886). The work of the first, second, and fourth authors are supported in part by a grant from Total Energies.
Metrics & Citations
Metrics
Citations
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.
Cited By
There are no citations for this item
View Options
- Access via your Institution
- Questions about how to access this content? Contact SIAM at [email protected].