An algebraic multigrid based shifted-Laplacian preconditioner for the Helmholtz equation

Journal of Computational Physics

Volumn 226, Issue 1, 2007, Pages 1196-1210

  Airaksinen, Tuomas ^a   Heikkola, Erkki ^b   Pennanen, Anssi ^a   Toivanen, Jari ^c  

^a UNIVERSITY OF JYVÄSKYLÄ   (Finland)

^b Numerola Oy   (Finland)

^c Stanford University   (United States)

Author keywords

Algebraic multigrid method; Finite element method; GMRES; Helmholtz equation; Preconditioner

Indexed keywords

ACOUSTIC WAVE SCATTERING; ALGEBRA; HELMHOLTZ EQUATION;

ACOUSTIC SCATTERING PROBLEM; ALGEBRAIC MULTIGRID METHODS; ALGEBRAIC MULTIGRIDS; GMRES; HELMHOLTZ OPERATORS; HELMHOLTZ'S EQUATIONS; LAPLACIANS; LOWER FREQUENCIES; MULTIGRID CYCLES; PRECONDITIONERS;

FINITE ELEMENT METHOD;

EID: 34548405076     PISSN: 00219991     EISSN: 10902716     Source Type: Journal    
DOI: 10.1016/j.jcp.2007.05.013     Document Type: Article

References (36)

1. Thompson L.L. A review of finite-element methods for time-harmonic acoustics. J. Acoust. Soc. Am. 119 3 (2006) 1315-1330
2. Ihlenburg F. Finite element analysis of acoustic scattering. Applied Mathematical Sciences vol. 132 (1998), Springer-Verlag, New York
3. Nefske D.J., Wolf J., and Howell L. Structural-acoustic finite element analysis of the automobile passenger compartment: a review of current practice. J. Sound Vib. 80 2 (1982) 247-266
4. Engquist B., and Majda A. Absorbing boundary conditions for numerical simulation of waves. Math. Comput. 31 (1977) 629-651
5. Saad Y., and Schultz M.H. GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J. Sci. Statist. Comput. 7 3 (1986) 856-869
6. van der Vorst H.A. Bi-CGSTAB: a fast and smoothly converging variant of Bi-CG for the solution of nonsymmetric linear systems. SIAM J. Sci. Statist. Comput. 13 2 (1992) 631-644
7. Farhat C., Macedo A., Lesoinne M., Roux F.-X., Magoulés F., and de La Bourdonnaie A. Two-level domain decomposition methods with Lagrange multipliers for the fast iterative solution of acoustic scattering problems. Comput. Methods Appl. Mech. Eng. 184 2-4 (2000) 213-239
8. Farhat C., Avery P., Tezaur R., and Li J. FETI-DPH: a dual-primal domain decomposition method for acoustic scattering. J. Comput. Acoust. 13 3 (2005) 499-524
9. Gander M.J., Magoulès F., and Nataf F. Optimized Schwarz methods without overlap for the Helmholtz equation. SIAM J. Sci. Comput. 24 1 (2002) 38-60
10. Ito K., and Toivanen J. A fast iterative solver for scattering by elastic objects in layered media. Appl. Numer. Math. 57 5-7 (2007) 811-820
11. Otto K., and Larsson E. Iterative solution of the Helmholtz equation by a second-order method. SIAM J. Matrix Anal. Appl. 21 1 (1999) 209-229
12. Brandt A., and Livshits I. Wave-ray multigrid method for standing wave equations. Electron. Trans. Numer. Anal. 6 (1997) 162-181
13. Elman H.C., Ernst O.G., and O'Leary D.P. A multigrid method enhanced by Krylov subspace iteration for discrete Helmholtz equations. SIAM J. Sci. Comput. 23 4 (2001) 1291-1315
14. Erlangga Y.A., Oosterlee C.W., and Vuik C. A novel multigrid based preconditioner for heterogeneous Helmholtz problems. SIAM J. Sci. Comput. 27 4 (2006) 1471-1492
15. Ernst O.G. A finite-element capacitance matrix method for exterior Helmholtz problems. Numer. Math. 75 2 (1996) 175-204
16. Heikkola E., Kuznetsov Y.A., Neittaanmäki P., and Toivanen J. Fictitious domain methods for the numerical solution of two-dimensional scattering problems. J. Comput. Phys. 145 1 (1998) 89-109
17. Heikkola E., Rossi T., and Toivanen J. A parallel fictitious domain method for the three-dimensional Helmholtz equation. SIAM J. Sci. Comput. 24 5 (2003) 1567-1588
18. Monga Made M.M. Incomplete factorization-based preconditionings for solving the Helmholtz equation. Int. J. Numer. Methods Eng. 50 (2001) 1077-1101
19. Plessix R.E., and Mulder W.A. Separation-of-variables as a preconditioner for an iterative Helmholtz solver. Appl. Numer. Math. 44 3 (2003) 385-400
20. Bristeau M.O., Glowinski R., and Périaux J. Controllability methods for the computation of time-periodic solutions; application to scattering. J. Comput. Phys. 147 2 (1998) 265-292
21. Heikkola E., Mönkölä S., Pennanen A., and Rossi T. Controllability method for acoustic scattering with spectral elements. J. Comput. Appl. Math. 204 2 (2007) 344-355
22. Turkel E. Numerical methods and nature. J. Sci. Comput. 28 2-3 (2006) 549-570
23. Bayliss A., Goldstein C.I., and Turkel E. An iterative method for the Helmholtz equation. J. Comput. Phys. 49 3 (1983) 443-457
24. A.L. Laird, M.B. Giles, Preconditioned iterative solution of the 2D Helmholtz equation, Tech. Rep. 02/12, Oxford Computer Laboratory, Oxford, UK, 2002.
25. Erlangga Y.A., Vuik C., and Oosterlee C.W. On a class of preconditioners for solving the Helmholtz equation. Appl. Numer. Math. 50 3-4 (2004) 409-425
26. E. Turkel, Y. Erlangga, Preconditioning a finite element solver of the exterior Helmholtz equation, in: P. Wesseling, E. Oñate, J. Périaux (Eds.), ECCOMAS CFD, ECCOMAS, 2006.
27. Y. Erlangga, A preconditioner for the Helmholtz equation with perfectly matched layer, in: P. Wesseling, E. Oñate, J. Périaux (Eds.), ECCOMAS CFD, ECCOMAS, 2006.
28. Bamberger A., Joly P., and Roberts J.E. Second-order absorbing boundary conditions for the wave equation: a solution for the corner problem. SIAM J. Numer. Anal. 27 2 (1990) 323-352
29. Bayliss A., Gunzburger M., and Turkel E. Boundary conditions for the numerical solution of elliptic equations in exterior regions. SIAM J. Appl. Math. 42 2 (1982) 430-451
30. Babuška I.M., and Sauter S.A. Is the pollution effect of the FEM avoidable for the Helmholtz equation considering high wave numbers?. SIAM J. Numer. Anal. 34 6 (1997) 2392-2423
31. Bayliss A., Goldstein C.I., and Turkel E. On accuracy conditions for the numerical computation of waves. J. Comput. Phys. 59 3 (1985) 396-404
32. Saad Y. Iterative Methods for Sparse Linear Systems. second ed. (2003), Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA
33. Briggs W.L., Henson V.E., and McCormick S.F. A Multigrid Tutorial. second ed. (2000), Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA
34. Kickinger F. Algebraic multi-grid for discrete elliptic second-order problems. Multigrid Methods V (Stuttgart, 1996) (1998), Springer, Berlin 157-172
35. J. Martikainen, A. Pennanen, T. Rossi, Application of an algebraic multigrid method to incompressible flow problems, Tech. Rep. B2/2006, Department of Mathematical Information Technology, University of Jyväskylä, Jyväskylä, Finland, 2006.
36. Schöberl J. NETGEN - an advancing front 2D/3D-mesh generator based on abstract rules. Comput. Visual. Sci. 1 (1997) 41-52