Multigrid based preconditioners for the numerical solution of two-dimensional heterogeneous problems in geophysics

Abstract

We study methods for the numerical solution of the Helmholtz equation for two-dimensional applications in geophysics. The common framework of the iterative methods in our study is a combination of an inner iteration with a geometric multigrid method used as a preconditioner and an outer iteration with a Krylov subspace method. The preconditioning system is based on either a pure or shifted Helmholtz operator. A multigrid iteration is used to approximate the inverse of this operator. The proposed solution methods are evaluated on a complex benchmark in geophysics involving highly variable coefficients and high wavenumbers. We compare this preconditioned iterative method with a direct method and a hybrid method that combines our iterative approach with a direct method on a reduced problem. We see that the hybrid method outperforms both the iterative and the direct approach.

Keywords:

• Direct methods
• Helmholtz equation
• Iterative methods
• Multigrid
• Preconditioning

Acknowledgements

This work has been made possible thanks to public domain research software. The authors would like to gratefully thank Y. Erlangga, C. Vuik and C. Oosterlee (multigrid preconditioner for two-dimensional Helmholtz problems), P. Amestoy, I. Duff, A. Guermouche, J. Koster, J.-Y. L'Excellent and S. Pralet (MUMPS direct solver) and V. Frayssé, L. Giraud, S. Gratton and J. Langou (set of GMRES routines for real and complex arithmetics), respectively, for making their software publicly available. We would also like to thank the anonymous referee for his/her helpful comments and encouragement to include the Fourier analysis.

Notes

^†Software available at http://www.cerfacs.fr/algor/Softs/GMRES/index.html

^†Software available at http://mumps.enseeiht.fr/