Influence of perturbations in transmission conditions on the convergence of the domain decomposition method for the Helmholtz equation

Authors

  • A.F. Zaitseva Trofimuk Institute of Petroleum Geology and Geophysics of SB RAS
  • V.V. Lisitsa Trofimuk Institute of Petroleum Geology and Geophysics of SB RAS https://orcid.org/0000-0003-3544-4878

Keywords:

domain decomposition method, Helmholtz equation, Dirichlet-to-Neumann map, finite difference method, method of small perturbations, convergence of iterative processes

Abstract

The domain decomposition is a widely used technique applied in parallel iterative solvers of the Helmholtz equation with the convergence rate controlled by the quality of transmission conditions. The optimal conditions are those based on the Dirichlet-to-Neumann map. However, if grid-based numerical methods are used to solve the Helmholtz equation, this map needs to be localized, which induces artificial grid-independent perturbations in the transmission conditions. In this paper it is proved that if the perturbations are nonsymmetric, i.e. different errors are induced in the adjoint subdomains, then the domain decomposition method converges to a solution of a problem differing from the original one. In other words, there exists an irreducible error.

Author Biographies

A.F. Zaitseva

V.V. Lisitsa

References

  1. Белоносов М.А., Костов К., Решетова Г.В., Соловьев С.А., Чеверда В.А. Организация параллельных вычислений для моделирования сейсмических волн с использованием аддитивного метода Шварца // Вычислительные методы и программирование. 2012. 13. 525-535.
  2. Вишневский Д.М., Лисица В.В., Решетова Г.В. Численное моделирование распространения сейсмических волн в средах с вязкоупругими включениями // Вычислительные методы и программирование. 2013. 14. 155-165.
  3. Asvadurov S., Druskin V., Guddati M.N., Knizhnerman L. On optimal finite-difference approximation of PML // SIAM J. Numer. Anal. 2003. 41, N 1. 287-305.
  4. Babuvska I., Strouboulis T., Gangaraj S.K., Upadhyay C.S. Pollution error in the h-version of the finite element method and the local quality of the recovered derivatives // Computer Methods in Applied Mechanics and Engineering. 1997. 140, N 1-2. 1-37.
  5. Baldassari C., Barucq H., Calandra H., Diaz J. Numerical performances of a hybrid local-time stepping strategy applied to the reverse time migration // Geophysical Prospecting. 2011. 59, N 5. 907-919.
  6. Berenger J.-P. A perfectly matched layer for the absorption of electromagnetic waves // J. Comput. Phys. 1994. 114, N 2. 185-200.
  7. Christensen R.M. Theory of viscoelasticity: an introduction. New York: Academic, 1971.
  8. Collino F., Ghanemi S., Joly P. Domain decomposition method for harmonic wave propagation: A general presentation // Computer Methods in Applied Mechanics and Engineering. 2000. 184, N 2-4. 171-211.
  9. Engquist B., Ying L. Fast algorithms for high frequency wave propagation // Lecture Notes in Computational Science and Engineering. Vol. 83. Heidelberg: Springer, 2012. 127-161.
  10. Fornberg B. The pseudospectral method: Comparisons with finite differences for the elastic wave equation // Geophysics. 1987. 52, N 4. 483-501.
  11. Fornberg B. The pseudospectral method: Accurate representation of interfaces in elastic wave calculations // Geophysics. 1988. 53, N 5. 625-637.
  12. Gander M.J., Halpern L., Magoulés F. An optimized Schwarz method with two-sided Robin transmission conditions for the Helmholtz equation // International Journal for Numerical Methods in Fluids. 2007. 55, N 2. 163-175.
  13. Gander M.J., Halpern L., Nataf F. Optimal Schwarz waveform relaxation for the one dimensional wave equation // SIAM Journal on Numerical Analysis. 2003. 41, N 5. 1643-1681.
  14. Grote M.J., Schneebeli A., Schötzau D. Discontinuous Galerkin finite element method for the wave equation // SIAM Journal on Numerical Analysis. 2006. 44, N 6. 2408-2431.
  15. Hagstrom T., Lau S. Radiation boundary conditions for Maxwell’s equations: A review of accurate time-domain formulations // J. Comput. Math. 2007. 25, N 3. 305-336.
  16. Hagstrom T., Givoli D., Rabinovich D., Bielak J. The double absorbing boundary method // Journal of Computational Physics. 2014. 259. 220-241.
  17. Lisitsa V. Optimal discretization of PML for elasticity problems // Electron. Trans. Numer. Anal. 2008. 30. 258-277.
  18. Magoulés F., Putanowicz R. Optimal convergence of non-overlapping Schwarz methods for the Helmholtz equation // Journal of Computational Acoustics. 2005. 13, N 3. 525-545.
  19. Melenk J.M., Parsania A., Sauter S. General DG-methods for highly indefinite Helmholtz problems // Journal of Scientific Computing. 2013. 57, N 3. 536-581.
  20. Neklyudov D., Silvestrov I., Tcheverda V. A Helmholtz iterative solver with semianalytical preconditioner for the frequency-domain full-waveform inversion // SEG Technical Program Expanded Abstracts 2010. Denver, 2010. 1070-1074
    doi 10.1190/1.3513031
  21. Plessix R.-’E. Three-dimensional frequency-domain full-waveform inversion with an iterative solver // Geophysics. 2009. 74, N 6. 149-157.
  22. Pradhan D., Shalini B., Nataraj N., Pani A.K. A Robin-type non-overlapping domain decomposition procedure for second order elliptic problems // Advances in Computational Mathematics. 2011. 34, N 4. 339-368.
  23. Rabinovich D., Givoli D., Bécache E. Comparison of high-order absorbing boundary conditions and perfectly matched layers in the frequency domain // International Journal for Numerical Methods in Biomedical Engineering. 2010. 26, N 10. 1351-1369.
  24. Stolk C.C. A rapidly converging domain decomposition method for the Helmholtz equation // Journal of Computational Physics. 2013. 241. 240-252.
  25. Vion A., Geuzaine C. Double sweep preconditioner for optimized Schwarz methods applied to the Helmholtz problem // Journal of Computational Physics. 2014. 266. 171-190.
  26. Virieux J., Calandra H., Plessix R.-’E. A review of the spectral, pseudo-spectral, finite-difference and finite-element modelling techniques for geophysical imaging // Geophysical Prospecting. 2011. 59, N 5. 794-813.
  27. Virieux J., Operto S., Ben-Hadj-Ali H., Brossier R., Etienne V., Sourbier F., Giraud L., Haidar A. Seismic wave modeling for seismic imaging // The Leading Edge. 2009. 28, N 5. 538-544.
  28. White R.E. The accuracy of estimating Q from seismic data // Geophysics. 1992. 57, N 11. 1508-1511.

Published

11-08-2014

How to Cite

Зайцева А., Лисица В. Influence of Perturbations in Transmission Conditions on the Convergence of the Domain Decomposition Method for the Helmholtz Equation // Numerical Methods and Programming (Vychislitel’nye Metody i Programmirovanie). 2014. 15. 476-486

Issue

Section

Section 1. Numerical methods and applications

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