Abstract
The capabilities of modern computers and mathematical software are entirely sufficient for the numerical solution of 2D PDEs. Two-dimensional problems naturally arise when the Fourier method is applied to a three-dimensional boundary value problem for PDEs with coefficients independent of one variable. Such specially constructed solvers can be used as preconditioners for iterative solvers of boundary value problems for PDEs with general case coefficients. In this paper, we develop the outlined above idea for the numerical solution of the Helmholtz equation.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
D is assumed to contain the origin (0, 0, 0).
- 2.
The density is constant and excluded from further considerations.
References
Belonosov, M., Dmitriev, M., Kostin, V., Neklyudov, D., Tcheverda, V.: An iterative solver for the 3D Helmholtz equation. J. Comput. Phys. 345, 330–344 (2017). https://doi.org/10.1016/j.jcp.2017.05.026
Belonosov, M., Kostin, V., Neklyudov, D., Tcheverda, V.: 3D numerical simulation of elastic waves with a frequency-domain iterative solver. Geophysics 83(6), T333–T344 (2018). https://doi.org/10.1190/geo2017-0710.1
Berenger, J.-P.: A perfectly matched layer for the absorption of electromagnetic waves. J. Comput. Phys. 114(2), 185–200 (1994)
BLAS (Basic Linear Algebra Subprograms). http://www.netlib.org/blas/
Erlangga, Y.A., Herrmann, F.J.: Full-waveform inversion with Gauss–Newton–Krylov method. In: 79th SEG Annual Meeting Expanded Abstracts, pp. 2357–2361. SEG (2009)
Erlangga, Y.A., Nabben, R.: On a multilevel Krylov method for the Helmholtz equation preconditioned by shifted Laplacian. Electron. Trans. Numer. Anal. 31, 403–424 (2008)
Intel® Math Kernel Library (Intel® MKL). https://software.intel.com/en-us/intel-mkl
Kostin, V., Solovyev, S., Bakulin, A., Dmitriev, M.: Direct frequency-domain 3D acoustic solver with intermediate data compression benchmarked against time-domain modeling for FWI applications. Geophysics 84(4), T207–T219 (2019). https://doi.org/10.1190/geo2018-0465.1
Mulder, W.A., Plessix, R.E.: How to choose a subset of frequencies in frequency-domain finite-difference migration. Geophys. J. Int. 158(3), 801–812 (2004). https://doi.org/10.1111/j.1365-246X.2004.02336.x
Oosterlee, C.W., Vuik, C., Mulder, W.A., Plessix, R.-E.: Shifted-Laplacian preconditioners for heterogeneous Helmholtz problems. In: Advanced Computational Methods in Science and Engineering, pp. 21–46. Springer (2010). https://doi.org/10.1007/978-3-642-03344-5_2
Plessix, R.E.: Three-dimensional frequency-domain full-waveform inversion with an iterative solver. Geophysics 74, WCC149—WCC157 (2009). https://doi.org/10.1190/1.3211198
Saad, Y.: Iterative Methods for Sparse Linear Systems, 2nd edn. SIAM, Philadelphia (2003). https://doi.org/10.1137/1.9780898718003.bm
Singer, I., Turkel, E.: A perfectly matched layer for the Helmholtz equation in a semi-infinite strip. J. Comput. Phys. 201, 439–465 (2004). https://doi.org/10.1016/j.jcp.2004.06.010
Vainberg, B.: Principles of radiation, vanishing attenuation and limit amplitude in the general theory of partial differential equations. Usp. Mat. Nauk 21, 115–194 (1966). (in Russian)
Virieux, J., Operto, S., Ben-Hadj-Ali, H., Brosssier, R., Etienne, V., Sourber, F., Guraud, L., Haidar, A.: Seismic wave modeling for seismic imaging. Lead. Edge 28(5), 538–544 (2009). https://doi.org/10.1190/1.3124928
Wang, S., Li, X., Xia, J., Situ, Y., De Hoop, M.: Efficient scalable algorithms for solving dense linear systems with hierarchically semiseparable structures. SIAM J. Sci. Comput. 35(6), C519–C544 (2013). https://doi.org/10.1137/110848062
Acknowledgements
The main results of this paper were obtained by Victor Kostin during his stay at the Tel-Aviv University in the period 03.11–04.10 of 2018. Victor Kostin thanks the Russian Science Foundation for the financial support of the trip to Tel-Aviv University under the project 17-17-01128.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2020 Springer Nature Switzerland AG
About this chapter
Cite this chapter
Klyuchinskiy, D., Kostin, V., Landa, E. (2020). New Efficient Preconditioner for Helmholtz Equation. In: Demidenko, G., Romenski, E., Toro, E., Dumbser, M. (eds) Continuum Mechanics, Applied Mathematics and Scientific Computing: Godunov's Legacy. Springer, Cham. https://doi.org/10.1007/978-3-030-38870-6_32
Download citation
DOI: https://doi.org/10.1007/978-3-030-38870-6_32
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-38869-0
Online ISBN: 978-3-030-38870-6
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)