Sweeping preconditioners for stratified media in the presence of reflections
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ORIGINAL PAPER
Sweeping preconditioners for stratified media in the presence of reflections Janosch Preuß1
•
Thorsten Hohage1,2 • Christoph Lehrenfeld2
Received: 26 February 2020 / Accepted: 24 June 2020 Ó The Author(s) 2020. This article is published with open access at Springerlink.com
Abstract In this paper we consider sweeping preconditioners for time harmonic wave propagation in stratified media, especially in the presence of reflections. In the most famous class of sweeping preconditioners Dirichlet-to-Neumann operators for half-space problems are approximated through absorbing boundary conditions. In the presence of reflections absorbing boundary conditions are not accurate resulting in an unsatisfactory performance of these sweeping preconditioners. We explore the potential of using more accurate Dirichlet-to-Neumann operators within the sweep. To this end, we make use of the separability of the equation for the background model. While this improves the accuracy of the Dirichlet-to-Neumann operator, we find both from numerical tests and analytical arguments that it is very sensitive to perturbations in the presence of reflections. This implies that even if accurate approximations to Dirichlet-to-Neumann operators can be devised for a stratified medium, sweeping preconditioners are limited to very small perturbations. Keywords Helmholtz equation Dirichlet-to-Neumann operator Preconditioning Domain decomposition High-frequency waves Computational seismology Perfectly matched layers Sweeping preconditioner Mathematics Subject Classification 65F08 65N30 35L05 86-08 86A15
This article is part of the topical collection‘‘Waves 2019 – invited papers’’ edited by Manfred Kaltenbacher andMarkus Melenk. & Janosch Preuß [email protected] Thorsten Hohage [email protected] Christoph Lehrenfeld [email protected] 1
Max-Planck-Institut fu¨r Sonnensystemforschung, Justus-von-Liebig-Weg 3, 37077 Go¨ttingen, Germany
2
Institut fu¨r Numerische und Angewandte Mathematik, Lotzestraße 16-18, 37083 Go¨ttingen, Germany SN Partial Differential Equations and Applications
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SN Partial Differ. Equ. Appl. (2020)1:17
1 Introduction Time harmonic wave equations arise in various applications. Their numerical discretization leads to large linear systems which are difficult to solve with classical iterative methods [6]. Recently, sweeping preconditioners have emerged as a promising approach to overcome this problem [5]. Since the introduction of the moving perfectly matched layer (PML) preconditioner by Engquist and Ying [4], numerous impressive results and further developments of this technique have been published. We refer to [6] for a comprehensive review. Unfortunately, the range of wave propagation problems in which sweeping preconditioners can be used is limited. We are not aware of any publication in which sweeping preconditioners have been successfully applied to media that contain strong resonant cavities. In fact,
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