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Stability estimate for the Helmholtz equation with rapidly jumping coefficients


Sauter, Stefan A; Torres, Céline (2018). Stability estimate for the Helmholtz equation with rapidly jumping coefficients. Zeitschrift für Angewandte Mathematik und Physik (ZAMP), 69(6):139.

Abstract

The goal of this paper is to investigate the stability of the Helmholtz equation in the high-frequency regime with non-smooth and rapidly oscillating coefficients on bounded domains. Existence and uniqueness of the problem can be proved using the unique continuation principle in Fredholm’s alternative. However, this approach does not give directly a coefficient-explicit energy estimate. We present a new theoretical approach for the one-dimensional problem and find that for a new class of coefficients, including coefficients with an arbitrary number of discontinuities, the stability constant (i.e. the norm of the solution operator) is bounded by a term independent of the number of jumps. We emphasize that no periodicity of the coefficients is required. By selecting the wave speed function in a certain “resonant” way, we construct a class of oscillatory configurations, such that the stability constant grows exponentially in the frequency. This shows that our estimates are sharp.

Abstract

The goal of this paper is to investigate the stability of the Helmholtz equation in the high-frequency regime with non-smooth and rapidly oscillating coefficients on bounded domains. Existence and uniqueness of the problem can be proved using the unique continuation principle in Fredholm’s alternative. However, this approach does not give directly a coefficient-explicit energy estimate. We present a new theoretical approach for the one-dimensional problem and find that for a new class of coefficients, including coefficients with an arbitrary number of discontinuities, the stability constant (i.e. the norm of the solution operator) is bounded by a term independent of the number of jumps. We emphasize that no periodicity of the coefficients is required. By selecting the wave speed function in a certain “resonant” way, we construct a class of oscillatory configurations, such that the stability constant grows exponentially in the frequency. This shows that our estimates are sharp.

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Additional indexing

Item Type:Journal Article, not_refereed, original work
Communities & Collections:07 Faculty of Science > Institute of Mathematics
Dewey Decimal Classification:510 Mathematics
Uncontrolled Keywords:General Physics and Astronomy, Applied Mathematics, General Mathematics
Language:English
Date:1 December 2018
Deposited On:23 Jan 2019 15:36
Last Modified:24 Jan 2019 08:36
Publisher:Birkhäuser
ISSN:0044-2275
OA Status:Closed
Publisher DOI:https://doi.org/10.1007/s00033-018-1031-9
Project Information:
  • : FunderSNSF
  • : Grant ID200021_172803
  • : Project TitleGalerkin Discretization of High-Frequency Helmholtz Problems with Variable Coefficients

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