Communications in Mathematical Sciences

Volume 20 (2022)

Number 1

Wavenumber-explicit convergence analysis for finite element discretizations of time-harmonic wave propagation problems with perfectly matched layers

Pages: 1 – 52

DOI:  https://dx.doi.org/10.4310/CMS.2022.v20.n1.a1

Authors

Théophile Chaumont-Frelet (Inria Sophia Antipolis-Méditerranée Research Centre, Nachos Project Team and University of Nice, J.A. Dieudonné Mathematics Laboratory Sophia Antipolis, France)

Dietmar Gallistl (Institut für Mathematik, Universität Jena, Germany)

Serge Nicaise (Laboratoire de Mathématiques et leurs Applications, Université Polytechnique des Hauts-de-France, Valenciennes, France)

Jérôme Tomezyk (Laboratoire de Mathématiques et leurs Applications, Université Polytechnique des Hauts-de-France, Valenciennes, France)

Abstract

The first part of this paper is devoted to a wavenumber-explicit stability analysis of a planar Helmholtz problem with a perfectly matched layer. We prove that, for a model scattering problem, the $H^1$ norm of the solution is bounded by the right-hand side, uniformly in the wavenumber $k$ in the high wavenumber regime. The second part proposes two numerical discretizations, namely, a high-order finite element method and a multiscale method based on local subspace correction. We establish a priori error estimates, based on the aforementioned stability result, that permit to properly select the discretization parameters with respect to the wavenumber. Numerical experiments assess the sharpness of our key results.

Keywords

Helmholtz problems, perfectly matched layers, high order methods, finite elements, multiscale method, pollution effect

2010 Mathematics Subject Classification

Primary 35J05, 65N12, 65N30. Secondary 78A40.

The full text of this article is unavailable through your IP address: 3.239.4.127

Received 28 February 2019

Received revised 29 May 2021

Accepted 29 May 2021

Published 10 December 2021