Communications in Mathematical Sciences
Volume 20 (2022)
Wavenumber-explicit convergence analysis for finite element discretizations of time-harmonic wave propagation problems with perfectly matched layers
Pages: 1 – 52
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.
Helmholtz problems, perfectly matched layers, high order methods, finite elements, multiscale method, pollution effect
2010 Mathematics Subject Classification
Primary 35J05, 65N12, 65N30. Secondary 78A40.
Received 28 February 2019
Received revised 29 May 2021
Accepted 29 May 2021
Published 10 December 2021