High-frequency behaviour of corner singularities in Helmholtz problems^★

¹ BCAM – Basque Center for Applied Mathematics, Alameda Mazarredo, 14, 48009 Bilbo, Bizkaia, Spain.
² Univ. Valenciennes, EA 4015, LAMAV – Laboratoire de Mathématiques et leurs Applications de Valenciennes, FR CNRS 2956, 59313 Valenciennes, France.

^* Corresponding author: tchaumont@bcamath.org

Received: 23 February 2017
Accepted: 30 April 2018

Abstract

We analyze the singular behaviour of the Helmholtz equation set in a non-convex polygon. Classically, the solution of the problem is split into a regular part and one singular function for each re-entrant corner. The originality of our work is that the “amplitude” of the singular parts is bounded explicitly in terms of frequency. We show that for high frequency problems, the “dominant” part of the solution is the regular part. As an application, we derive sharp error estimates for finite element discretizations. These error estimates show that the “pollution effect” is not changed by the presence of singularities. Furthermore, a consequence of our theory is that locally refined meshes are not needed for high-frequency problems, unless a very accurate solution is required. These results are illustrated with numerical examples that are in accordance with the developed theory.