On the discontinuous Galerkin method for solving boundary value problems for the Helmholtz equation: A priori and a posteriori error analyses

TL;DR

This paper analyzes the local discontinuous Galerkin method for the Helmholtz equation, providing error estimates, stability conditions, and numerical validation for boundary value problems in 2D polygonal domains.

Contribution

It offers new a priori and a posteriori error analyses for LDG applied to Helmholtz problems, including explicit meshsize conditions related to the wavenumber.

Findings

The scheme is well-posed under certain regularity assumptions.

Convergence rates depend on meshsize and wavenumber.

Numerical experiments support theoretical results.

Abstract

We apply the local discontinuous Galerkin (LDG for short) method to solve a mixed boundary value problems for the Helmholtz equation in bounded polygonal domain in 2D. Under some assumptions on regularity of the solution of an adjoint problem, we prove that: (a) the corresponding indefinite discrete scheme is well posed; (b) there is convergence with the expected convergence rates as long as the meshsize h is small enough. We give precise information on how small h has to be in terms of the size of the wavenumber and its distance to the set of eigenvalues for the same boundary value problem for the Laplacian. We also present an a posteriori error estimator showing both the reliability and efficiency of the estimator complemented with detailed information on the dependence of the constants on the wavenumber. We finish presenting extensive numerical experiments which illustrate the theoretical results proven in this paper and suggest that stability and convergence may occur under less restrictive assumptions than those taken in the present work.