Robust adaptive hp discontinuous Galerkin finite element methods for the Helmholtz equation

TL;DR

This paper develops a robust $hp$ a posteriori error analysis for the 2D Helmholtz equation using discontinuous Galerkin methods, enabling effective adaptive refinement even at high wave numbers.

Contribution

It introduces a new robust a posteriori error estimator for $hp$ discontinuous Galerkin methods applied to the Helmholtz equation, valid across polynomial degrees and wave numbers.

Findings

Error estimator is reliable and efficient.

Adaptive refinement strategy improves accuracy.

Method is robust against pollution effects.

Abstract

This paper presents an $hp$ a posteriori error analysis for the 2D Helmholtz equation that is robust in the polynomial degree $p$ and the wave number $k$. For the discretization, we consider a discontinuous Galerkin formulation that is unconditionally well posed. The a posteriori error analysis is based on the technique of equilibrated fluxes applied to a shifted Poisson problem, with the error due to the nonconformity of the discretization controlled by a potential reconstruction. We prove that the error estimator is both reliable and efficient, under the condition that the initial mesh size and polynomial degree is chosen such that the discontinuous Galerkin formulation converges, i.e., it is out of the regime of pollution. We confirm the efficiency of an $hp$-adaptive refinement strategy based on the presented robust a posteriori error estimator via several numerical examples.