A Posteriori Error Estimation of hp-dG Finite Element Methods for Highly Indefinite Helmholtz Problems (extended version)

TL;DR

This paper develops an a posteriori error estimator for hp-dG finite element methods applied to highly indefinite Helmholtz problems, demonstrating stability, efficiency, and robustness through numerical experiments.

Contribution

It introduces a reliable and efficient a posteriori error estimator for hp-dG methods on Helmholtz problems, explicitly accounting for wavenumber and discretization parameters.

Findings

Estimator is reliable and efficient

Method is unconditionally stable

Numerical experiments confirm robustness

Abstract

In this paper, we will consider an $hp$-finite elements discretization of a highly indefinite Helmholtz problem by some dG formulation which is based on the ultra-weak variational formulation by Cessenat and Depr\'{e}s. We will introduce an a posteriori error estimator and derive reliability and efficiency estimates which are explicit with respect to the wavenumber and the discretization parameters $h$ and $p$. In contrast to the conventional conforming finite element method for indefinite problems, the dG formulation is unconditionally stable and the adaptive discretization process may start from a very coarse initial mesh. Numerical experiments will illustrate the efficiency and robustness of the method.