A Posteriori Error Estimation of hp-dG Finite Element Methods for Highly Indefinite Helmholtz Problems (extended version)
For researchers solving high-frequency Helmholtz problems, this provides a rigorous error estimator enabling adaptive mesh refinement for unconditionally stable dG methods.
The paper introduces an a posteriori error estimator for hp-dG finite element methods applied to highly indefinite Helmholtz problems, proving reliability and efficiency bounds explicit in wavenumber and discretization parameters. Numerical experiments demonstrate the method's efficiency and robustness.
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 Deprés. 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.