An a posteriori error estimator for hp-adaptive discontinuous Galerkin methods for elliptic eigenvalue problems

Abstract

In this paper we present a residual-based {\em a posteriori} error estimator for $hp$-adaptive discontinuous Galerkin (DG) methods for elliptic eigenvalue problems. In particular we use as a model problem the Laplace eigenvalue problem on bounded domains in $\mathbb{R}^d$, $d=2,3$, with homogeneous Dirichlet boundary conditions. Analogous error estimators can be easily obtained for more complicated elliptic eigenvalue problems. We prove the reliability and efficiency of the residual based error estimator and use numerical experiments to show that, under an $hp$-adaptation strategy driven by the error estimator, exponential convergence can be achieved, even for non--smooth eigenfunctions