DPG approximation of eigenvalue problems

TL;DR

This paper investigates the discontinuous Petrov-Galerkin method for Laplace eigenvalue problems, providing convergence proofs, error estimates, and adaptive schemes validated through numerical experiments.

Contribution

It introduces and analyzes primal and ultra weak formulations of the DPG method for eigenvalue problems, including new a posteriori error estimators and adaptive strategies.

Findings

Convergence of DPG for eigenvalue problems is proven.

Two effective a posteriori error estimators are proposed.

Numerical results confirm the optimal convergence of the adaptive scheme.

Abstract

In this paper, the discontinuous Petrov--Galerkin approximation of the Laplace eigenvalue problem is discussed. We consider in particular the primal and ultra weak formulations of the problem and prove the convergence together with a priori error estimates. Moreover, we propose two possible error estimators and perform the corresponding a posteriori error analysis. The theoretical results are confirmed numerically and it is shown that the error estimators can be used to design an optimally convergent adaptive scheme.