Duality analysis of interior penalty discontinuous Galerkin methods under minimal regularity and application to the a priori and a posteriori error analysis of Helmholtz problems

TL;DR

This paper develops a duality-based analysis for interior penalty discontinuous Galerkin methods applied to Helmholtz problems, providing new error estimates that work under minimal regularity assumptions and extend known results.

Contribution

It introduces a novel duality analysis for DG methods on Helmholtz problems that requires minimal regularity, leading to generalized a priori and a posteriori error estimates.

Findings

Optimal error bounds in energy norm for DG solutions

Explicit a posteriori error control with fine mesh

Extension of conforming discretization results to DG methods

Abstract

We consider interior penalty discontinuous Galerkin discretizations of time-harmonic wave propagation problems modeled by the Helmholtz equation, and derive novel a priori and a posteriori estimates. Our analysis classically relies on duality arguments of Aubin-Nitsche type, and its originality is that it applies under minimal regularity assumptions. The estimates we obtain directly generalize known results for conforming discretizations, namely that the discrete solution is optimal in a suitable energy norm and that the error can be explicitly controlled by a posteriori estimators, provided the mesh is sufficiently fine.