A posteriori error estimates for discontinuous Galerkin methods on polygonal and polyhedral meshes

TL;DR

This paper develops a new residual-based a posteriori error estimate for interior penalty discontinuous Galerkin methods applicable to general polygonal and polyhedral meshes, including irregular and hanging-node configurations.

Contribution

It introduces a novel error analysis that accommodates highly irregular meshes and small faces, extending existing bounds to more general mesh geometries.

Findings

The new error bounds are effective for meshes with arbitrary polygonal/polyhedral shapes.

The analysis includes cases with many small faces and irregular hanging nodes.

Numerical experiments confirm the practical usefulness of the error estimators.

Abstract

We present a new residual-type energy-norm a posteriori error analysis for interior penalty discontinuous Galerkin (dG) methods for linear elliptic problems. The new error bounds are also applicable to dG methods on meshes consisting of elements with very general polygonal/polyhedral shapes. The case of simplicial and/or box-type elements is included in the analysis as a special case. In particular, for the upper bounds, an arbitrary number of very small faces are allowed on each polygonal/polyhedral element, as long as certain mild shape regularity assumptions are satisfied. As a corollary, the present analysis generalizes known a posteriori error bounds for dG methods, allowing in particular for meshes with an arbitrary number of irregular hanging nodes per element. The proof hinges on a new conforming recovery strategy in conjunction with a Helmholtz decomposition formula. The resulting a posteriori error bound involves jumps on the tangential derivatives along elemental faces. Local lower bounds are also proven for a number of practical cases. Numerical experiments are also presented, highlighting the practical value of the derived a posteriori error bounds as error estimators.