A subcell-enriched Galerkin method for advection problems

TL;DR

This paper introduces an adaptive two-mesh enriched Galerkin method for advection problems, combining conforming and non-conforming discretizations with local enrichment to improve stability and accuracy.

Contribution

It generalizes the enriched Galerkin method with an adaptive two-mesh approach, allowing arbitrary enrichment levels and unifying several finite element methods.

Findings

Proves stability and error estimates for the proposed scheme.

Numerical experiments confirm robustness and accuracy.

Method encompasses standard finite element and discontinuous Galerkin methods.

Abstract

In this work, we introduce a generalization of the enriched Galerkin (EG) method. The key feature of our scheme is an adaptive two-mesh approach that, in addition to the standard enrichment of a conforming finite element discretization via discontinuous degrees of freedom, allows to subdivide selected (e.g. troubled) mesh cells in a non-conforming fashion and to use further discontinuous enrichment on this finer submesh. We prove stability and sharp a priori error estimates for a linear advection equation by using a specially tailored projection and conducting some parts of a standard convergence analysis for both meshes. By allowing an arbitrary degree of enrichment on both, the coarse and the fine mesh (also including the case of no enrichment), our analysis technique is very general in the sense that our results cover the range from the standard continuous finite element method to the standard discontinuous Galerkin (DG) method with (or without) local subcell enrichment. Numerical experiments confirm our analytical results and indicate good robustness of the proposed method.