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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, inaddition to the standard enrichment of a conforming finite elementdiscretization via discontinuous degrees of freedom, allows to subdivideselected (e.g. troubled) mesh cells in a non-conforming fashion and to usefurther discontinuous enrichment on this finer submesh. We prove stability andsharp a priori error estimates for a linear advection equation by using aspecially tailored projection and conducting some parts of a standardconvergence analysis for both meshes. By allowing an arbitrary degree ofenrichment on both, the coarse and the fine mesh (also including the case of noenrichment), our analysis technique is very general in the sense that ourresults cover the range from the standard continuous finite element method tothe standard discontinuous Galerkin (DG) method with (or without) local subcellenrichment. Numerical experiments confirm our analytical results and indicategood robustness of the proposed method.

A subcell-enriched Galerkin method for advection problems