Interior over-penalized enriched Galerkin methods for second order elliptic equations

TL;DR

This paper introduces a novel enriched Galerkin method with interior over-penalization for second order elliptic equations, providing optimal error estimates and improved preconditioning strategies.

Contribution

It develops a new variant of enriched Galerkin methods with over-penalization, offering better preconditioners and error analysis for elliptic problems.

Findings

Optimal a priori error estimates achieved.

Enhanced preconditioners for mesh refinement.

Numerical results confirm convergence and preconditioning effectiveness.

Abstract

In this paper we propose a variant of enriched Galerkin methods for second order elliptic equations with over-penalization of interior jump terms. The bilinear form with interior over-penalization gives a non-standard norm which is different from the discrete energy norm in the classical discontinuous Galerkin methods. Nonetheless we prove that optimal a priori error estimates with the standard discrete energy norm can be obtained by combining a priori and a posteriori error analysis techniques. We also show that the interior over-penalization is advantageous for constructing preconditioners robust to mesh refinement by analyzing spectral equivalence of bilinear forms. Numerical results are included to illustrate the convergence and preconditioning results.