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

In this paper we propose a variant of enriched Galerkin methods for second order elliptic equations with over-stabilization of interior jump terms. The bilinear form with interior over-stabilization 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-stabilization 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.