A Penalty-Free and Essentially Stabilization-Free DG Method for Convection-Dominated Second-Order Elliptic Problems

Abstract

A new discontinuous Galerkin (DG) method is proposed and analyzed for general second-order elliptic problems. It features that local \(L^2\) projections are used to reconstruct the diffusion term and the convection term and that it does not need any penalty and even does not need any stabilization in the formulation. The Babus̆ka inf-sup stability is proven. The error estimates are established. More importantly, the new DG method can hold the SUPG-type stability for the convection; the SUPG-type optimal error estimates \({{\mathcal {O}}}(h^{\ell +1/2})\) is obtained for the problem with a dominating convection for the \(\ell \)-th order (\(\ell \ge 0\)) discontinuous element. Numerical results are provided.