An Extended Galerkin Analysis for Elliptic Problems

A general analysis framework is presented in this paper for many different types of finite element methods (including various discontinuous Galerkin methods). For second order elliptic equation, this framework employs $4$ different discretization variables, $uh, \bm{p}h, \check uh$ and $\check ph$, where $uh$ and $\bm{p}h$ are for approximation of $u$ and $\bm{p}=-\alpha \nabla u$ inside each element, and $ \check uh$ and $\check ph$ are for approximation of residual of $u$ and $\bm{p} \cdot \bm{n}$ on the boundary of each element. The resulting 4-field discretization is proved to satisfy inf-sup conditions that are uniform with respect to all discretization and penalization parameters. As a result, most existing finite element and discontinuous Galerkin methods can be analyzed using this general framework by making appropriate choices of discretization spaces and penalization parameters.