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 the second-order elliptic equation −div(α∇u) = f, this framework employs four different discretization variables, u_h, p_h, ŭ_h and p⌣ _h, where u_h and p_h are for approximation of u and p = −α∇u inside each element, and ŭ_h and p⌣ _h are for approximation of residual of u and p · n on the boundary of each element. The resulting 4-field discretization is proved to satisfy two types of inf-sup conditions that are uniform with respect to all discretization and penalization parameters. As a result, many existing finite element and discontinuous Galerkin methods can be analyzed using this general framework by making appropriate choices of discretization spaces and penalization parameters.