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.