Extended Galerkin Method
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A general framework, known as extended Galerkin method, is presented in this paper for the derivation and analysis of many different types of finite element methods (including various discontinuous Galerkin methods). For second order elliptic equation, this framework employs 4 different discretization variables, u_h, p_h, ǔ_h and p̌_h, where u_h and p_h are for approximation of u and p=-α∇ u inside each element, and ǔ_h and p̌_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 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 derived and analyzed using this general theory by making appropriate choices of discretization spaces and penalization parameters.
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