Analysis of an incongruity in the standard Galerkin Finite Element Method

Abstract

In the standard Galerkin finite element method applied to equations of groundwater flow, the development to incorporate second and third type boundary conditions implicitly assumes that the basis functions satisfy these boundary conditions. Because the basis functions normally employed are not required to satisfy these boundary conditions, a theoretical incongruity is created. If the Galerkin procedure is reformulated by adding a term to explicitly consider failure of the basis functions to satisfy the boundary conditions, the incongruity is eliminated; however, the resulting set of operational equations is unchanged from the set resulting from applying the boundary conditions in the normal manner. An analysis demonstrates that if the differential equation has a variational equivalent, the same error functional is minimized whether or not the basis functions satisfy second and third type boundary conditions.