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Analysis of an incongruity in the standard Galerkin Finite Element Method
Author(s) -
Cooley Richard L.
Publication year - 1983
Publication title -
water resources research
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.863
H-Index - 217
eISSN - 1944-7973
pISSN - 0043-1397
DOI - 10.1029/wr019i001p00289
Subject(s) - galerkin method , mathematics , boundary value problem , basis (linear algebra) , mathematical analysis , finite element method , basis function , boundary (topology) , boundary knot method , term (time) , boundary element method , geometry , physics , thermodynamics , quantum mechanics
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.