An a posteriori residual error estimator for the finite element method on anisotropic tetrahedral meshes

A new a posteriori residual error estimator is defined and rigorously analy sed for anisotropic tetrahedral finite element meshes. All considerations c arry over to anisotropic triangular meshes with minor changes only. The low er error bound is obtained by means of bubble functions and the correspondi ng anisotropic inverse inequalities. In order to prove the upper error boun d, it is vital that an anisotropic mesh corresponds to the anisotropic func tion under consideration. To measure this correspondence, a so-called match ing function is defined, and its discussion shows it to be a useful tool. W ith its help anisotropic interpolation estimates and subsequently the upper error bound are proven. Additionally it is pointed out how to treat Robin boundary conditions ina posteriori error analysis on isotropic and anisotro pic meshes. A numerical example supports the anisotropic error analysis. Ma thematics Subject Classification (1991): 65N15, 65N30.