A local problem error estimator for anisotropic tetrahedral finite elementmeshes

The Poisson problem is solved by the finite element method on anisotropic t etrahedral or triangular meshes. The focus is on adaptive algorithms and, i n particular, on a posteriori error estimators based on the solution of a l ocal problem. On anisotropic meshes, such estimators cannot be analyzed in the common way known from isotropic meshes. The first estimator proposed here is a slight modi cation of a familiar iso tropic counterpart. By a rigorous analysis it is proven that this estimator is equivalent to a known anisotropic residual error estimator. Hence the l ocal problem error estimator also yields reliable upper and lower error bou nds on anisotropic meshes. The local problems are shown to be well-conditio ned. Two further local problem error estimators originally defined for isotropic meshes are investigated in the anisotropic context here. The results revea l the significance of the proper choice of local problem. Furthermore the a nalysis covers Dirichlet, Neumann, and Robin boundary conditions. Particula r attention is paid to the Robin boundary conditions since they require a d ifferent treatment than the Neumann boundary conditions. A numerical exampl e supports the error analysis.