A Local Problem Error Estimator for Anisotropic Tetrahedral Finite Element Meshes

Abstract

The Poisson problem is solved by the finite element method on anisotropic tetrahedral or triangular meshes. The focus is on adaptive algorithms and, in particular, on a posteriori error estimators based on the solution of a local 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 modification of a familiar isotropic counterpart. By a rigorous analysis it is proven that this estimator is equivalent to a known anisotropic residual error estimator. Hence the local problem error estimator also yields reliable upper and lower error bounds on anisotropic meshes. The local problems are shown to be well-conditioned. Two further local problem error estimators originally defined for isotropic meshes are investigated in the anisotropic context here. The results reveal the significance of the proper choice of local problem. Furthermore the analysis covers Dirichlet, Neumann, and Robin boundary conditions. Particular attention is paid to the Robin boundary conditions since they require a different treatment than the Neumann boundary conditions. A numerical example supports the error analysis.