On the finite element method in anistropic Sobolev spaces

The anisotropic case differs from the isotropic case mainly in regard of the boundary. Anisotropic problems are usually posed on product domains, but isotropic problems are often treated under the assumption of a boundary without corners. For this reason the proof of a global regularity result for the problems considered here is of some independent interest.

As in the standard theory, the error of a Galerkin approximation is quasi-optimal when measured in the natural norm of the problem. An improved rate of convergence for the Lg-norm of the error is shown by the technique of Aubin-Nitsche. This estimate is not quasi-optimal in general, but is still best possible in the sense of agreement with experiments.