Elliptic Problems in Domains with Edges: Anisotropic Regularity and Anisotropic Finite Element Meshes

Abstract

This paper is concerned with the anisotropic singular behaviour of the solution of elliptic boundary value problems in domains with edges and its consequences for anisotropic FEM. We first deal with the description of the analytic properties of the solution in newly defined anisotropic weighted Sobolev spaces. The finite element method with anisotropic, graded meshes and piecewise linear shape functions is then investigated for such problems; the schemes exhibit optimal convergence rates with decreasing mesh size. For the proof, new local interpolation error estimates in anisotropically weighted Sobolev spaces are derived.