Apel, Thomas; Nicaise, Serge Elliptic problems in domains with edges: Anisotropic regularity and anisotropic finite element meshes. (English) Zbl 0854.35005 Cea, Jean (ed.) et al., Partial differential equations and functional analysis. In memory of Pierre Grisvard. Proceedings of a conference held in November 1994. Boston, MA: Birkhäuser. Prog. Nonlinear Differ. Equ. Appl. 22, 18-34 (1996). Summary: 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.For the entire collection see [Zbl 0840.00032]. Cited in 4 Documents MSC: 65M99 Numerical methods for partial differential equations, initial value and time-dependent initial-boundary value problems 65N99 Numerical methods for partial differential equations, boundary value problems 35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs 35B65 Smoothness and regularity of solutions to PDEs Keywords:anisotropic weighted Sobolev spaces; local interpolation error estimates PDFBibTeX XMLCite \textit{T. Apel} and \textit{S. Nicaise}, Prog. Nonlinear Differ. Equ. Appl. 22, 18--34 (1996; Zbl 0854.35005)