Finite Element Approximations for PDEs with Irregular Dirichlet Boundary Data on Boundary Concentrated Meshes

Abstract

Abstract This paper is concerned with finite element error estimates for second order elliptic PDEs with inhomogeneous Dirichlet boundary data in convex polygonal domains. The Dirichlet boundary data are assumed to be irregular such that the solution of the PDE does not belong to H 2 ⁢ ( Ω ) {H^{2}(\Omega)} but only to H r ⁢ ( Ω ) {H^{r}(\Omega)} for some r ∈ ( 1 , 2 ) {r\in(1,2)} . As a consequence, a discretization of the PDE with linear finite elements exhibits a reduced convergence rate in L 2 ⁢ ( Ω ) {L^{2}(\Omega)} and H 1 ⁢ ( Ω ) {H^{1}(\Omega)} . In order to restore the best possible convergence rate we propose and analyze in detail the usage of boundary concentrated meshes. These meshes are gradually refined towards the whole boundary. The corresponding grading parameter does not only depend on the regularity of the Dirichlet boundary data and their discrete implementation but also on the norm, which is used to measure the error. In numerical experiments we confirm our theoretical results.