Anisoptropic mesh refinement in polyhedral domains: Error estimates with data in $L^2(\Omega)$

THOMAS APEL; ARIEL L. LOMBARDI; MAX WINKLER

ESAIM-MATHEMATICAL MODELLING AND NUMERICAL ANALYSIS-MODELISATION MATHEEMATIQUE ET ANALYSE NUMERIQUE

EDP SCIENCES S A

Lugar: Paris; Año: 2014 vol. 48 p. 1117 - 1145

0764-583X

Abstract . The paper is concerned with the finite element solution of the Poisson equation with homogeneous Dirichlet boundary condition in a three-dimensional domain. Anisotropic, graded meshes from a former paper are reused for dealing with the singular behaviour of the solution in the vicinity of the non-smooth parts of the boundary. The discretization Error is analyzed for the piecewise linear approximation in the H^1(§Ù)- and L^2(§Ù)-norms by using a new quasi-interpolation operator. This new interpolant is introduced in order to prove the estimates for L^2(§Ù)-data in the differential equation which is not possible for the standard nodal interpolant. These new estimates allow for the extension of certain error estimates for optimal control problems with elliptic partial differential equations and for a simpler proof of the discrete compactness property for edge elements of any order on this kind of finite element meshes.