Anisotropic mesh refinement in polyhedral domains: error estimates with data in L^2(\Omega)

TL;DR

This paper develops new error estimates for finite element solutions of the Poisson equation on 3D polyhedral domains with L^2 data, using anisotropic graded meshes and a novel quasi-interpolation operator.

Contribution

It introduces a new quasi-interpolation operator enabling L^2 data error estimates, extending previous results for anisotropic mesh refinement in polyhedral domains.

Findings

Error estimates in L^2( abla u) and L^2( abla^2 u) norms achieved

Extension of error estimates to optimal control problems

Simplified proof of discrete compactness property for edge elements

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(\Omega)- and L^2(\Omega)-norms by using a new quasi-interpolation operator. This new interpolant is introduced in order to prove the estimates for L^2(\Omega)-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 equation and for a simpler proof of the discrete compactness property for edge elements of any order on this kind of finite element meshes.