Anisotropic regularity and optimal rates of convergence for the Finite Element Method on three dimensional polyhedral domains

TL;DR

This paper reviews anisotropic weighted regularity for the Laplace problem on 3D polyhedral domains and demonstrates how these results inform the design of efficient finite element methods for elliptic PDEs.

Contribution

It provides a comprehensive review of anisotropic regularity results and their application to finite element discretization of elliptic problems on polyhedral domains.

Findings

Regularity results in weighted Sobolev spaces for 3D polyhedral domains.

Enhanced finite element discretization strategies based on anisotropic regularity.

Extension of results to interface problems in elasticity.

Abstract

We consider the model Poisson problem $-\Delta u = f \in \Omega$, $u = g$ on $\pa \Omega$, where $ \Omega $ is a bounded polyhedral domain in $\RR^n$. The objective of the paper is twofold. The first objective is to review the well posedness and the regularity of our model problem using appropriate weighted spaces for the data and the solution. We use these results to derive the domain of the Laplace operator with zero boundary conditions on a concave domain, which seems not to have been fully investigated before. We also mention some extensions of our results to interface problems for the Elasticity equation. The second objective is to illustrate how anisotropic weighted regularity results for the Laplace operator in 3D are used in designing efficient finite element discretizations of elliptic boundary value problems, with the focus on the efficient discretization of the Poisson problem on polyhedral domains in $\RR^3$, following {\em Numer. Funct. Anal. Optim.}, 28(7-8):775--824, 2007. The anisotropic weighted regularity results described and used in the second part of the paper are a consequence of the well-posedness results in (isotropically) weighted Sobolev spaces described in the first part of the paper. The paper is based on the talk by the last named author at the Congress of Romanian Mathematicians, Brasov 2011, and is largely a survey paper.