Higher order energy-corrected finite element methods

TL;DR

This paper extends energy-corrected finite element methods to higher orders, effectively eliminating pollution effects and achieving optimal convergence in weighted and standard norms for elliptic PDEs in polygonal domains with re-entrant corners.

Contribution

It introduces a parameter-dependent local modification of the stiffness matrix for higher order finite elements, removing pollution effects and enabling optimal error estimates.

Findings

Pollution effects are eliminated with the modified methods.

Optimal order convergence is achieved in weighted L2-norms.

A simple post-processing step recovers optimal convergence in standard L2-norm.

Abstract

The regularity of the solution of elliptic partial differential equa- tions in a polygonal domain with re-entrant corners is, in general, reduced compared to the one on a smooth convex domain. This results in a best approximation property for standard norms which depend on the re-entrant corner but does not increase with the polynomial degree. Standard Galerkin approximations are moreover affected by a global pollution effect. Even in the far field no optimal error reduction can be observed. Here, we generalize the energy-correction method for higher order finite elements. It is based on a parameter-dependent local modification of the stiffness matrix. We will show firstly that for such modified finite element approximation the pollution effect does not occur and thus optimal order estimates in weighted L2-norms can be obtained. Two different modification techniques are introduced and illustrated numerically. Secondly we propose a simple post-processing step such that even with respect to the standard L2-norm optimal order convergence can be recovered.