Finite element method with local damage on the mesh

TL;DR

This paper analyzes finite element methods on meshes with local damage, demonstrating that standard error estimates hold and proposing an alternative scheme with optimal convergence and good conditioning.

Contribution

It introduces a finite element scheme that remains accurate and well-conditioned on damaged meshes with distorted cells, extending the applicability of FEM.

Findings

Standard error estimates remain valid on damaged meshes.

Proposed scheme is optimally convergent.

Scheme has conditioning comparable to regular meshes.

Abstract

We consider the finite element method on locally damaged meshes allowing for some distorted cells which are isolated from one another. In the case of the Poisson equation and piecewise linear Lagrange finite elements, we show that the usual a priori error estimates remain valid on such meshes. We also propose an alternative finite element scheme which is optimally convergent and, moreover, well conditioned, i.e. its conditioning is of the same order as that of a standard finite element method on a regular mesh of comparable size.