Local energy estimates for the finite element method on sharply varying grids

TL;DR

This paper extends local energy error estimates for finite element methods to shape regular, highly graded meshes, improving upon classical results that were limited to quasi-uniform grids, thus better reflecting practical computational scenarios.

Contribution

It introduces local a priori energy estimates valid on shape regular, graded meshes, with a key innovation in superapproximation analysis, broadening the applicability of error bounds.

Findings

Estimates hold on highly graded, shape regular meshes.

Superapproximation results are improved.

Error bounds include local approximation and global pollution terms.

Abstract

Local energy error estimates for the finite element method for elliptic problems were originally proved in 1974 by Nitsche and Schatz. These estimates show that the local energy error may be bounded by a local approximation term, plus a global "pollution" term that measures the influence of solution quality from outside the domain of interest and is heuristically of higher order. However, the original analysis of Nitsche and Schatz is restricted to quasi-uniform grids. We present local a priori energy estimates that are valid on shape regular grids, an assumption which allows for highly graded meshes and which much more closely matches the typical practical situation. Our chief technical innovation is an improved superapproximation result.