Error estimates in balanced norms of finite element methods on Shishkin meshes for reaction-diffusion problems

TL;DR

This paper develops error estimates in a balanced norm for finite element methods applied to reaction-diffusion problems on Shishkin meshes, effectively capturing layer behavior in singularly perturbed cases.

Contribution

It introduces a balanced norm for error analysis, addressing limitations of traditional energy norms in singularly perturbed reaction-diffusion problems, including anisotropic and semilinear cases.

Findings

Balanced norm accurately reflects layer behavior.

Error estimates are established for various problem types.

The approach improves understanding of finite element performance in singular perturbations.

Abstract

Error estimates of finite element methods for reaction-diffusion Problems are often realized in the related energy norm. In the singularly perturbed case, however, this norm is not adequate. A different scaling of the $H^1$ seminorm leads to a balanced norm which reflects the layer behavior correctly. We discuss also anisotropic problems, semilinear equations, supercloseness and a combination technique.