Improved energy-norm a posteriori error estimates for singularly perturbed reaction-diffusion problems on anisotropic meshes

TL;DR

This paper improves energy-norm a posteriori error estimates for singularly perturbed reaction-diffusion problems on anisotropic meshes by refining the jump residual weights using a new scaled trace theorem involving hat basis functions.

Contribution

It introduces a sharper scaled trace theorem for anisotropic elements, enhancing the weights in the jump residual part of the error estimator.

Findings

Error constants are independent of mesh element size and aspect ratio.

The improved estimator provides more accurate error bounds.

The method is applicable to anisotropic triangulations in reaction-diffusion problems.

Abstract

In the recent article [Kopteva, N., Numer. Math., 137, 607--642 (2017)] the author obtained residual-type a posteriori error estimates in the energy norm for singularly perturbed semilinear reaction-diffusion equations on anisotropic triangulations. The error constants in these estimates are independent of the diameters and the aspect ratios of mesh elements and of the small perturbation parameter. The purpose of this note is to improve the weights in the jump residual part of the estimator. This is attained by using a novel sharper version of the scaled trace theorem for anisotropic elements, in which the hat basis functions are involved as weights.