Robust consistent a posteriori error majorants for approximate solutions of diffusion-reaction equations

TL;DR

This paper introduces a robust a posteriori error majorant for diffusion-reaction equations that remains consistent across all nonnegative reaction coefficients, improving accuracy and efficiency of error estimation in finite element solutions.

Contribution

The paper presents a new guaranteed error majorant that is consistent for any nonnegative reaction coefficient, enhancing existing methods especially for large reaction terms.

Findings

Majorant is effective for any nonnegative reaction coefficient.

For large reaction coefficients, the majorant aligns with Aubin's classical majorant.

The proposed majorant improves upon previous bounds for Poisson and reaction-diffusion equations.

Abstract

Efficiency of the error control of numerical solutions of partial differential equations entirely depends on the two factors: accuracy of an a posteriori error majorant and the computational cost of its evaluation for some test function/vector-function plus the cost of the latter. In the paper, consistency of an a posteriori bound implies that it is the same in the order with the respective unimprovable a priori bound. Therefore, it is the basic characteristic related to the first factor. The paper is dedicated to the elliptic diffusion-reaction equations. We present a guaranteed robust a posteriori error majorant effective at any nonnegative constant reaction coefficient (r.c.). For a wide range of finite element solutions on a quasiuniform meshes the majorant is consistent. For big values of r.c. the majorant coincides with the majorant of Aubin (1972), which, as it is known, for not big r.c. ($<ch^{-2}$) is inconsistent and loses its sense at r.c. approaching zero. Our majorant improves also some other majorants derived for the Poisson and reaction-diffusion equations.