On Error Control in the Numerical Solution of Reaction–Diffusion Equation

Abstract

A novel method for deriving a posteriori error bounds for approximate solutions of reaction–diffusion equations is proposed. As a model problem, the problem $- \Delta u + \sigma u = f$ in $\Omega$, ${{\left. u \right|}_{{\partial \Omega }}} = 0$ with an arbitrary constant reaction coefficient $\sigma \geqslant 0$ is studied. For the solutions obtained by the finite element method, bounds, which are called consistent for brevity, are proved. The order of accuracy of these bounds is the same as the order of accuracy of unimprovable a priori bounds. The consistency also assumes that the order of accuracy of such bounds is ensured by test fluxes that satisfy only the corresponding approximation requirements but are not required to satisfy the balance equations. The range of practical applicability of consistent a posteriori error bounds is very wide because the test fluxes appearing in these bounds can be calculated using numerous flux recovery procedures that were intensively developed for error indicators of the residual method. Such recovery procedures often ensure not only the standard approximation orders but also the superconvergency of the recovered fluxes. The advantages of the proposed family of a posteriori bounds are their guaranteed sharpness, no need for satisfying the balance equations in flux recovery procedures, and a much wider range of efficient applicability compared with other a posteriori bounds.