Functional A Posteriori Error Control for Conforming Mixed Approximations of the Reaction-Convection-Diffusion Problem

TL;DR

This paper introduces a method to precisely measure the global error in conforming mixed approximations of the reaction-convection-diffusion problem using functional a posteriori error control, applicable under specific conditions.

Contribution

It provides an exact error formula for conforming mixed approximations and extends to two-sided estimates when certain operator conditions are not met.

Findings

Exact global error value expressed by a functional with known quantities

Error equalities hold under isometry conditions of solution operators

Two-sided error estimates derived for general cases

Abstract

In this paper we show how to obtain the exact value of the global error of a conforming mixed approximation of the reaction-convection-diffusion problem. We operate in the framework of functional type a posteriori error control. The error is measured in a combined norm which takes into account both the primal and dual variables. Our main results state that the exact global error value of a conforming mixed approximation is given by a functional which includes only known quantities. The presented error equalities hold with certain restrictions on the reaction coefficient and the convection vector, namely, under the conditions when the solution operators of the corresponding problems are isometries. For the case where these restrictions are not satisfied we derive a two-sided error estimate.