Functional A Posteriori Error Equalities for Conforming Mixed Approximations of Elliptic Problems

TL;DR

This paper presents a method to compute the exact error of conforming mixed approximations for elliptic problems using functional a posteriori estimates, eliminating the need for auxiliary data and enabling straightforward error calculation.

Contribution

It introduces a novel approach to determine the exact global error in mixed approximations of elliptic PDEs using only problem data and the approximation itself.

Findings

Exact error can be computed without auxiliary data.

Method verified through numerical experiments.

Applicable to a class of elliptic PDEs.

Abstract

In this paper we show how to find the exact error (not just an estimate of the error) of a conforming mixed approximation by using the functional type a posteriori error estimates in the spirit of Repin. The error is measured in a mixed norm which takes into account both the primal and dual variables. We derive this result for elliptic partial differential equations of a certain class. We first derive a special version of our main result by using a simplified reaction-diffusion problem to demonstrate the strong connection to the classical functional a posteriori error estimates of Repin. After this we derive the main result in an abstract setting. Our main result states that in order to obtain the exact global error value of a conforming mixed approximation one only needs the problem data and the mixed approximation of the exact solution. There is no need for calculating any auxiliary data. The calculation of the exact error consists of simply calculating two (usually integral) quantities where all the quantities are known after the approximate solution has been obtained by any conforming method. We also show some numerical computations to confirm the results.