An Elementary method of Deriving A Posteriori Error Equalities and Estimates for Linear Partial Differential Equations

TL;DR

This paper introduces a straightforward, continuous-level method for deriving a posteriori error equalities and estimates for linear elliptic and parabolic PDEs, considering both primal and dual variables without relying on specific discretizations.

Contribution

It presents a novel, simple approach to obtain a posteriori error estimates at the functional level for linear PDEs, independent of numerical discretization properties.

Findings

Provides error equalities and estimates in combined norms

Applicable to elliptic and parabolic PDEs

Does not depend on specific numerical methods

Abstract

In this paper we present a simple method of deriving a posteriori error equalities and estimates for linear elliptic and parabolic partial differential equations. The error is measured in a combined norm taking into account both the primal and dual variables. We work only on the continuous (often called functional) level and do not suppose any specific properties of numerical methods and discretizations.