A Posteriori Error Estimates for Nonconforming Approximations of Evolutionary Convection-Diffusion Problems

TL;DR

This paper develops computable a posteriori error bounds for nonconforming numerical approximations of evolutionary convection-diffusion problems, enabling reliable error estimation without requiring specific solution properties.

Contribution

It introduces a general approach to derive upper bounds for the error in nonconforming approximations of convection-diffusion equations, applicable to various approximation classes.

Findings

Provides explicit computable error estimates

Applicable to nonconforming and piecewise continuous approximations

Dependence only on global embedding constants

Abstract

We derive computable upper bounds for the difference between an exact solution of the evolutionary convection-diffusion problem and an approximation of this solution. The estimates are obtained by certain transformations of the integral identity that defines the generalized solution. These estimates depend on neither special properties of the exact solution nor its approximation, and involve only global constants coming from embedding inequalities. The estimates are first derived for functions in the corresponding energy space, and then possible extensions to classes of piecewise continuous approximations are discussed.