Functional a posteriori estimates for the reaction-diffusion problem

Abstract

The Note is concerned with functional type a posteriori estimates for stationary reaction–diffusion problems. Functional a posteriori estimates are derived on purely functional grounds without using any type of the Galerkin orthogonality condition and special properties of approximation spaces. Therefore, they contain no mesh-dependent constants and provide guaranteed error bounds for any conforming approximation. Generalizations to non-conforming approximations are also possible. Estimates derived in the Note are equally efficient for the problems with constant reaction parameter and for those admitting a high variability of it in different parts of the domain. Such a robustness with respect to the reaction parameter is important because in applications the reaction parameter my often be large in one subdomain and almost zero in another one. It is shown that the a posteriori bounds obtained are directly computable and provide sharp error bounds.