Estimates of the distance to the exact solution of evolutionary reaction-diffusion problems based on local Poincare type inequalities

TL;DR

This paper develops two-sided bounds for the error in solutions to evolutionary reaction-diffusion problems, utilizing local Poincare inequalities and domain decomposition to handle complex geometries and boundary conditions.

Contribution

It introduces a novel approach to estimate the distance to the exact solution using local Poincare inequalities and domain decomposition, providing bounds that are equivalent to the energy norm of the error.

Findings

Derived two-sided bounds for the solution error.

Utilized classical and Poincare type inequalities for error estimation.

Bound constants are estimated based on existing inequalities.

Abstract

The goal of the paper is to derive two-sided bounds of the distance between the exact solution of the evolutionary reaction-diffusion problem with mixed Dirichlet--Robin boundary conditions and any function in the admissible energy space. The derivation is based upon transformation of the integral identity, which defines the generalized solution, and exploits classical Poincare inequalities and Poincare type inequalities for functions with zero mean boundary traces. The corresponding constants are estimated due to Payne and Weinberger, 1960, and Nazarov and Repin, 2013. To handle problems with complex domains and mixed boundary conditions, domain decomposition is used. The corresponding bounds of the distance to the exact solution, contain only constants in local Poincare type inequalities associated with subdomains. Moreover, it is proved that the bounds are equivalent to the energy norm of the error.