Two-sided estimates of the solution set for the reaction-diffusion problem with uncertain data

TL;DR

This paper develops computable bounds for the solution set of reaction-diffusion problems with uncertain data, using a posteriori error estimates to quantify the impact of data variations on solutions.

Contribution

It introduces a method to estimate the diameter of the solution set for reaction-diffusion problems with uncertain parameters using functional error bounds.

Findings

Derived explicit bounds for the solution set diameter.

Utilized a posteriori error majorants and minorants.

Applicable to problems with bounded data variations.

Abstract

We consider linear reaction--diffusion problems with mixed Diriclet-Neumann-Robin conditions. The diffusion matrix, reaction coefficient, and the coefficient in the Robin boundary condition are defined with an uncertainty which allow bounded variations around some given mean values. A solution to such a problem cannot be exactly determined (it is a function in the set of "possible solutions" formed by generalized solutions related to possible data). The problem is to find parameters of this set. In this paper, we show that computable lower and upper bounds of the diameter (or radius) of the set can be expressed throughout problem data and parameters that regulate the indeterminacy range. Our method is based on using a posteriori error majorants and minorants of the functional type (see monographs Neittaanm\"aki&Repin 2004, Repin 2008), which explicitly depend on the coefficients and allow to obtain the corresponding lower and upper bounds by solving the respective extremal problems generated by indeterminacy of coefficients.