A posteriori verification of the positivity of solutions to elliptic boundary value problems

TL;DR

This paper introduces a unified a posteriori method for verifying the positivity of solutions to elliptic boundary value problems, broadening applicability without requiring $L^{ abla}error$ estimation or $H^2$-regularity.

Contribution

It extends previous approaches by combining a priori eigenvalue bounds with $H^1_0$-error estimates, enabling positivity verification in more general elliptic problems.

Findings

Successfully verified positivity for several elliptic problems previously inaccessible

Developed a method to evaluate constants needed for positivity verification

Extended applicability of positivity verification methods to broader problem classes

Abstract

The purpose of this paper is to develop a unified a posteriori method for verifying the positivity of solutions of elliptic boundary value problems by assuming neither $H^2$-regularity nor $ L^{\infty} $-error estimation, but only $ H^1_0 $-error estimation. In [J. Comput. Appl. Math, Vol. 370, (2020) 112647], we proposed two approaches to verify the positivity of solutions of several semilinear elliptic boundary value problems. However, some cases require $ L^{\infty} $-error estimation and, therefore, narrow applicability. In this paper, we extend one of the approaches and combine it with a priori error bounds for Laplacian eigenvalues to obtain a unified method that has wide application. We describe how to evaluate some constants required to verify the positivity of desired solutions. We apply our method to several problems, including those to which the previous method is not applicable.