Numerical verification method for positive solutions of elliptic problems

TL;DR

This paper introduces a verification method to confirm the positivity of weak solutions to elliptic problems using error bounds from numerical approximations, with applications to polynomial nonlinearities.

Contribution

It provides a new sufficient condition for positivity verification of elliptic solutions based on $H^1_0$-error estimates and analyzes its applicability to polynomial nonlinear problems.

Findings

Method successfully verifies positivity in numerical examples.

Applicable to elliptic problems with polynomial nonlinearities.

Provides explicit error bounds for positivity confirmation.

Abstract

The purpose of this paper is to propose methods for verifying the positivity of a weak solution $ u $ of an elliptic problem assuming $ H^1_0 $-error estimation $ \left\|u-\hat{u}\right\|_{H_{0}^{1}} \leq \rho $ given some numerical approximation $ \hat{u} $ and an explicit error bound $ \rho $. We provide a sufficient condition for the solution to be positive and analyze the range of application of our method for elliptic problems with polynomial nonlinearities. We present numerical examples where our method is applied to some important problems.