Numerical verification of solutions for nonlinear parabolic problems

TL;DR

This paper introduces a numerical verification method for solutions of nonlinear parabolic problems, combining finite element techniques with theoretical heat equation analysis, and demonstrates its effectiveness through verified examples.

Contribution

It develops a novel combined approach using Nakao's projection method and heat equation analysis for verifying solutions of nonlinear parabolic problems.

Findings

Successfully verified solutions near stationary states

Confirmed the effectiveness of the proposed method

Provided concrete verified examples

Abstract

In this paper, we present a numerical verification method of solutions for nonlinear parabolic initial boundary value problems. Decomposing the problem into a nonlinear part and an initial value part, we apply Nakao's projection method, which is based on the full-discrete finite element method with constructive error estimates, to the nonlinear part and use the theoretical analysis for the heat equation to the initial value part, respectively. We show some verified examples for solutions of nonlinear problems from initial value to the neighborhood of the stationary solutions, which confirm us the actual effectiveness of our method.