Pointwise a posteriori error bounds for blow-up in the semilinear heat equation

TL;DR

This paper develops a space-time adaptive finite element method with rigorous a posteriori error bounds for the semilinear heat equation, effectively handling solutions that may blow up in finite time.

Contribution

It introduces a novel a posteriori error estimation technique for blow-up problems in the semilinear heat equation, applicable in both blow-up and non blow-up scenarios.

Findings

Effective error bounds for blow-up solutions

Numerical experiments confirm theoretical results

Method applicable to general Lipschitz reaction terms

Abstract

This work is concerned with the development of a space-time adaptive numerical method, based on a rigorous a posteriori error bound, for the semilinear heat equation with a general local Lipschitz reaction term whose solution may blow-up in finite time. More specifically, conditional a posteriori error bounds are derived in the $L^{\infty}L^{\infty}$ norm for a first order in time, implicit-explicit (IMEX), conforming finite element method in space discretization of the problem. Numerical experiments applied to both blow-up and non blow-up cases highlight the generality of our approach and complement the theoretical results.