Conditional a posteriori error bounds for high order DG time stepping approximations of semilinear heat models with blow-up

TL;DR

This paper develops an adaptive high-order discontinuous Galerkin method with a posteriori error bounds for semilinear heat equations that may blow up, validated by numerical experiments.

Contribution

It introduces a novel adaptive scheme with conditional a posteriori error bounds for high-order DG time stepping in semilinear heat models with potential blow-up.

Findings

Error bounds effectively guide adaptive refinement.

Numerical experiments confirm theoretical accuracy.

Method captures blow-up behavior accurately.

Abstract

This work is concerned with the development of an adaptive numerical method for semilinear heat flow models featuring a general (possibly) nonlinear reaction term that may cause the solution to blow up in finite time. The fully discrete scheme consists of a high order discontinuous Galerkin (dG) time stepping method and a conforming finite element discretisation (cG) in space. The proposed adaptive procedure is based on rigorously devised conditional a posteriori error bounds in the $L^{\infty}(L^{\infty})$ norm. Numerical experiments complement the theoretical results.