A mass-lumping finite element method for radially symmetric solution of a multidimensional semilinear heat equation with blow-up

TL;DR

This paper introduces two mass-lumping finite element schemes for radially symmetric solutions of a multidimensional semilinear heat equation, with one accurately capturing blow-up behavior and blow-up time.

Contribution

The paper develops and analyzes two new mass-lumping finite element schemes for semilinear heat equations, including error estimates and blow-up time approximation.

Findings

Scheme (ML-1) achieves optimal convergence in weighted L2 norm.

Scheme (ML-2) reproduces blow-up and accurately estimates blow-up time.

Numerical experiments validate theoretical error estimates and blow-up behavior.

Abstract

This study presents a new mass-lumping finite element method for computing the radially symmetric solution of a semilinear heat equation in an $N$ dimensional ball ($N\ge 2$). We provide two schemes, (ML-1) and (ML-2), and derive their error estimates through the discrete maximum principle. In the weighted $L^{2}$ norm, the convergence of (ML-1) was at the optimal order but that of (ML-2) was only at sub-optimal order. Nevertheless, scheme (ML-2) reproduces a blow-up of the solution of the original equation. In fact, in scheme (ML-2), we could accurately approximate the blow-up time. Our theoretical results were validated in numerical experiments.