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

Toru Nakanishi, Norikazu Saito

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.

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