Preserving invariance properties of reaction–diffusion systems on stationary surfaces

Frittelli, Massimo, Madzvamuse, Anotida, Sgura, Ivonne and Venktaraman, Chandrasekhar (2017) Preserving invariance properties of reaction–diffusion systems on stationary surfaces. IMA Journal of Numerical Analysis, 2017. pp. 1-36. ISSN 0272-4979

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Abstract

We propose and analyse a lumped surface finite element method for the numerical approximation of reaction–diffusion systems on stationary compact surfaces in R3. The proposed method preserves the invariant regions of the continuous problem under discretization and, in the special case of scalar equations, it preserves the maximum principle. On the application of a fully discrete scheme using the implicit–explicit Euler method in time, we prove that invariant regions of the continuous problem are preserved (i) at the spatially discrete level with no restriction on the meshsize and (ii) at the fully discrete level under a timestep restriction. We further prove optimal error bounds for the semidiscrete and fully discrete methods, that is, the convergence rates are quadratic in the meshsize and linear in the timestep. Numerical experiments are provided to support the theoretical findings. We provide examples in which, in the absence of lumping, the numerical solution violates the invariant region leading to blow-up.

Item Type: Article
Schools and Departments: School of Mathematical and Physical Sciences > Mathematics
Subjects: Q Science > QA Mathematics
Depositing User: Billy Wichaidit
Date Deposited: 31 Oct 2017 09:47
Last Modified: 22 Feb 2018 12:05
URI: http://sro.sussex.ac.uk/id/eprint/70777

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