Preserving invariance properties of reaction-diffusion systems on stationary surfaces
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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.
Frittelli , M , Madzvamuse , A , Sgura , I & Venkataraman , C 2017 , ' Preserving invariance properties of reaction-diffusion systems on stationary surfaces ' IMA Journal of Numerical Analysis , vol Advance articles . DOI: https://doi.org/10.1093/imanum/drx058
IMA Journal of Numerical Analysis
© The authors 2017. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted reuse, distribution, and reproduction in any medium, provided the original work is properly cited.
This work (AM, CV) is partly supported by the EPSRC grant number EP/J016780/1 and the Leverhulme Trust Research Project Grant (RPG-2014-149). The authors (MF, AM, IS CV) would like to thank the Isaac Newton Institute for Mathematical Sciences for its hospitality during the programme [Coupling Geometric PDEs with Physics for Cell Morphology, Motility and Pattern Formation] supported by EPSRC Grant Number EP/K032208/1. AM acknowledges funding from the European Union Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No 642866 and was partially supported by a grant from the Simons Foundation. AM is a Royal Society Wolfson Research Merit Award Holder funded generously by the Wolfson Foundation.
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