Lumped finite elements for reaction–cross-diffusion systems on stationary surfaces

Frittelli, Massimo, Madzvamuse, Anotida, Sgura, Ivonne and Venkataraman, Chandrasekhar (2017) Lumped finite elements for reaction–cross-diffusion systems on stationary surfaces. Computers & Mathematics with Applications. ISSN 0898-1221

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Abstract

We consider a lumped surface finite element method (LSFEM) for the spatial approximation of reaction–diffusion equations on closed compact surfaces in R3R3 in the presence of cross-diffusion. We provide a fully-discrete scheme by applying the Implicit–Explicit (IMEX) Euler method. We provide sufficient conditions for the existence of polytopal invariant regions for the numerical solution after spatial and full discretisations. Furthermore, we prove optimal error bounds for the semi- and fully-discrete methods, that is the convergence rates are quadratic in the meshsize and linear in the timestep. To support our theoretical findings, we provide two numerical tests. The first test confirms that in the absence of lumping numerical solutions violate the invariant region leading to blow-up due to the nature of the kinetics. The second experiment is an example of Turing pattern formation in the presence of cross-diffusion on the sphere.

Item Type: Article
Schools and Departments: School of Mathematical and Physical Sciences > Mathematics
Subjects: Q Science > QA Mathematics
Depositing User: Billy Wichaidit
Date Deposited: 01 Sep 2017 15:17
Last Modified: 04 Sep 2017 10:24
URI: http://sro.sussex.ac.uk/id/eprint/69966

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