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Preserving invariance properties of reaction–diffusion systems on stationary surfaces

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posted on 2023-06-09, 08:30 authored by Massimo Frittelli, Anotida Madzvamuse, Ivonne Sgura, Chandrasekhar VenkataramanChandrasekhar Venkataraman
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.

Funding

InCeM: Research Training Network on Integrated Component Cycling in Epithelial Cell Motility; G1546; EUROPEAN UNION; 642866 - InCeM

New predictive mathematical and computational models in experimental sciences; G1949; ROYAL SOCIETY; WM160017

Unravelling new mathematics for 3D cell migration; G1438; LEVERHULME TRUST; RPG-2014-149

History

Publication status

  • Published

File Version

  • Published version

Journal

IMA Journal of Numerical Analysis

ISSN

0272-4979

Publisher

Oxford University Press

Issue

1

Volume

39

Page range

235-270

Department affiliated with

  • Mathematics Publications

Full text available

  • Yes

Peer reviewed?

  • Yes

Legacy Posted Date

2017-10-31

First Open Access (FOA) Date

2017-10-31

First Compliant Deposit (FCD) Date

2017-10-31

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