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Preserving invariance properties of reaction–diffusion systems on stationary surfaces
journal contribution
posted on 2023-06-09, 08:30 authored by Massimo Frittelli, Anotida MadzvamuseAnotida Madzvamuse, Ivonne Sgura, Chandrasekhar VenkataramanChandrasekhar VenkataramanWe 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
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Publication status
- Published
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- Published version
Journal
IMA Journal of Numerical AnalysisISSN
0272-4979Publisher
Oxford University PressExternal DOI
Issue
1Volume
39Page range
235-270Department affiliated with
- Mathematics Publications
Full text available
- Yes
Peer reviewed?
- Yes
Legacy Posted Date
2017-10-31First Open Access (FOA) Date
2017-10-31First Compliant Deposit (FCD) Date
2017-10-31Usage metrics
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