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Numerical preservation of velocity induced invariant regions for reaction-diffusion systems on evolving surfaces

journal contribution
posted on 2023-06-09, 13:17 authored by Massimo Frittelli, Anotida Madzvamuse, Ivonne Sgura, Chandrasekhar VenkataramanChandrasekhar Venkataraman
We propose and analyse a finite element method with mass lumping (LESFEM) for the numerical approximation of reaction-diffusion systems (RDSs) on surfaces in R3 that evolve under a given velocity field. A fully-discrete method based on the implicit-explicit (IMEX) Euler time-discretisation is formulated and dilation rates which act as indicators of the surface evolution are introduced. Under the assumption that the mesh preserves the Delaunay regularity under evolution, we prove a sufficient condition, that depends on the dilation rates, for the existence of invariant regions (i) at the spatially discrete level with no restriction on the mesh size and (ii) at the fully-discrete level under a timestep restriction that depends on the kinetics, only. In the specific case of the linear heat equation, we prove a semi- and a fully-discrete maximum principle. For the well-known activator-depleted and Thomas reaction-diffusion models we prove the existence of a family of rectangles in the phase space that are invariant only under specific growth laws. Two numerical examples are provided to computationally demonstrate (i) the discrete maximum principle and optimal convergence for the heat equation on a linearly growing sphere and (ii) the existence of an invariant region for the LESFEM-IMEX Euler discretisation of a RDS on a logistically growing surface.

Funding

Coupling Geometric PDEs with Physics; ISAAC NEWTON INSTITUTE FOR MATHEMATICAL SCIENCES; EP/K032208/1

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

Mathematical Modelling and Analysis of Spatial Patterning on Evolving Surfaces; G0872; EPSRC-ENGINEERING & PHYSICAL SCIENCES RESEARCH COUNCIL; EP/J016780/1

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

History

Publication status

  • Published

File Version

  • Published version

Journal

Journal of Scientific Computing

ISSN

0885-7474

Publisher

Springer Verlag

Issue

2

Volume

77

Page range

971-1000

Department affiliated with

  • Mathematics Publications

Full text available

  • Yes

Peer reviewed?

  • Yes

Legacy Posted Date

2018-05-15

First Open Access (FOA) Date

2018-06-26

First Compliant Deposit (FCD) Date

2018-05-14

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