Projected finite elements for systems of reaction-diffusion equations on closed evolving spheroidal surfaces

Tuncer, Necibe and Madzvamuse, Anotida (2017) Projected finite elements for systems of reaction-diffusion equations on closed evolving spheroidal surfaces. Communications in Computational Physics, 21 (3). pp. 718-747. ISSN 1815-2406

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The focus of this article is to present the projected finite element method for solving systems of reaction-diffusion equations on evolving closed spheroidal surfaces with applications to pattern formation. The advantages of the projected finite element method are that it is easy to implement and that it provides a conforming finite element discretization which is ``logically'' rectangular. Furthermore, the surface is not approximated but described exactly through the projection. The surface evolution law is incorporated into the projection operator resulting in a time-dependent operator. The time-dependent projection operator is composed of the radial projection with a Lipschitz continuous mapping. The projection operator is used to generate the surface mesh whose connectivity remains constant during the evolution of the surface. To illustrate the methodology several numerical experiments are exhibited for different surface evolution laws such as uniform isotropic (linear, logistic and exponential), anisotropic, and concentration-driven. This numerical methodology allows us to study new reaction-kinetics that only give rise to patterning in the presence of surface evolution such as the activator-activator and short-range inhibition; long-range activation.

Item Type: Article
Keywords: Time-dependent projection operators, Projected finite elements, Non-autonomous partial differential equations, Reaction-diffusion systems, Evolving surfaces, Turing diffusively-driven instability; Pattern formation
Schools and Departments: School of Mathematical and Physical Sciences > Mathematics
Research Centres and Groups: Mathematics Applied to Biology Research Group
Subjects: Q Science
Q Science > QA Mathematics
Q Science > QA Mathematics > QA0297 Numerical analysis
Q Science > QA Mathematics > QA0299 Analysis. Including analytical methods connected with physical problems
Depositing User: Anotida Madzvamuse
Date Deposited: 08 Sep 2016 10:53
Last Modified: 15 Aug 2017 12:33

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Project NameSussex Project NumberFunderFunder Ref
Mathematical Modelling and Analysis of Spatial Patterning on Evolving SurfacesG0872EPSRC-ENGINEERING & PHYSICAL SCIENCES RESEARCH COUNCILEP/J016780/1
Unravelling new mathematics for 3D cell migrationG1438LEVERHULME TRUSTRPG-2014-149
InCeM: Research Training Network on Integrated Component Cycling in Epithelial Cell MotilityG1546EUROPEAN UNION642866 - InCeM
Simons Foundation FellowUnsetSimons FoundationUnset