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Projected Finite Elements for Systems of Reaction-Diffusion Equations on Closed Evolving Spheroidal Surfaces

Part of: Partial differential equations, initial value and time-dependent initial-boundary value problems Parabolic equations and systems

Published online by Cambridge University Press:  07 February 2017

Necibe Tuncer  and
Anotida Madzvamuse
Necibe Tuncer*
Affiliation:
Department of Mathematics, Florida Atlantic University, 777 Glades Road, Boca Raton, Fl 33431, USA
Anotida Madzvamuse*
Affiliation:
University of Sussex, School of MPS, Department of Mathematics, BN1 9QH, Brighton, UK
*
*Corresponding author. Email addresses:[email protected] (N. Tuncer), [email protected] (A. Madzvamuse)
*Corresponding author. Email addresses:[email protected] (N. Tuncer), [email protected] (A. Madzvamuse)
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Abstract

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.

Keywords

Time-dependent projection operatorsprojected finite elementsnon-autonomous partial differential equationsreaction-diffusion systemsevolving surfacesTuring diffusively-driven instabilitypattern formation

MSC classification

Secondary: 65M12: Stability and convergence of numerical methods 65M60: Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods 35K57: Reaction-diffusion equations 35K58: Semilinear parabolic equations
Type
Research Article
Information
Communications in Computational Physics , Volume 21 , Issue 3 , March 2017 , pp. 718 - 747
Copyright
Copyright © Global-Science Press 2017 

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