Article (Published Version) http://sro.sussex.ac.uk Tuncer, N, Madzvamuse, A and Meir, A J (2015) Projected finite elements for reaction-diffusion systems on stationary closed surfaces. Applied Numerical Mathematics, This version is available from Sussex Research Online: http://sro.sussex.ac.uk/id/eprint/54034/ This document is made available in accordance with publisher policies and may differ from the published version or from the version of record. If you wish to cite this item you are advised to consult the publisher's version. Please see the URL above for details on accessing the published version. Sussex Research Online is a digital repository of the research output of the University. Copyright and all moral rights to the version of the paper presented here belong to the individual author(s) and/or other copyright owners. To the extent reasonable and practicable, the material made available in SRO has been checked for eligibility before being made available. Copies of full text items generally can be reproduced, displayed or performed and given to third parties in any format or medium for personal research or study, educational, or not-for-profit purposes without prior permission or charge, provided that the authors, title and full bibliographic details are credited, a hyperlink and/or URL is given for the original metadata page and the content is not changed in any way. In this paper we present a robust, efficient and accurate finite element method for solving reaction-diffusion systems on stationary spheroidal surfaces (these are surfaces which are deformations of the sphere such as ellipsoids, dumbbells, and heart-shaped surfaces) with potential applications in the areas of developmental biology, cancer research, wound healing, tissue regeneration, and cell motility among many others where such models are routinely used. Our method is inspired by the radially projected finite element method introduced in [39] . (Hence the name "projected" finite element method.) 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. To demonstrate the robustness, applicability and generality of this numerical method, we present solutions of reaction-diffusion systems on various spheroidal surfaces. We show that the method presented in this paper preserves positivity of the solutions of reaction-diffusion equations which is not generally true for Galerkin type methods. We conclude that surface geometry plays a pivotal role in pattern formation. For a fixed set of model parameter values, different surfaces give rise to different pattern generation sequences of either spots or stripes or a combination (observed as circular spot-stripe patterns). These results clearly demonstrate the need for detailed theoretical analytical studies to elucidate how surface geometry and curvature influence pattern formation on complex surfaces.
Recently, REF have introduced a projected Finite Elements Method for studying pattern formation by RD systems on surfaces that can be approximated analytically and later mapped with Lipschitz continuity onto a sphere.
Projected finite elements for reaction-diffusion systems on stationary closed surfaces
Applied Numerical Mathematics
Mathematics