Projected finite elements for reaction–diffusion systems on stationary closed surfaces

• Published in: Applied numerical mathematics 2015-10, Vol.96, p.45-71
• Main Authors: Tuncer, N., Madzvamuse, A., Meir, A.J.
• Format: Article
• Language: English
• Subjects: Discretization; Finite element method; Formations; Mathematical analysis; Mathematical models; Pattern formation; Pattern generation; Preserves; Radially projected finite elements; Reaction–diffusion systems; Surface geometry; Surface partial differential equations
• ISSN: 0168-9274; 1873-5460
• DOI: 10.1016/j.apnum.2014.12.012

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. •To our knowledge first article to present the radially projected finite element for reaction–diffusion systems on surfaces.•The method gives a geometrically exact discretization of the surfaces.•We demonstrate that surface geometry and curvature play pivotal roles in pattern formation.