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

Tuncer, N, Madzvamuse, A and Meir, A J (2015) Projected finite elements for reaction-diffusion systems on stationary closed surfaces. Applied Numerical Mathematics, 96. pp. 45-71. ISSN 0168-9274

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This paper presents 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 by Meir and Tuncer (2009), 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.

Item Type: Article
Keywords: radially projected finite elements, reaction-diffusion systems, pattern formation, surface geometry, surface partial differential equations.
Schools and Departments: School of Mathematical and Physical Sciences > Mathematics
Subjects: Q Science
Q Science > QA Mathematics
Q Science > QA Mathematics > QA0297 Numerical analysis
Depositing User: Anotida Madzvamuse
Date Deposited: 18 May 2015 09:47
Last Modified: 11 Mar 2017 07:32

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