Tuncer_Madzvamuse_Meir_2015.pdf (4.42 MB)
Projected finite elements for reaction-diffusion systems on stationary closed surfaces
journal contribution
posted on 2023-06-08, 20:48 authored by N Tuncer, Anotida Madzvamuse, A J MeirThis 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.
Funding
Unravelling new mathematics for 3D cell migration; G1438; LEVERHULME TRUST; RPG-2014-149
Mathematical Modelling and Analysis of Spatial Patterning on Evolving Surfaces; G0872; EPSRC-ENGINEERING & PHYSICAL SCIENCES RESEARCH COUNCIL; EP/J016780/1
History
Publication status
- Published
File Version
- Published version
Journal
Applied Numerical MathematicsISSN
0168-9274Publisher
ElsevierExternal DOI
Volume
96Page range
45-71Department affiliated with
- Mathematics Publications
Full text available
- Yes
Peer reviewed?
- Yes
Legacy Posted Date
2015-05-18First Open Access (FOA) Date
2015-05-18First Compliant Deposit (FCD) Date
2015-05-17Usage metrics
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