Projected Finite Elements for Systems of Reaction-Diffusion Equations on Closed Evolving Spheroidal Surfaces. (7th February 2017)

• Record Type: Journal Article
• Authors: Tuncer, Necibe; Madzvamuse, Anotida
• Is Part Of: Communications in computational physics. Volume 21:Number 3(2017:Mar.)
• Page Start: 718
• Page End: 747
• Publication Date: 2017-02-07
• Subjects: 65M12, -- 65M60, -- 35K57, -- 35K58, -- 9208; Time-dependent projection operators, -- projected finite elements, -- non-autonomous partial differential equations, -- reaction-diffusion systems, -- evolving surfaces, -- Turing diffusively-driven instability, -- pattern formation; Mathematical physics -- Data processing -- Periodicals; Physics -- Data processing -- Periodicals; 530.150285
• DOI: 10.4208/cicp.OA-2016-0029
• Languages: English
• ISSNs: 1815-2406

Abstract

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 .