Projected Finite Elements for Systems of Reaction-Diffusion Equations on Closed Evolving Spheroidal Surfaces (Communications in Computational Physics, 2018)
The focus of this article is to present the projected finite element methodfor solving systems of reaction-diffusion equations on evolving closed spheroidal surfaceswith applications to pattern formation. The advantages of the projected finiteelement method are that it is easy to implement and that it provides a conforming finite element discretization which is "logically" rectangular. Furthermore, the surfaceis not approximated but described exactly through the projection. The surface evolutionlaw is incorporated into the projection operator resulting in a time-dependentoperator. The time-dependent projection operator is composed of the radial projectionwith a Lipschitz continuous mapping. The projection operator is used to generate thesurface mesh whose connectivity remains constant during the evolution of the surface.To illustrate the methodology several numerical experiments are exhibited for differentsurface evolution laws such as uniform isotropic (linear, logistic and exponential),anisotropic, and concentration-driven. This numerical methodology allows us to studynew reaction-kinetics that only give rise to patterning in the presence of surface evolutionsuch as the activator-activator and short-range inhibition; long-range activation.