Full Finite Element Scheme for Reaction-Diffusion Systems on Embedded Curved Surfaces in ℝ 3

The purpose of this article is to study numerically the Turing diffusion-driven instability mechanism for pattern formation on curved surfaces embedded in ℝ3, specifically the surface of the sphere and the torus with some well-known kinetics. To do this, we use Euler’s backward scheme for discretizing time. For spatial discretization, we parameterize the surface of the torus in the standard way, while for the sphere, we do not use any parameterization to avoid singularities. For both surfaces, we use finite element approximations with first-order polynomials.