Lumped finite elements for reaction-cross-diffusion systems on stationary surfaces

We consider a lumped surface finite element method (LSFEM) for the spatial approximation of reaction-diffusion equations on closed compact surfaces in R3 in the presence of cross-diffusion. We provide a fully-discrete scheme by applying the Implicit-Explicit (IMEX) Euler method. We provide sufficient conditions for the existence of polytopal invariant regions for the numerical solution after spatial and full discretisations. Furthermore, we prove optimal error bounds for the semi- and fully-discrete methods, that is the convergence rates are quadratic in the meshsize and linear in the timestep. To support our theoretical findings, we provide two numerical tests. The first test confirms that in the absence of lumping numerical solutions violate the invariant region leading to blow-up due to the nature of the kinetics. The second experiment is an example of Turing pattern formation in the presence of cross-diffusion on the sphere.