On the Numerical Simulation of Schnakenberg Model on Evolving Surface

The Schnakenberg model can be used to describe the emergence of patterns on the animal skin. The problem is numerically challenging for two reasons. First the organism grows so the computational domain changes. Second the domain is topologically a sphere and hence cannot be considered as a subset of the plane. In our approach we consider the computational domain as a sphere whose Riemannian metric changes and use special parametrization of the sphere to formulate the discrete problem. Our choice of parametrization allows a very convenient way to treat a large class of surfaces in a straightforward way and in a similar way one could treat other PDE systems on surfaces. The same kind of ideas can be used also to compute on surfaces which are not diffeomorphic to a sphere. We have used finite elements in the discretization. We have also analyzed how the eigenfunctions of the Laplacian are related to the emergence of patterns.