|Reaction-diffusion problems finite elements unstructured grid grid adaption W-method stiffness local partitioning excitable medium spiral wave drift|
We consider numerical methods for reaction-diffusion problems in the plane as well as on curved surfaces. The spatial discretization is carried out with linear finite elements on unstructured triangular grids. Several algorithms for an adaptive grid refinement are presented. A projection technique is proposed for the construction of grids on curved surfaces. The semi-discrete problems are often stiff, either due to strong grid refinement in the presence of diffusion or due to steep gradients of the reaction function. Since in many cases the problem is only locally stiff local partitioning methods might increase the efficiency of numerical solving. We present several partitioning methods based on a W-method. A comparison of numerous partitioning strategies is carried out on three test problems. The numerical methods presented are used for the simulation of excitable media on curved surfaces. These reaction-diffusion problems can have solutions with rotating spiral waves that show an additional drift if the surface is nonuniformly curved. In a numerical study we qualitatively investigate the spiral wave drift on several ellipsoids. As a result a relation between excitability, Gauss curvature, and drift velocity is obtained.