On the performance of anisotropic mesh adaptation for scroll wave turbulence dynamics in reaction-diffusion systems

Nonlinear reaction-diffusion systems are widely employed to study the spatio-temporal chaotic behavior that occurs in excitable media such as cardiac tissue where sufficiently strong perturbations can excite nonlinear propagating waves which can form spiral waves in two dimensions or scroll waves in three dimensions. The numerical simulation of these waves calls for grids that are extremely fine over the whole computational domain to accurately predict the trajectories and multiplication of wave fronts and therefore leads to huge computational challenges. Mesh adaptation methods can reduce the number of degrees of freedom required for a given accuracy but they also have a cost and it is not clear if they are competitive with respect to very fine uniform meshes. Previous mesh adaptation techniques applied to spatio-temporal chaotic behavior have been mostly limited to the two-dimensional case. The purpose of this paper is to explore the efficiency of a three-dimensional anisotropic finite element mesh adaptivity for simulating scroll wave turbulence. The computational efficiency of the proposed method is assessed using reference solutions obtained on a uniform refined mesh with more than 44 millions degrees of freedom. The proposed method reduces significantly the number of elements leading to huge saving in memory as well as in computational time. Examples of the dynamics of ventricular fibrillation in cardiac tissue will be presented illustrating the performance of the overall methodology.