A hybrid multilevel Schwarz method for the bidomain model

A hybrid multilevel Schwarz method is studied numerically for the anisotropic Bidomain model in both two and three dimensions. This multiscale system models the electrical activity of the heart and it consists of two degenerate parabolic non-linear reaction–diffusion equations, coupled with a stiff system of ordinary differential equations. The numerical discretization of the whole system by finite elements in space and semi-implicit methods in time generates ill-conditioned linear systems that must be solved at each time step. The multilevel algorithm studied employs a hierarchy of nested meshes with overlapping Schwarz preconditioners on each level and is additive within the levels and multiplicative among the levels. We perform several parallel tests on two Linux clusters, showing that the convergence of the method is independent of the number of subdomains (scalability), the discretization parameters and the number of levels (optimality). Moreover the comparison with the traditional Block Jacobi ILU parallel preconditioner and the V-cycle Multigrid parallel preconditioner shows that, on a whole heart cycle simulation, the proposed method attains the best performances in terms of CPU times.