Multilevel Schwarz and multigrid preconditioners for the Bidomain system

Two parallel and scalable multilevel preconditioners for the Bidomain system in computational electrocardiology are introduced and studied. The Bido- main system, consisting of two degenerate parabolic reaction-diffusion equations coupled with a stiff system of several ordinary differential equations, generates very ill-conditioned discrete systems when discretized with semi-implicit methods in time and finite elements in space. The multilevel preconditioners presented in this paper attain the best performance to date, both in terms of convergence rate and solution time and outperform the simpler one-level preconditioners previously introduced. Parallel numerical results, using the PETSc library and run on Linux Clusters, show the scalability of the proposed preconditioners and their efficiency on large- scale simulations of a complete cardiac cycle.