Decoupled Schwarz algorithms for implicit discretizations of nonlinear monodomain and bidomain systems

Novel decoupled Schwarz algorithms for the implicit discretizations of the Monodomain and Bidomain systems in three dimensions are constructed and analyzed. Both implicit Euler and linearly implicit Rosenbrock time discretizations are considered. Convergence rate estimates are proven for a domain decomposition preconditioner based on overlapping additive Schwarz techniques and employed in a Newton–Krylov–Schwarz method for the Euler scheme. An analogous result is proven for the same preconditioner applied to the linear systems originated in the Rosenbrock scheme. Several parallel numerical results in three dimensions confirm the convergence rates predicted by the theory and study the performance of our algorithms for a complete heartbeat, for both Bidomain and Monodomain models, Euler and Rosenbrock schemes, fixed time step and adaptive strategies. The results also show the considerable CPU-time reduction of our decoupled Schwarz solvers with respect to fully-implicit Schwarz solvers.