The propagation of the action potential in the heart chambers is accurately described bythe Bidomain model, which is commonly accepted and used in the specialistic literature.However, its mathematical structure of a degenerate parabolic system entails highcomputational costs in the numerical solution of the associated linear system. Domaindecomposition methods are a natural way to reduce computational costs, and OptimizedSchwarz Methods have proven in the recent years their effectiveness in accelerating theconvergence of such algorithms. The latter are based on interface matching conditions moreefficient than the classical Dirichlet or Neumann ones. In this paper we analyze anOptimized Schwarz approach for the numerical solution of the Bidomain problem. We assessthe convergence of the iterative method by means of Fourier analysis, and we investigatethe parameter optimization in the interface conditions. Numerical results in 2D and 3D aregiven to show the effectiveness of the method.

The Bidomain model is nowadays one of the most accurate mathematical descriptions of the action potential propagation in the heart. However, its numerical approximation is in general fairly expensive as a consequence of the mathematical features of this system. For this reason, a simplification of this model, called Monodomain problem is quite often adopted in order to reduce computational costs. Reliability of this model is however questionable, in particular in the presence of applied currents and in the regions where the upstroke or the late recovery of the action potential is occurring. In this paper we investigate a domain decomposition approach for this problem, where the entire computational domain is suitably split and the two models are solved in different subdomains. Since the mathematical features of the two problems are rather different, the heterogeneous coupling is non trivial. Here we investigate appropriate interface matching conditions for the coupling on non overlapping domains. Moreover, we pursue an Optimized Schwarz approach for the numerical solution of the heterogeneous problem. Convergence of the iterative method is analyzed by means of a Fourier analysis. We investigate the parameters to be selected in the matching radiation-type conditions to accelerate the convergence. Numerical results both in two and three dimensions illustrate the effectiveness of the coupling strategy.