NON-OVERLAPPING DOMAIN DECOMPOSITION METHODS FOR CARDIAC REACTION-DIFFUSION MODELS AND APPLICATIONS

In this thesis we consider different aspects related to the mathematical modeling of cardiac electrophysiology, either from the cellular or from the tissue perspective, and we develope novel numerical methods for the parallel iterative solution of the resulting reaction-diffusion models. In Chapter one we develope and validate the HHRd model, which accounts for transmural cellular heterogeneities of the canine left ventricle. Next, we introduce the reaction-diffusion models describing the spread of excitation in cardiac tissue, namely the anisotropic Bidomain and Monodomain models. For their discretization, we consider trilinear isoparametric finite elements in space and a semi-implicit (IMEX) method in time. In order to reduce the computational costs of parallel three-dimensional cardiac simulations, in Chapter three we consider different strategies to accelerate convergence of the Preconditioned Conjugate Gradient method. We consider novel choices for the Krylov initial guess in order to reduce the number of iterations per time step, either lagrangian interpolants in time or the Proper Orthogonal Decomposion technique combined with a usual Galerkin projection. In the last three chapters we construct non-overlapping domain decomposition methods for both cardiac reaction-diffusion models. In Chapter four we deal with preconditioners of the Neumann-Neumann type, in particular we consider the additive Neumann-Neumann method for the Monodomain model and the Balancing Neumann-Neumann method for the Bidomain model. In Chapter five we construct a Balancing Domain Decomposition by Constraint (BDDC) method for the Bidomain model, whereas in Chapter six we investigate the use of an approximate BDDC method for the Bidomain model. For all preconditioners considered, we develope novel theoretical estimates for the condition number of the preconditioned systems with respect to the spatial discretization, to the subdomains' diameter and to the time step, also in case of discontinuity in the conductivity coefficients of the cardiac tissue, with jumps aligned with the interface among subdomains. We prove scalability and quasi-optimality for the balancing methods, providing parallel numerical results confirming the theoretical estimates.