DD29 Advanced Courses

We start by introducing general and abstract principles, both at the continuous and at the numerical level, about the coupling of heterogeneous problems, arising from different physics, different geometric description or different mathematical modeling of the same physics. Then we describe three example of such heterogeneous domain decomposition problems for hemodynamic applications:
i) the fluid-structure interaction problems arising for compliant vessels and for the cardiac muscle; 
ii) the 3D-1D-0D coupling of the vascular tree where different geometric details are used in different portion of the system;
iii) the Navier-Stokes/multi-compartmen Darcy coupling for the cardiac perfusion.
In all the cases, we start by writing the continuous problem with the coupling conditions and then we address the numerical solution by means of DD methods.

Abstract: The spread of electrical impulse in the cardiac tissue is mathematically described by systems of nonlinear partial differential equations (PDEs). The most complete model is the Bidomain model, a degenerate system of two parabolic reaction-diffusion PDEs for the intra- and extracellular electric potentials. The numerical solution of this system is highly computational demanding, thus reduced model have been developed in the literature. The most popular one is the Monodomain model, consisting of a parabolic reaction-diffusion PDE for the transmembrane potential. Both models are coupled through the nonlinear reaction term with stiff systems of ordinary differential equations modeling the electrical behavior of the cellular membrane.
The space and time discretizations of the Bidomain and Monodomain models yield at each time step the solution of large scale ill conditioned linear systems. In this lecture we will focus on the parallel solvers that have been developed in the last decade to speed up the solution of such models, including mainly multilevel overlapping Schwarz, dual primal nonoverlapping and algebraic multigrid preconditioners.