Multilevel additive Schwarz preconditioners for the Bidomain reaction-diffusion system

Multilevel additive Schwarz methods are analyzed and studied numerically for the anisotropic cardiac Bidomain model in three dimensions. This is the most complete model to date of the bioelectrical activity of the heart tissue, consisting of a degenerate parabolic system of nonlinear reaction-diffusion equations coupled with a stiff system of several ordinary differential equations describing the ionic currents through the cellular membrane. Due to the presence of very different scales in both space and time, the numerical discretization of this system by finite elements in space and semi-implicit methods in time produces very ill-conditioned linear systems that must be solved at each time step. The proposed multilevel algorithm employs a hierarchy of nested meshes with overlapping Schwarz preconditioners on each level and is fully additive, hence parallel, within and among levels. Convergence estimates are proved for the resulting multilevel algorithm, showing that its convergence rate is independent of the number of subdomains (scalability), of the mesh sizes of each level and of the number of levels (optimality). Several parallel tests on a Linux cluster confirm the scalability and optimality of the method, as well as its parallel efficiency on both Cartesian and deformed domains in three dimensions.