Balancing Neumann-Neumann methods for the cardiac Bidomain model

Balancing Neumann-Neumann preconditioners are constructed, analyzed and numerically studied for the cardiac Bidomain model in three-dimensions. This reaction-diffusion system is discretized by low-order finite elements in space and implicit-explicit methods in time, yielding very ill-conditioned linear systems that must be solved at each time step. The proposed algorithm is based on decomposing the domain into nonoverlapping subdomains and on solving iteratively the Bidomain Schur complement obtained by implicitly eliminating the degrees of freedom interior to each subdomain. The iteration is preconditioned by a Balancing Neumann-Neumann method employing local Neumann solves on each subdomain and a coarse Bidomain solve. A novel approach for the estimation of the average operator of the nonoverlapping decomposition provides a framework for designing coarse spaces for Balancing Neumann-Neumann methods. The theoretical estimates obtained show that the proposed method is scalable, quasi-optimal and robust with respect to possible coefficient discontinuities of the Bidomain operator. The results of extensive parallel numerical tests in three dimensions confirm the convergence rates predicted by the theory.