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Two novel parallel Newton-Krylov Balancing Domain Decomposition byConstraints (BDDC) and Dual-Primal Finite Element Tearing and Interconnecting(FETI-DP) solvers are here constructed, analyzed and tested numerically forimplicit time discretizations of the three-dimensional Bidomain system ofequations.  This model represents the most advanced mathematical description of thecardiac bioelectrical activity and it consists of a degenerate system of twonon-linear reaction-diffusion partial differential equations (PDEs), coupledwith a stiff system of ordinary differential equations (ODEs).  A finite element discretization in space and a segregated implicitdiscretization in time, based on decoupling the PDEs from the ODEs, yields ateach time step the solution of a non-linear algebraic system.  The Jacobian linear system at each Newton iteration is solved by a Krylovmethod, accelerated by BDDC or FETI-DP preconditioners, both augmented with therecently introduced {\em deluxe} scaling of the dual variables.  A polylogarithmic convergence rate bound is proven for the resulting parallelBidomain solvers.  Extensive numerical experiments on linux clusters up to two thousandsprocessors confirm the theoretical estimates, showing that the proposedparallel solvers are scalable and quasi-optimal.

Parallel Newton-Krylov-BDDC and FETI-DP deluxe solvers for implicit time discretizations of the cardiac Bidomain equations