Scalable Newton-Krylov-BDDC and FETI-DP Deluxe Solvers for Decoupled Cardiac Reaction-Diffusion Models

Abstract

Two parallel Newton-Krylov Balancing Domain Decomposition by Constraints (BDDC) and Dual-Primal Finite Element Tearing and Interconnecting (FETI-DP) solvers are analyzed and numerically studied for implicit time discretizations of the Bidomain equations. This system models the cardiac bioelectrical activity and it consists of a degenerate system of two non-linear reaction-diffusion partial differential equations (PDEs), coupled with a stiff system of ordinary differential equations (ODEs). A non-linear algebraic system arises from a finite element discretization in space and an implicit discretization in time, based on decoupling the PDEs from the ODEs. Within each Newton iteration, the Jacobian linear system is solved by a Krylov method, accelerated by BDDC or FETI-DP preconditioners, both augmented with the recently introduced deluxe scaling of the dual variables. Several parallel numerical tests on Linux clusters confirm a novel polylogarithmic convergence rate bound, showing scalability and quasi-optimality of the proposed solvers.

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