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

TL;DR

This paper introduces and tests new parallel Newton-Krylov BDDC and FETI-DP solvers for implicit time discretizations of the complex 3D cardiac Bidomain equations, demonstrating their scalability and efficiency on large clusters.

Contribution

The paper develops and analyzes novel parallel BDDC and FETI-DP solvers with deluxe scaling for the Bidomain equations, including theoretical convergence bounds and extensive numerical validation.

Findings

Solvers are scalable up to two thousand processors.

Theoretical convergence rate bounds are confirmed numerically.

Proposed methods are quasi-optimal for large-scale cardiac simulations.

Abstract

Two novel parallel Newton-Krylov Balancing Domain Decomposition by Constraints (BDDC) and Dual-Primal Finite Element Tearing and Interconnecting (FETI-DP) solvers are here constructed, analyzed and tested numerically for implicit time discretizations of the three-dimensional Bidomain system of equations. This model represents the most advanced mathematical description of 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 finite element discretization in space and a segregated implicit discretization in time, based on decoupling the PDEs from the ODEs, yields at each time step the solution of a non-linear algebraic system. The Jacobian linear system at each Newton iteration is solved by a Krylov method, accelerated by BDDC or FETI-DP preconditioners, both augmented with the recently introduced {\em deluxe} scaling of the dual variables. A polylogarithmic convergence rate bound is proven for the resulting parallel Bidomain solvers. Extensive numerical experiments on linux clusters up to two thousands processors confirm the theoretical estimates, showing that the proposed parallel solvers are scalable and quasi-optimal.