Newton-Krylov-BDDC deluxe solvers for non-symmetric fully implicit time discretizations of the Bidomain model

TL;DR

This paper introduces a new convergence analysis for a domain decomposition solver applied to the nonlinear Bidomain model, supported by numerical tests demonstrating robustness and efficiency.

Contribution

It provides the first theoretical convergence rate estimate for a BDDC solver on the nonlinear Bidomain model with implicit discretization.

Findings

The convergence rate bound is proven for the GMRES iterations.

Numerical tests confirm the theoretical predictions.

The solver is robust and efficient for large-scale cardiac simulations.

Abstract

A novel theoretical convergence rate estimate for a Balancing Domain Decomposition by Constraints algorithm is proven for the solution of the cardiac Bidomain model, describing the propagation of the electric impulse in the cardiac tissue. The non-linear system arises from a fully implicit time discretization and a monolithic solution approach. The preconditioned non-symmetric operator is constructed from the linearized system arising within the Newton-Krylov approach for the solution of the non-linear problem; we theoretically analyze and prove a convergence rate bound for the Generalised Minimal Residual iterations' residual. The theory is confirmed by extensive parallel numerical tests, widening the class of robust and efficient solvers for implicit time discretizations of the Bidomain model.