Analysis and numerical validation of robust parallel nonlinear solvers for implicit time discretizations of the Bidomain equations

In this work, we study the convergence and performance of nonlinear solvers for the Bidomain equations after decoupling the ordinary and partial differential equations of the cardiac system. We first rigorously prove that Quasi-Newton methods such as BFGS and nonlinear Conjugate-Gradient such as Fletcher-Reeves methods are globally convergent, by studying an auxiliary variational problem under physically reasonable hypotheses. Then, we compare several nonlinear solvers in terms of execution time, robustness with respect to the data and parallel scalability. Our results suggest that Quasi-Newton methods are the best choice for this type of problem, being faster than standard Newton-Krylov methods without hindering their robustness or scalability. In addition, first order methods are also competitive, and represent a better alternative for matrix-free implementations, which are suitable for GPU computing.