  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. Firstly, we provide a rigorous
proof of the global convergence of Quasi-Newton methods, such as BFGS, and
nonlinear Conjugate-Gradient methods, such as Fletcher--Reeves, for the
Bidomain system, by analyzing an auxiliary variational problem under physically
reasonable hypotheses. Secondly, we compare several nonlinear Bidomain solvers
in terms of execution time, robustness with respect to the data and parallel
scalability. Our findings indicate that Quasi-Newton methods are the best
choice for nonlinear Bidomain systems, since they exhibit faster convergence
rates compared to standard Newton-Krylov methods, while maintaining robustness
and scalability. Furthermore, first-order methods also demonstrate
competitiveness and serve as a viable alternative, particularly for matrix-free
implementations that are well-suited for GPU computing.
