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.

翻译：本文研究了分离心脏系统常微分与偏微分方程后，Bidomain方程非线性求解器的收敛性与性能。首先，通过分析具有物理合理假设的辅助变分问题，我们严格证明了拟牛顿法（如BFGS）和非线性共轭梯度法（如Fletcher-Reeves）对Bidomain系统的全局收敛性。其次，从执行时间、数据鲁棒性和并行可扩展性三个方面比较了多种非线性Bidomain求解器。结果表明，拟牛顿法是Bidomain非线性系统的最佳选择——相比标准Newton-Krylov方法，其不仅具有更快的收敛速度，同时能保持鲁棒性与可扩展性。此外，一阶方法也展现出竞争力，特别适用于面向GPU计算的免矩阵实现方案。