Stimulus Protocol Determines the Most Computationally Efficient Preconditioner for the Bidomain Equations

The efficient solution of the bidomain equations is a fundamental tool in the field of cardiac electrophysiology. When choosing a finite element discretization of the coupled system, one has to deal with the solution of a large, highly sparse system of linear equations. The conjugate gradient algorithm, along with suitable preconditioning, is the natural choice in this scenario. In this study, we identify the optimal preconditioners with respect to both stimulus protocol and mesh geometry. The results are supported by a comprehensive study of the mesh-dependence properties of several preconditioning techniques found in the literature. Our results show that when only intracellular stimulus is considered, incomplete LU factorization remains a valid choice for current cardiac geometries. However, when extracellular shocks are delivered to tissue, preconditioners that take into account the structure of the system minimize execution time and ensure mesh-independent convergence.