Modeling excitable cells with the EMI equations: spectral analysis and iterative solution strategy

Abstract: In this work, we are interested in solving large linear systems stemming from the Extra-Membrane-Intra (EMI) model, which is employed for simulating excitable tissues at a cellular scale. After setting the related systems of partial differential equations (PDEs) equipped with proper boundary conditions, we provide numerical approximation schemes for the EMI PDEs and focus on the resulting large linear systems. We first give a relatively complete spectral analysis using tools from the theory of Generalized Locally Toeplitz matrix sequences. The obtained spectral information is used for designing appropriate (preconditioned) Krylov solvers. We show, through numerical experiments, that the presented solution strategy is robust w.r.t. problem and discretization parameters, efficient and scalable.