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.

翻译：本文致力于求解源自膜外-膜内（EMI）模型的大规模线性系统，该模型用于在细胞尺度模拟可兴奋组织。在设定配备适当边界条件的相关偏微分方程组后，我们提出了EMI偏微分方程的数值近似格式，并聚焦于由此产生的大规模线性系统。我们首先利用广义局部Toeplitz矩阵序列理论工具给出了相对完备的谱分析。利用所获得的谱信息来设计合适的（预条件）Krylov求解器。通过数值实验表明，所提出的求解策略对问题和离散化参数具有鲁棒性，且高效、可扩展。