Mathematical models for excitable tissue with explicit representation of individual cells are highly detailed and can, unlike classical homogenized models, represent complex cellular geometries and local membrane variations. However, these cell-based models are challenging to approximate numerically, partly due to their mixed-dimensional nature with unknowns both in the bulk and at the lower-dimensional cellular membranes. We here develop and evaluate a novel solution strategy for the cell-based KNP-EMI model describing ionic electrodiffusion in and between intra- and extracellular compartments with explicit representation of individual cells. The strategy is based on operator splitting, a multiplier-free formulation of the coupled dynamics across sub-regions, and a discontinuous Galerkin discretization. In addition to desirable theoretical properties, such as local mass conservation, the scheme is practical as it requires no specialized functionality in the finite element assembly and order optimal solvers for the resulting linear systems can be realized with black-box algebraic multigrid preconditioners. Numerical investigations show that the proposed solution strategy is accurate, robust with respect to discretization parameters, and that the parallel scalability of the solver is close to optimal - both for idealized and realistic two and three dimensional geometries.

翻译：具有显式单细胞表征的可兴奋组织数学模型细节丰富，与传统均质化模型不同，它能呈现复杂的细胞几何结构及局部膜电位变化。然而，这类基于细胞的模型在数值逼近上极具挑战性，部分原因在于其混合维度特性——未知量同时存在于体区域和低维的细胞膜区域。本文针对基于细胞的KNP-EMI模型，提出并验证了一种新型求解策略。该模型描述了细胞内、外以及跨膜区域的离子电扩散过程，并对单个细胞进行显式表征。该策略基于算子分裂、跨子区域耦合动力学的无乘子公式以及间断伽辽金离散化。除了局部质量守恒等理想的理论特性外，该方案具有实用性：它无需有限元组装中的特殊功能，且可通过黑箱代数多重网格预条件子实现所得线性系统的最优阶求解器。数值实验表明，该求解策略具有高精度、对离散化参数的鲁棒性，且在理想化与真实的三维几何中均展现出近乎最优的并行扩展性。