A Balancing Domain Decomposition by Constraints (BDDC) preconditioner is constructed and analyzed for the solution of composite Discontinuous Galerkin discretizations of reaction-diffusion systems of ordinary and partial differential equations arising in cardiac cell-by-cell models. Unlike classical Bidomain and Monodomain cardiac models, which rely on homogenized descriptions of cardiac tissue at the macroscopic level, the cell-by-cell models enable the representation of individual cardiac cells, cell aggregates, damaged tissues, and nonuniform distributions of ion channels on the cell membrane. The resulting discrete cell-by-cell models exhibit discontinuous global solutions across the cell boundaries. Therefore, the proposed BDDC preconditioner employs appropriate dual and primal spaces with additional constraints to transfer information between cells (subdomains) without affecting the overall discontinuity of the global solution. A scalable convergence rate bound is proved for the resulting BDDC cell-by-cell preconditioned operator, while numerical tests validate this bound and investigate its dependence on the discretization parameters.

翻译：本文针对心脏逐细胞模型中出现的常微分和偏微分方程反应扩散系统，构造并分析了基于约束平衡域分解（BDDC）预处理器的复合间断伽辽金离散化求解方法。传统心脏双域和单域模型依赖于宏观层面的均质化组织描述，而逐细胞模型则能表征单个心肌细胞、细胞聚集体、受损组织以及细胞膜上离子通道的非均匀分布。由此产生的离散逐细胞模型在细胞边界处呈现非连续全局解。为此，本文提出的BDDC预处理器采用带有附加约束的适当对偶和主空间，在不影响全局解整体非连续性的前提下，实现细胞间（子域间）信息传递。理论分析证明了该BDDC逐细胞预处理算子具有可扩展的收敛率界，数值实验验证了该界并探讨了离散化参数对其的影响。