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：本文针对心脏单元-单元模型中出现的常微分方程与偏微分方程反应扩散系统，构建并分析了用于求解复合间断伽辽金离散化的平衡域分解约束（BDDC）预条件子。与依赖宏观层面心脏组织均质化描述的传统双域和单域心脏模型不同，单元-单元模型能够表征单个心肌细胞、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。由此得到的离散单元-单元模型在细胞边界处呈现全局解的不连续性。因此，所提出的BDDC预条件子采用具有附加约束的适当对偶空间与原始空间，在单元（子域）间传递信息的同时不影响全局解的整体不连续性。本文证明了所得BDDC单元-单元预条件算子具有可扩展的收敛速率界，数值实验验证了该界限，并研究了其对离散化参数的依赖关系。