The aim of the present work is to design, analyze theoretically, and test numerically, a generalized Dryja-Smith-Widlund (GDSW) preconditioner for composite Discontinuous Galerkin discretizations of multicompartment parabolic reaction-diffusion equations, where the solution can exhibit natural discontinuities across the domain. We prove that the resulting preconditioned operator for the solution of the discrete system arising at each time step converges with a scalable and quasi-optimal upper bound for the condition number. The GDSW preconditioner is then applied to the EMI (Extracellular - Membrane - Intracellular) reaction-diffusion system, recently proposed to model microscopically the spatiotemporal evolution of cardiac bioelectrical potentials. Numerical tests validate the scalability and quasi-optimality of the EMI-GDSW preconditioner, and investigate its robustness with respect to the time step size as well as jumps in the diffusion coefficients.

翻译：本研究旨在为多室抛物型反应-扩散方程的复合间断伽辽金离散化设计、理论分析和数值测试一种广义Dryja-Smith-Widlund（GDSW）预条件子，该方程的解在计算域中可呈现自然间断性。我们证明，用于求解每个时间步离散系统的预条件算子具有可扩展且准最优的条件数上界。随后将GDSW预条件子应用于EMI（细胞外-膜-细胞内）反应-扩散系统，该系统最近被提出用于微观模拟心脏生物电位的时空演化。数值实验验证了EMI-GDSW预条件子的可扩展性与准最优性，并考察了其对时间步长及扩散系数突变的鲁棒性。