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.

中文翻译：

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用于多室反应-扩散问题的复合间断伽辽金离散的 GDSW 预调节器


本研究的目的是设计、理论分析和数值测试广义的 Dryja-Smith-Widlund （GDSW） 预条件器，用于多隔室抛物线反应-扩散方程的复合间断伽辽金离散化，其中解可以在整个域中表现出自然不连续性。我们证明，在每个时间步长出现的离散系统解的预条件运算符与条件数的可扩展和准最优上限收敛。然后将 GDSW 预处理剂应用于 EMI （细胞外 - 膜 - 细胞内） 反应-扩散系统，该系统最近提出用于在显微镜下模拟心脏生物电势的时空演变。数值测试验证了 EMI-GDSW 预调节器的可扩展性和准最优性，并研究了其在时间步长和扩散系数跳跃方面的鲁棒性。