We formulate and analyze interior penalty discontinuous Galerkin methods for coupled elliptic PDEs modeling excitable tissue, represented by intracellular and extracellular domains sharing a common interface. The PDEs are coupled through a dynamic boundary condition, posed on the interface, that relates the normal gradients of the solutions to the time derivative of their jump. This system is referred to as the Extracellular Membrane Intracellular model or the cell-by-cell model. Due to the dynamic nature of the interface condition and to the presence of corner singularities, the analysis of discontinuous Galerkin methods is non-standard. We prove the existence and uniqueness of solutions by a reformulation of the problem to one posed on the membrane. Convergence is shown by utilizing face-to-element lifting operators and notions of weak consistency suitable for solutions with low spatial regularity. Further, we present parameter-robust preconditioned iterative solvers. Numerical examples in idealized geometries demonstrate our theoretical findings, and simulations in multiple cells portray the robustness of the method.

翻译：我们针对模拟可兴奋组织的耦合椭圆型偏微分方程，提出并分析了内部惩罚不连续伽辽金方法。该组织由共享公共界面的细胞内域和细胞外域表示。偏微分方程通过设置在界面上的动态边界条件耦合，该条件将解的法向梯度与其跳跃的时间导数相关联。该系统被称为细胞外膜细胞内模型或逐细胞模型。由于界面条件的动态特性以及角点奇点的存在，不连续伽辽金方法的分析具有非标准特性。我们通过将问题重新表述为在膜上求解的问题，证明了解的存在唯一性。通过利用面到单元的提升算子以及适用于低空间正则性解的弱相容性概念，证明了方法的收敛性。此外，我们提出了参数鲁棒的预条件迭代求解器。理想化几何结构中的数值算例验证了我们的理论结果，多细胞模拟则展示了该方法的鲁棒性。