In this article, a fast numerical method is developed for solving the FitzHugh-Nagumo (FHN) model by combining two-grid finite element (TGFE) algorithm in space with a linearized variable-time-step (VTS) two-step backward differentiation formula (BDF2) in time. This algorithm mainly included two steps: firstly, the nonlinear coupled system on the coarse grid is solved by a nonlinear iteration; secondly, a linearized coupled system on the fine grid by making use of a Taylor formula is formulated, and the numerical solution pair is solved directly. The techniques of the discrete orthogonal convolution (DOC) kernels and the discrete complementary convolution (DCC) kernels are used to derive the optimal error estimations in -norm and the stability analysis for the fully discrete scheme on the coarse and fine grids, and to prove the optimal -norm error estimation for the fully discrete TGFE scheme. These analyses hold for adjacent time-step ratios ( being an arbitrarily small constant). Finally, the effectiveness and computing efficiency of the proposed algorithm are verified through several numerical examples.

中文翻译：

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变时间步长BDF2结合快速二网格有限元算法对FitzHugh-Nagumo模型的分析


在本文中，通过将空间中的两网格有限元（TGFE）算法与线性化变时间步长（VTS）两步后向微分公式相结合，开发了一种用于求解 FitzHugh-Nagumo（FHN）模型的快速数值方法（BDF2）及时。该算法主要包括两个步骤：首先，通过非线性迭代求解粗网格上的非线性耦合系统；其次，利用泰勒公式建立细网格上的线性化耦合系统，并直接求解数值解对。利用离散正交卷积（DOC）核和离散互补卷积（DCC）核技术推导出全离散方案在粗网格和细网格上的最优范数误差估计和稳定性分析，并证明全离散 TGFE 方案的最优范数误差估计。这些分析适用于相邻时间步长比（任意小的常数）。最后，通过多个数值算例验证了所提算法的有效性和计算效率。