Analysis of variable-time-step BDF2 combined with the fast two-grid finite element algorithm for the FitzHugh-Nagumo model

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 $L^2$-norm and the stability analysis for the fully discrete scheme on the coarse and fine grids, and to prove the optimal $H^1$-norm error estimation for the fully discrete TGFE scheme. These analyses hold for adjacent time-step ratios $0<r_k\le 4.8645-\delta$ (δ being an arbitrarily small constant). Finally, the effectiveness and computing efficiency of the proposed algorithm are verified through several numerical examples.