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<p>This article mainly researches the two-grid Crank–Nicolson (CN) mixed finite element (FE) (TGCNMFE) model dimension reduction (TGCNMFEMDR) of the two-dimensional (2D) nonlinear convective FitzHugh–Nagumo equation with a variable parameter. To accomplish this study, a temporal semi-discrete CN mixed (TSDCNM) scheme, a TGCNMFE model, and the existence, stability, and estimation of errors for the TSDCNM and TGCNMFE solutions are firstly reviewed, whereafter a new TGCNMFEMRD algorithm is created, and the existence, stability, and estimate of errors for the TGCNMFEMRD solutions are attested. The new TGCNMFEMRD algorithm, like the classic TGCNMFE model, consists of a nonlinear system on coarser meshes as well as a linear system on sufficiently fine meshes, thus being solved easily. Finally, the rightness of the attained theoretic conclusions and the efficacy of the TGCNMFEMRD algorithm are confirmed by two sets of numerical experiments. The results show that the new TGCNMFEMRD algorithm can greatly reduce the unknowns, lessen the computation burden, and improve calculation efficiency.</p>

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Two-grid model dimension reduction for the nonlinear convective FitzHugh–Nagumo equation

  • 期刊论文

  • 发表日期：

    2026年7月14日

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  • Jie Chu,
  • Yue-Jie Li,
  • Zhen-Dong Luo

摘要

This article mainly researches the two-grid Crank–Nicolson (CN) mixed finite element (FE) (TGCNMFE) model dimension reduction (TGCNMFEMDR) of the two-dimensional (2D) nonlinear convective FitzHugh–Nagumo equation with a variable parameter. To accomplish this study, a temporal semi-discrete CN mixed (TSDCNM) scheme, a TGCNMFE model, and the existence, stability, and estimation of errors for the TSDCNM and TGCNMFE solutions are firstly reviewed, whereafter a new TGCNMFEMRD algorithm is created, and the existence, stability, and estimate of errors for the TGCNMFEMRD solutions are attested. The new TGCNMFEMRD algorithm, like the classic TGCNMFE model, consists of a nonlinear system on coarser meshes as well as a linear system on sufficiently fine meshes, thus being solved easily. Finally, the rightness of the attained theoretic conclusions and the efficacy of the TGCNMFEMRD algorithm are confirmed by two sets of numerical experiments. The results show that the new TGCNMFEMRD algorithm can greatly reduce the unknowns, lessen the computation burden, and improve calculation efficiency.