In this article, a spatial two‐grid finite element (TGFE) algorithm is used to solve a two‐dimensional nonlinear space–time fractional diffusion model and improve the computational efficiency. First, the second‐order backward difference scheme is used to formulate the time approximation, where the time‐fractional derivative is approximated by the weighted and shifted Grünwald difference operator. In order to reduce the computation time of the standard FE method, a TGFE algorithm is developed. The specific algorithm is to iteratively solve a nonlinear system on the coarse grid and then to solve a linear system on the fine grid. We prove the scheme stability of the TGFE algorithm and derive a priori error estimate with the convergence result O(Δt² + h^{r + 1 − η} + H^{2r + 2 − 2η}). Finally, through a two‐dimensional numerical calculation, we improve the computational efficiency and reduce the computation time by the TGFE algorithm.

中文翻译：

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基于空间二网格有限元算法的非线性时空分数扩散模型的快速计算

在本文中，空间二维网格有限元（TGFE）算法用于求解二维非线性时空分数扩散模型，并提高了计算效率。首先，使用二阶后向差分方案来表示时间近似，其中时间分数导数由加权和移位的Grünwald差分算子近似。为了减少标准有限元方法的计算时间，开发了一种TGFE算法。具体算法是迭代求解粗网格上的非线性系统，然后求解细网格上的线性系统。我们证明了TGFE算法的方案稳定性，并利用收敛结果O（Δt ²  +  h^{r  + 1-  η}  +  H ^{2 r  + 2-2η}）。最后，通过二维数值计算，我们通过TGFE算法提高了计算效率并减少了计算时间。