This paper introduces a novel temporal second-order fully discrete approach of finite element method (FEM) and its fast two-grid FEM on non-uniform meshes, which aims to solve non-linear time-fractional variable coefficient mobile/immobile (MIM) equations with a solution exhibiting weak regularity. The proposed method utilizes the averaged L1 formula on graded meshes in the temporal domain to handle the weak initial singularity. In the spatial domain, a two-grid approach based on FEM and its associated fast algorithm are employed to optimize computational efficiency. To ensure fast and accurate calculations of kernels, an innovative algorithm is developed. The stability and optimal error estimates in L2-norm and H1-norm are rigorously established for the non-uniform averaged L1-based FEM, two-grid FEM and their associated fast algorithms, respectively. The numerical findings clearly showcase the validity of our theoretical discoveries, highlighting the enhanced effectiveness of our two-grid approach in contrast to the conventional approach. An important point to mention is that this work is the pioneering effort in addressing both H1-stability and second-order H1-norm error analysis for the fractional MIM problem with weak regularity, as well as temporal second-order approaches of two-grid for the fractional MIM equation with a weakly singular solution.

中文翻译：

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二阶非均匀和快速双网格有限元方法，用于具有弱正则性的非线性时间分数移动/不动方程


本文介绍了一种新颖的时间二阶全离散有限元法 （FEM） 及其在非均匀网格上的快速双网格 FEM，旨在求解具有弱正则性的解的非线性时间分数可变系数移动/固定 （MIM） 方程。所提出的方法利用时间域中渐变网格上的平均 L1 公式来处理弱初始奇点。在空间域，采用基于 FEM 的双网格方法及其相关的快速算法来优化计算效率。为了确保快速准确地计算内核，开发了一种创新算法。分别针对非均匀平均基于 L1 的 FEM、双网格 FEM 及其相关的快速算法，严格建立了 L2-norm 和 H1-norm 中的稳定性和最优误差估计。数值研究结果清楚地证明了我们理论发现的有效性，突显了与传统方法相比，我们的双网格方法的有效性更高。值得一提的重要一点是，这项工作是解决弱正则性分数 MIM 问题的 H1 稳定性和二阶 H1 范数误差分析，以及弱奇异解的分数 MIM 方程的双网格时间二阶方法的开创性工作。