Second-order non-uniform and fast two-grid finite element methods for non-linear time-fractional mobile/immobile equations with weak regularity

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 $L^2$-norm and $H^1$-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 $H^1$-stability and second-order $H^1$-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.