Fast second-order time two-mesh mixed finite element method for a nonlinear distributed-order sub-diffusion model

In this article, a time two-mesh (TT-M) algorithm combined with the H1-Galerkin mixed finite element (FE) method is introduced to numerically solve the nonlinear distributed-order sub-diffusion model, which is faster than the H1-Galerkin mixed FE method. The Crank-Nicolson scheme with TT-M algorithm is used to discretize the temporal direction at time tn+12\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$t_{n+\frac {1}{2}}$\end{document}, the FBN-𝜃 formula is developed to approximate the distributed-order derivative, and the H1-Galerkin mixed FE method is used to approximate the spatial direction. TT-M mixed element algorithm mainly covers three steps: first, the mixed finite element solution of the nonlinear coupled system on the time coarse mesh ΔtC is calculated; next, based on the numerical solution obtained in the first step, the numerical solution of the nonlinear coupled system on time fine mesh ΔtF is obtained by using Lagrange’s interpolation formula; finally, the numerical solution of the linearized system on time fine mesh ΔtF is solved by using the results in the second step. The existence and uniqueness of the solution for our numerical scheme are shown. Moreover, the stability and a priori error estimate are analyzed in detail. Furthermore, numerical examples with smooth and nonsmooth solutions are given to validate our method.