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 H 1 -Galerkin mixed finite element (FE) method is introduced to numerically solve the nonlinear distributed-order sub-diffusion model, which is faster than the H 1 -Galerkin mixed FE method. The Crank-Nicolson scheme with TT-M algorithm is used to discretize the temporal direction at time t_n+1/2 , the FBN- 𝜃 formula is developed to approximate the distributed-order derivative, and the H 1 -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 Δ t C 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 Δ t F is obtained by using Lagrange’s interpolation formula; finally, the numerical solution of the linearized system on time fine mesh Δ t F 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.