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 \(t_{n+\frac {1}{2}}\), 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 Δt_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 Δt_F is obtained by using Lagrange’s interpolation formula; finally, the numerical solution of the linearized system on time fine mesh Δt_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.

在本文中，引入时间二网格（TT-M）算法结合H ¹ -Galerkin混合有限元（FE）方法对非线性分布阶次扩散模型进行数值求解，该算法比H ^1-伽辽金混合有限元法。使用带有 TT-M 算法的 Crank-Nicolson 方案来离散时间\(t_{n+\frac {1}{2}}\)的时间方向，开发了 FBN- 𝜃公式来近似分布阶导数, 和H ¹-Galerkin混合有限元方法用于近似空间方向。TT-M混合元件算法主要包括三个步骤：首先，非线性耦合系统的对时的混合有限元解粗网Δ吨_Ç被计算; 接下来，基于在所述第一步骤中获得的数值解，非线性耦合系统的时间精细的数值解啮合Δ吨_˚F通过使用Lagrange内插公式而获得; 最后，线性化系统在时间细网格上的数值解 Δ t _F使用第二步的结果求解。显示了我们数值方案的解的存在性和唯一性。此外，详细分析了稳定性和先验误差估计。此外，给出了具有光滑和非光滑解的数值例子来验证我们的方法。