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Fast second-order time two-mesh mixed finite element method for a nonlinear distributed-order sub-diffusion model
Numerical Algorithms ( IF 1.7 ) Pub Date : 2021-01-08 , DOI: 10.1007/s11075-020-01048-8
Cao Wen , Yang Liu , Baoli Yin , Hong Li , Jinfeng Wang

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 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.



中文翻译:

非线性分布阶次扩散模型的快速二阶时间双网格混合有限元法

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ΔÇ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˚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。

更新日期:2021-01-08
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