本项目拟围绕分数阶 Navier-Stokes 耦合方程的有限元方法展开研究，主要研究内容包括三个方面：（1）对于受到磁场作用且介质具有非局部性质的磁流体模型，本项目拟引入基于两局部高斯积分的变分多尺度 (LG-VMS) 有限元方法和两水平方法，建立空间分数阶非定常磁流体动力学方程的全离散两水平 LG-VMS 有限元格式；（2）对于受到温度场作用且介质具有非局部性质的流体模型，本项目拟结合 LG-VMS 有限元方法、两水平方法和解耦技术，构造空间分数阶非定常自然对流问题的全离散两水平 LG-VMS 有限元解耦算法；（3）对上述所构造的数值格式给出稳定性和收敛性分析，并对相关的磁流体力学和流体力学问题进行数值模拟，探讨分数阶导数对解的动力学性质的影响以及多物理场耦合的作用规律。

This project is to study finite element method for the fractional coupled Navier-Stokes equations. The research content of this project includes the following three aspects: (1) For the magnetic fluid which is affected by magnetic field and the medium has nonlocal property, this project intends to establish fully discrete two level LG-VMS finite element scheme of the space fractional nonstationary magnetohydrodynamics (MHD) equation by bringing two local Gauss integrations variational multiscale finite element method and two-level method. (2) For the fluid model which is affected by temperature field and the medium has nonlocal property, this project intends to construct fully discrete two-level LG-VMS finite element decoupling algorithm for the space fractional nonstationary natural convection problem by combining LG-VMS finite element method, two-level method and decoupling technique. (3) We will give the stability and convergence analysis of the the proposed numerical scheme, make the numerical simulations of the related magneto-fluidmechanics and fluid mechanics problems, discuss the influence of fractional derivatives on the dynamic properties of solutions, and explore the effect of fractional derivative on multi physical field coupling.