He Yinnian
School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an 710049, China.
Entropy (Basel). 2021 Dec 9;23(12):1659. doi: 10.3390/e23121659.
In this work, a finite element (FE) method is discussed for the 3D steady Navier-Stokes equations by using the finite element pair Xh×Mh. The method consists of transmitting the finite element solution (uh,ph) of the 3D steady Navier-Stokes equations into the finite element solution pairs (uhn,phn) based on the finite element space pair Xh×Mh of the 3D steady linearized Navier-Stokes equations by using the Stokes, Newton and Oseen iterative methods, where the finite element space pair Xh×Mh satisfies the discrete inf-sup condition in a 3D domain Ω. Here, we present the weak formulations of the FE method for solving the 3D steady Stokes, Newton and Oseen iterative equations, provide the existence and uniqueness of the FE solution (uhn,phn) of the 3D steady Stokes, Newton and Oseen iterative equations, and deduce the convergence with respect to (σ,h) of the FE solution (uhn,phn) to the exact solution (u,p) of the 3D steady Navier-Stokes equations in the H1-L2 norm. Finally, we also give the convergence order with respect to (σ,h) of the FE velocity uhn to the exact velocity of the 3D steady Navier-Stokes equations in the L2 norm.
在这项工作中，讨论了一种通过使用有限元对(X_h\times M_h)求解三维稳态纳维 - 斯托克斯方程的有限元（FE）方法。该方法包括利用斯托克斯、牛顿和奥森迭代方法，将三维稳态纳维 - 斯托克斯方程的有限元解((u_h,p_h))转换为基于三维稳态线性化纳维 - 斯托克斯方程的有限元空间对(X_h\times M_h)的有限元解对((u_{hn},p_{hn}))，其中有限元空间对(X_h\times M_h)在三维域(\Omega)中满足离散下 - 上条件。在此，我们给出了用于求解三维稳态斯托克斯、牛顿和奥森迭代方程的有限元方法的弱形式，证明了三维稳态斯托克斯、牛顿和奥森迭代方程的有限元解((u_{hn},p_{hn}))的存在性和唯一性，并推导了有限元解((u_{hn},p_{hn}))在(H^1 - L^2)范数下相对于三维稳态纳维 - 斯托克斯方程精确解((u,p))关于((\sigma,h))的收敛性。最后，我们还给出了有限元速度(u_{hn})在(L^2)范数下相对于三维稳态纳维 - 斯托克斯方程精确速度关于((\sigma,h))的收敛阶。
