混合边界条件下的三维定常旋转Navier-Stokes方程的原始变量有限元计算

THE FINITE ELEMENT METHOD OF THE THREE DIMENSIONAL STATIONARY ROTATING NAVIER-STOKES EQUATIONS IN THE PRIMITIVE VARIABLES WITH MIXED BOUNDARY CONDITIONS

本文利用原始变量有限元法求解混合边界条件下的三维定常旋转Navier-Stokes方程,证明了离散问题解的存在唯一性,得到了有限元解的最优误差估计.给出了求解原始变量有限元逼近解的简单迭代算法,并证明了算法的收敛性.针对三维情况下计算资源的限制,采用压缩的行存储格式存储刚度矩阵的非零元素,并利用不完全的LU分解作预处理的GMRES方法求解线性方程组.最后分析了简单迭代和牛顿迭代的优劣对比,数值算例表明在同样精度下简单迭代更节约计算时间.

In this paper, we adopt the finite element method to solve the three dimensional station- ary rotating Navier-Stokes Equations in primitive variables with mixed boundary conditions. The existence and uniqueness of the approximate problem are showed, and the optimal error estimate is obtained. We give the simple iterative algorithm for the approximate problem, and prove the convergence of this algorithm. To overcome the restriction of the computer resource in the case of three dimension, we use the compressed sparse row format to store the stiff matrix. Then the GMRES method with an incomplete LU preconditioner is adopted to solve the linear system of equations. We analyze the advantage of the simple iterative method over Newton method at last. The numerical examples show that the more computational time has saved by the simple iterative method than the Newton method.