解不可压缩流体力学问题的降阶法 Ⅳ.三维Navier—Stokes方程

A DIMENSIONAL REDUCTION METHOD FOR INCOMPRESSIBLE FLUID DYNAMICS Ⅳ.THREE DIMENSIONAL NAVIER-STOKES EQUATIONS

在本系列文章里,我们提出一种新的解不可压缩流体力学问题的有限元方法——降阶法。实现这种算法的关键是给出零散度空间V^h的一组简单基函数。求速度时,运动方程试函数空间取为V^h解函数空间也取V^h。压力项自动消掉。从而可先求出速度的近似解。之后再求压力解。 本文对于一大类数值求解三维k连通区域Ω上的Navier—Stokes方程(简记为N—S方程)边值问题的一阶有限元格式给出零散度空间V^h的一组简单基函数。与二维问题不同的是,直接给出的“基函数”线性相关。必须从中去掉一部分(对应于某“树”的)函数才能使之成为一组线性无关的基。

This paper is the 4th in a series of papers in which we propose a new finite element method for incompressible fluid dynamics-a dimensional reduction method. The divergence free space Vh is used as both the velocity solution space and the test function space in the momentum equation. Thus the pressure term disappear, the velocity vector can be solved(before the pressure).This paper presents a simple basis of Vh for a kind of first order finite element schemes solving three dimensional Navier-Stokes equations. It differs from the two dimensional problem on that the directly given 'basis' Bh is linearly dependent. Therefore we must remove some functions from Bh so that it becomes linearly independent.All the removed functions form a 'tree' in a sense.