在本篇論文中，我們主要研究二維穩態不可壓縮那維爾-史托克問題在給定速度邊界值情況下的最小平方有限元素法之片狀線性數值解。首先引入兩個新未知函數〝旋度和總壓力〞後，將那維爾-史托克問題分別改寫成〝速度-旋度-壓力〞和〝速度-旋度-總壓力〞兩個一階偏微分方程組。接著我們引用L2最小平方與網格依賴加權最小平方有限元素法求解源自於皮卡型迭代法所產生的系列奧辛問題。其中相對應的最小平方能量泛函定義在一個適當乘積函數空間下，分別取每項偏微分方程式殘餘的L2模未加權或加權之平方和。數值實驗證明，在雷諾數小的時候，L2最小平方法比網格依賴加權最小平方法更準確；然而對大雷諾數的問題，網格依賴加權最小平方法比L2最小平方法明顯地好很多。最後列出在不同雷諾數下凹槽驅動流場的數值結果。

In this thesis, we study the piecewise bilinear finite element approximations to the two-dimensional stationary incompressible Navier-Stokes equations with the velocity boundary condition by using the least-squares principles.
The Navier-Stokes problem is first recast into the velocity-vorticity-pressure and velocity-vorticity-total pressure first-order systems by introducing the vorticity variable and, in addition, total pressure variable. We then apply both the L2 least-squares and mesh-dependent weighted least-squares finite element schemes
to approximate the solutions of the sequence of Oseen problems arising from a Picard-type iteration associated with these first-order systems. The corresponding least-squares energy functionals are defined in terms of the sum of the squared L2 norms without or with mesh-dependent weights of the residual equations over a product function space. Numerical evidences show that, for low Reynolds number flows, the L2 least-squares method is more accurate than the mesh-dependent weighted least-squares method.
For flows with large Reynolds numbers, the mesh-dependent weighted least-squares method is apparently better than the L2 least-squares method. Some numerical results for driven cavity flows with various Reynolds numbers are also given.

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