本篇論文主要研究在均勻網格中使用兩種迭代最小平方有限元素法求解具速度邊界值的穩態不可壓縮那維爾-史托克方程組。引入旋度當作新的未知變數，那維爾-史托克問題可改寫成一個擬線性速度-旋度-壓力一階方程組。我們提出兩種皮卡型迭代最小平方有限元素法求取此非線性一階問題的數值近似解，在每次迭代過程中使用一般的L2最小平方法或加權L2最小平方法去求解其相對應的歐辛問題。我們主要專注於此兩種迭代最小平方法在均勻網格上使用連續片狀多項式有限元素求解二維模型問題。數值實驗證明，對於具有平滑正確解的相同問題，在雷諾數較小的時候，L2最小平方解比加權L2最小平方解更準確；然而當雷諾數相對較大時，加權L2最小平方近似解似乎比L2最小平方近似解更好。最後，我們報告凹槽驅動流場的數值結果以驗證此類迭代最小平方有限元素法之效力。

This thesis is devoted to a numerical study of two iterative least-squares finite element schemes on uniform meshes for solving the stationary incompressible Navier-Stokes equations with velocity boundary condition. Introducing vorticity as an additional unknown variable, the Navier-Stokes problem can be recast as a first-order quasilinear velocity-vorticity-pressure system. Two Picard-type iterative least-squares finite element schemes are proposed for approximating the solution to the nonlinear first-order problem. In each iteration, we apply the usual L2 least-squares scheme or a weighted L2 least-squares scheme to solve the corresponding Oseen problem. We concentrate on two-dimensional model problems using continuous piecewise polynomial finite elements on uniform meshes for both iterative least-squares schemes. Numerical evidences show that, for the same test problem with smooth exact solution, the L2 least-squares solutions are more accurate than the weighted L2 least-squares solutions for low Reynolds number flows, while for flows with relatively higher Reynolds numbers the weighted L2 least-squares approximations seem to be better than the L2 least-squares approximations. Finally, numerical results for driven cavity flows are also given to demonstrate the effectiveness of the iterative least-squares finite element approach.

[1] P. B. Bochev, Analysis of least-squares ¯nite element methods for the Navier-
Stokes equations, SIAM, J. Numer. Anal., 34 (1997), pp. 1817-1844.
[2] P. B. Bochev and M. D. Gunzburger, Analysis of least-squares ¯nite element
methods for the Stokes equations, Math. Comp., 63 (1994), pp 479-506.
[3] P. B. Bochev and M. D. Gunzburger, Finite element methods of least-squares
type, SIAM Rev., 40 (1998), pp 789-837.
[4] P. B. Bochev, Z. Cai, T. A. Manteu?el and S. F. McCormick, Analysis of
velocity-°ux ¯rst-order system least-squares principles for the Navier-Stokes
equations: Part I, SIAM J. Numer. Anal., 35 (1998), pp. 990-1009.
[5] S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element
Methods, Springer-Verlag, New York, 1994.
[6] F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods, Springer-
Verlag, New York, 1991.
[7] Z. Cai, T. Manteu?el, and S. McCormick, First-order system least squares
for velocity-vorticity-pressure form of the Stokes equations, with application
to linear elasticity, ETNA, 3 (1995), pp. 150-159.
[8] Z. Cai, T. A. Manteu?el, and S. F. McCormick, First-order system least
squares for the Stokes equations, with application to linear elasticity, SIAM
J. Numer. Anal., 34 (1997), pp. 1727-1741.
[9] C. L. Chang, An error estimate of the least squares ¯nite element method for
the Stokes problem in three dimensions, Math. Comp., 63 (1994), pp. 41-50.
[10] C. L. Chang and B.-N. Jiang, An error analysis of least-squares ¯nite ele-
ment method of velocity-pressure-vorticity formulation for Stokes problem,
Comput. Methods Appl. Mech. Engrg., 84 (1990), pp. 247-255.
[11] C. L. Chang and J. J. Nelson, Least-squares ¯nite element method for the
Stokes problem with zero residual of mass conservation, SIAM J. Numer.
Anal., 34 (1997), pp. 480-489.
[12] C. L. Chang, S.-Y. Yang, and C.-H. Hsu, A least-squares ¯nite element
method for incompressible °ow in stress-velocity-pressure version, Comput.
Methods Appl. Mech. Engrg., 128 (1995), pp. 1-9.
[13] C. L. Chang and S.-Y. Yang, Analysis of the L2 least-squares ¯nite ele-
ment method for the velocity-vorticity-pressure Stokes equations with veloc-
ity boundary conditions, Appl. Math. Comput., 130 (2002), pp. 121-144.
24
[14] M.-C. Chen, B.-W. Hsieh, C.-T. Li, Y.-T. Wang, and S.-Y. Yang, A compar-
ative study of two iterative least-squares ¯nite element schemes for solving
the stationary incompressible Navier-Stokes equations, preprint, 2007.
[15] J. M. Deang and M. D. Gunzburger, Issues related to least-squares ¯nite
element methods for the Stokes equations, SIAM J. Sci. comput., 20 (1998),
pp. 878-906.
[16] H.-Y. Duan and G.-P. Liang, On the velocity-pressure-vorticity least-squares
mixed ¯nite element method for the 3D Stokes equations, SIAM J. Numer.
Anal., 41 (2003), pp. 2114-2130.
[17] M. Feistauer, Mathematical Methods in Fluid Dynamics, Longman Group UK
Limited, 1993.
[18] U. Ghia, K. N. Ghia, and C. T. Shin, High-Re solutions for incompressible
°ow using the Navier-Stokes equations and a multigrid method, J. Comput.
Phys., 48 (1982), pp. 387-411.
[19] V. Girault and P. A. Raviart, Finite Element Methods for Navier-Stokes
Equations: Theory and Algorithms, Springer-Verlag, New York, 1986.
[20] B.-N. Jiang, The Least-Squares Finite Element Method, Springer-Verlag,
Berlin, 1998.
[21] S. D. Kim, Y. H. Lee, and S.-Y. Yang, Analysis of [H?1;L2;L2] ¯rst-order
system least squares for the incompressible Oseen type equations, Appl. Nu-
mer. Math., 52 (2005), pp. 77-88.
[22] C.-T. Li, Piecewise bilinear approximations to the 2-D stationary incompress-
ible Navier-Stokes problem by least-squares ¯nite element methods, Master
Thesis, May 2004, National Central University, Taiwan.
[23] C.-C. Tsai and S.-Y. Yang, On the velocity-vorticity-pressure least-squares
¯nite element method for the stationary incompressible Oseen problem, J.
Comp. Appl. Math., 182 (2005), pp. 211-232.
[24] S.-Y. Yang, Error analysis of a weighted least-squares ¯nite element method
for 2-D incompressible °ows in velocity-stress-pressure formulation, Math.
Meth. Appl. Sci., 21 (1998), pp. 1637-1654.