[Back to Results | New Search]

Student Number 942201021 Author Yun-Tsz Wang(王韻詞) Author's Email Address No Public. Statistics This thesis had been viewed 1258 times. Download 470 times. Department Mathematics Year 2007 Semester 2 Degree Master Type of Document Master's Thesis Language English Title On Two Iterative Least-Squares Finite Element Schemes for Solving the Incompressible Navier-Stokes Equations Date of Defense 2008-01-03 Page Count 25 Keyword driven cavity flows finite element methods iterative methods least squares Navier-Stokes equations Oseen equations Abstract 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. Table of Content 中文摘要 ……………………………………………………………… i

英文摘要 …………………………………………………………… ii

目錄 ………………………………………………………………… iii

Abstract ……………………………………………………………… 1

1. Problem formulation …………………………………………… 2

2. Least-squares finite element schemes …………………… 6

3. Analysis of the least-squares finite element schemes for the Oseen problem …………………………………………… 10

4. Numerical experiments ……………………………………… 13

5. Numerical results of driven cavity flows ……………… 17

6. Conclusions …………………………………………………… 23

References ………………………………………………………… 24Reference [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.Advisor Suh-Yuh Yang(楊肅煜)

Files approve immediately

942201021.pdf Date of Submission 2008-05-18

Our service phone is (03)422-7151 Ext. 57407,E-mail is also welcomed.