在本篇論文中，我們使用最小平方有限元素法來求解對流擴散方程中的對流佔優問題，發現到最小平方有限元素法中使用線性基底函數來求解對流佔優問題，其所求得的解並不理想，而傳統的網格加密的方法對最小平方有限元素法並不ㄧ個經濟的方法，因此我們使用RFB 方法對最小平方有限元素法進行改良的工作，這是ㄧ個新的應用。而數值結果顯示這個新的方法對於求解對流佔優問題有相當不錯的改良效果。

In this thesis, we formulate the least-squares finite element method using piececewise linears to solve the convection-diffusion equation which is convection-dominated and we find that the solution is diffusive and the classical mesh refinement for the least-squares finite element method is not an economical method. Then we use the
residual-free bubble method to enrich the least-squares finite element method. This is a new application of residual-free bubble method and we solve some test problems. The numerical results show that the residual-feee bubble method for the least-squares finite element method has a good effect of enrichment。

[1] B.N. Jiang, and L.A. Povinelli, Least-Squares Finite element method for fluid dynamics, Comput. Methods. Appl. Mech. Engrg. 81 (1990) 13-37.
[2] F. Brezzi, M.O. Bristeau, L.P. Franca, M. Mallet, and G. Roge, A relationship between stabilized finite element methods and the Galerkin method with bubble functions,Comput. Methods Appl. Mech. Engrg. 96 (1992) 117-129.
[3] L.P. Franca, S.L. Frey, and T.J.R Hughes, Stabilized-finite element methods: I. Application to advective-diffusive model, Comput. Methods Appl. Mech. Engrg.
95 (1992) 253-276.
[4] F. Brezzi and A. Russo, Choosing bubbles for advection-diffusion problems,Math.Models Methods Appl., 4 (1994)
571-587.
[5] L.P. Franca, and F. Charbel, On the Limitations of Bubble Functions, Comput. Methods Appl. Mech. Engrg. 117 (1994) 225-230.
[6] L.P. Franca, and F. Charbel, Bubble functions prompt unusual stabilized finite element methods, Comput. Methods. Appl. Mech. Engrg. 123 (1995) 299-308.
[7] T.F. Chen, G.J. Fix, and H.D. Yang, Numerical Studies of Optimal Grid Construction, Numer. Methods Partial Different. Eq. 12 (1996) 191-206.
[8] L.P. Franca, and A. Russo , Mass lumping emanating from residual-free bubbles, Comput. Methods Appl. Mech. Engrg. 142 (1997) 353-360.
[9] F. Brezzi, L.P. Franca, A. Russo, Further considerations on the residual-free bubbles for advective-diffusive equations, Comput. Methods Appl. Mech. Engrg.
166 (1998) 25-33.
[10] B.N. Jiang, On the least-squares method, Comput. Methods Appl. Mech. Engrg. 152 (1998) 239-257
[11] J.M. Fiard, T.A. Manteu®el, and S.F. Mccormick, First-order sustem leasts squares (FOSLS) for convection-diffusion problems: numerical results, SIAM
J. Sci. Comput. 19 (1998) 1958-1979.
[12] L.P. Franca, A. Nesliturk and M. Stynes, On the stability of residual-free bubbles for convection-diffusion problems and their approximation by a two-level finite element method, Comput. Methods Appl. Mech. Engrg. 166 (1998) 35-49.
[13] L.P. Franca, and A.P. Macedo, A two-level finite element method and its application to Helmholtz equation, Int. J. Numer. Methods Engrg. 43 (1998) 23-42.
[14] P.B. Bochev, and J. Choi, A Comparative Study of Least-squares, SUPG and Galerkin Methods for Convection Problem, Int. J. Comput. Fluids 15 (2001)
127-146.
[15] L.P. Franca, and A. Nesliturk, On a two-level finite element method to the incompressible Navier-Stokes equations, Int. J. Numer. Meth. Engrg 52 (2001)
433-453.
[16] L.P. Franca, and F.N. Hwang, Refining the submesh strategy in the two-level finite element method: Application to the advection-diffusion equation, Int. J.
Numer. Meth. Fluids 39 (2002) 161-187.
[17] L.P. Franca LP, G. Hauke and A. Masud, Revisiting stabilized finite element methods for the advective-diffusive equation, Comput. Methods Appl. Mech. En-
grg. 195 (2006) 1560-1572.
[18] M. Parvazinia, V. Nassehi, and R.J Wakeman, Multi-scale finite element modelling using bubble function method for a convection-diffusion problem , Chem.
Engrg. Sci. 61 (2006) 2742-2751.
[19] A. Russo, Streamline-upwind Petrov/Galerkin method (SUPG) vs residual-free bubble (RFB), Comput. Methods Appl. Mech. Engrg. 195 (2006) 1608-1620.
[20] C. Johnson, Numerical Solution of Partial Differential Equation by the Finite
Element Method, Cambridge University Press, Cambridge, 1987.
[21] K.W Morton, Numerical Solution of Convection-Diffusion Problems, Chapman & Hall Press, 1996
[22] S.A. Berger, W. Goldsmith, and E.R. Lewis, Introduction to bioengineering, Ox-
ford University Press, 1996
[23] B.N. Jiang. The Least-Squares Finite Element Method, Springer-Verlag, Berlin, 1998
[24] D. Jean, and H. Antonio, Finite Element Methods for Flow Problems, John Wiley & Sons Inc Press, 2003.
[25] S.C. Breneer, and L.R. Scott, The Mathmatical Theory of Finite Element Method, Springer-Verlag, New-York, 1994.