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標題: 對一種指數有限單元法與流線逆向基底於解擴散對流問題的比較.
A comparison between an exponential finite element scheme and streamline upwind schemes for convection-diffusion problems.
作者: 鄭俊勇
Cheng, Chun-yung
關鍵字: convection-diffusion equation;對流擴散問題;streamline upwind;exponential fitting;流線逆向法;指數有線單元法
出版社: 應用數學系所
引用: [1] O. Axelsson, On the numerical solution of convection dominated convectiondiffusion problems. In K. I. Gross, editor, Mathematics Methods in Energy Research, pages 3-21. SIAM, Philadelphia, 1984. [2] S.C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods. Springer-Verlag, New York, 1994. [3] B. Fischer, A. Ramage, D. Silvester, and A. J. Wathen, Towards parameter-free streamline upwinding for advection-diffusion problems. Technical Report 37, Department of Mathematics, University of Strathclyde, 1996. [4] P.W. Hemker, A numerical study of stiff two-point boundary problems. MC-Tract 80, Mathematical Centre, Amstrdam, 1977. [5] T.J.R. Hughes and A.N. Brooks, Streamline upwind petrov-galerkin formulations for convection-dominated flows with particular emphasis on the incompressible navier-stokes equations. Comput.Methods Appl. Mech. Engrg., 32(1982), pp. 199-259. [6] C. Johnson, Numerical Solution of Partial Differential Equations by the Finite Element Method. Cambridge University Press, New York, 1987. [7] C. Johnson, A.H. Schatz, and L.B. Wahlbin, Crosswing smear and pointwise errors in streamline diffusion finite element methods. Math. Comp., 49 (1987), pp. 25-38. [8] K.W. Morton, Numerical solution of Convection-Diffusion Problems. Applied Mathematics and Mathematics Computation 12 (1996) [9] E. O'Riordan and M. Stynes, A globally uniformly convergent finite element method for a singularly perturbed elliptic problem in two dimentionas. Math. Comp. 57 (1991), pp. 47-62. [10] Y. Shih and H.C. Elman, Modified Streamline Diffusion Schemes for Convection-Diffusion Problems, Computer Methods in Applied Mechanics and Engineering, 174 (1999), pp. 137-151. [11] M. Stynes and L. Tobiska, Necessary L2-uniform convergence conditions for difference schemes for two dimensional convection-diffusion problems. Computers Math. Applic., 29 (1998), pp. 45-53. [12] G. Zhou, How accurate is the streamline diffusion finite element method? Math. Comp., 66 (1997), pp. 31-44.

In this thesis, we consider an exponential fitting finite element scheme for the convection dominated convection-diffusion equation. The exponential trial function in this scheme provides an exponential approximation on the grid to stabilize the solution. We make a comparison for the accuracy between exponential fitting finite element scheme with several streamline upwind schemes in the bilinear elements. The exponential terms of the basis function
are kept in the flow direction for approaching the solution
accurately. We present an error analysis for this scheme and the error in the energy norm converges uniformly in order of O(h^{1/2}). Numerical results demonstrate the efficient and accuracy of this exponential fitting scheme.
其他識別: U0005-0307200818421800
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