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 在這篇論文裡，我們研究裁縫有限點法來解擴散對流反應方程式。裁縫有限點法的解空間的基底函數是由指數函數、修正的貝塞耳函數及三角函數相乘所組成。在理論分析上我們可以求得裁縫有限點法在對流控制問題的數值截尾誤差為O(e^-Pe)，這裡的Pe是一個網格貝克勒數。當擴散係數很小時，由我們的數值測試證明了在邊界層的誤差不管網格大小都會在機器誤差的精準之下，而內部層的誤差則在網格大小之下。這描述了在對流控制問題上，裁縫有限點法會有相當高的準確度。
 In this thesis, we study a tailored finite point method (TFP) for solving the convection-diffusion-reaction equation. The basis functions of solution space of the TFP are composed from the product of an exponential function, modified Bessel functions and trigonometric functions. The numerical truncation error for the TFP is in the order of O(e^-Pe) for the convection dominated problem, where Pe is the mesh Péclet number. Our numerical tests show that for small diffusion coefficient the numerical boundary layer is under the machine precision regardless the mesh size, and the internal layer is also within the mesh size. This depicts that the TFP method has high accuracy for convection-dominated problem.
 1 Introduction ........................................ 12 The Tailored Finite Point Methods......... 32.1Introduction....................................... 32.2 The five-point TFP scheme................. 52.3 The seven-point TFP scheme............. 82.4 The nine-point TFP scheme............... 113 Truncation Errors................................. 144 Numerical Experiments........................ 195 Conclusions........................................ 23Bibliography........................................... 33
 [1] 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), 199V259.[2] C. Johnson, A.H. Schatz and L.B. Wahlbin, Crosswing smear and pointwise errors in streamline diffusion finite element methods, Math. Comp., 49 (1987), 25-38.[3] Y. Shih and H.C. Elman, Modified streamline diffusion schemes for convectiondiffusion problems, Computer Methods in Applied Mechanics and Engineering, 174 (1999), 137-151.[4] H.G. Roos, M. Stynes and L. Tobiska, Numerical methods for singular perturbed differential equations, Springer-Verlag, New York, 1996.[5] H. Han, Z. Huang and R.B. Kellogg, The tailored finite point method and a problem of P. Hemker, Proceedings of the International Conference on Boundary and Interior Layers - Computational and Asymptotic Methods, Limerick, July 2008.[6] H. Han, Z. Huang and R.B. Kellogg, A tailored finite point method for a singular perturbation problem on an unbounded domain, J.Sci. Comp., 36 (2008), 243-261.[7] M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions, National Bureau of Standards, 1964.[8] H. Han and Z. Huang, Tailored Finite point method for a singular perturbation problem with variable coefficients in two dimensions, J.Sci. Comp., 2009.[9] M. Stynes and L. Tobiska, Necessary L2-uniform convergence conditions for difference schemes for two dimensional convection-diffusion problems, Computers Math. Applic., 29 (1998), 45-53.[10] H. Eckhaus, Boundary layers in linear elliptic singular pertubation problems, SIAM Rev., 14 (1972), 225-270.[11] C. Johnson, U. N¨avert, An analysis of some finite element methods for advectiondiffusion problems.Analytical and Numerical Approaches to Asymptotic Problem in Analysis (O. Axelsson, L.S. Frank, A. ver der Sluis, eds.), North-Holland, Amsterdam, 1981.
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 1 用特徵裁縫有限點法解對流佔優的擴散對流反應問題 2 發展求解不可壓縮Navier-Stokes及相關組成方程式之有限元素模型

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 1 使用自適應非等向性擴散方程式去除影像雜訊 2 以參數化區塊奇異值分解做壓縮和去除雜訊 3 用特徵裁縫有限點法解對流佔優的擴散對流反應問題 4 殘存價值對一個離散型運作產品在年齡置換策略及折價退費保固下之效應研究 5 偏微分方程與自適區塊截斷編碼結合用於影像去雜訊與壓縮 6 以由常微分方程式導出的狀態空間模型從時間序列基因表現資料對基因調控網路的推論 7 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. 8 核心邏輯斯迴歸模式之微陣列資料分析：次序類別的癌症分類 9 利用裁縫有限點法與延續法則解非線性薛丁格特徵值問題 10 比較多種數值方法在多種網格上解擴散對流問題之數值結果 11 對流擴散模型去雜訊法與小波和奇異值分解去雜訊法在影像上的比較 12 利用裁縫有限點法解決擴散對流問題 13 在小型基因調控網路使用時間序列基因表現資料辨識目標基因調控子：數學模型和電腦模擬 14 以邏輯斯迴歸模式探討在微陣列資料上的多類別癌症分類 15 徑向基底函數配點法解非線性薛丁格方程

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