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研究生:李偉任
研究生(外文):Lee, Wei-Jen
論文名稱:多重網格與自調適法於Laplace 方程角奇異解的數值計算
論文名稱(外文):Multigrid and Adaptive Methods for Computing Singular Solutions of Laplace Equation on Corner Domains
指導教授:吳金典
指導教授(外文):Wu, Chin-Tien
學位類別:碩士
校院名稱:國立交通大學
系所名稱:應用數學系數學建模與科學計算碩士班
學門:數學及統計學門
學類:數學學類
論文種類:學術論文
論文出版年:2010
畢業學年度:98
語文別:英文
論文頁數:56
中文關鍵詞:多重網格有限元素法奇異解
外文關鍵詞:Multigridsingular elementsingular funcitonFEMAdaptive mesh-refinementcorner singularitiesstress intensity factorscut-off function
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橢圓邊界值問題在凹角的地方會有奇異的行為,而這個奇異的行為對於用有限元素法離散的精確度會受到影響。對於給定Dirichlet 邊界條件的Poisson方程式和在定義域有凹角的情況之下,本論文利用一個奇異解的表示法,算出較準確的近似值,其中卡帕在工程上稱之為應力強度因子。這些量的精確計算在許多實際的工程問題上,是一門很重要的課題。
Elliptic boundary value problems on domain with corners have singular behavior near the corners. Such singular behavior affect the accuracy of the finite element method throughout the whole domain. For the Poisson equation with homogeneous Dirichlet boundary conditions defined on a polygonal domain with re-entrant corners, it is well known that the solution has the singular function u=w+ks representation ,where w is the regular part of the solution and s are known as singular functions that depend only on the corresponding re-entrant angles. Coefficients k known as the stress intensity factors in the context of mechanics can be expressed in terms of u by extraction formula, where s- are known as dual singular funciton. Accurate calculation of these quantities is of great importance in many practical engineering problems. Similar singular function representations hold for the solutions of interface,biharmonic,elasticity, and evolution problems in [1, 2].
Introduction 1
1 Finite Element Method 3
1.1 Introduction of Finite Element Method 3
1.2 Variational Formulation 5
1.2.1 Existence and Uniqueness of Solution 5
1.3 Finite Element Discretization 6
1.3.1 Linear Interpolation 7
1.3.2 Coordinate Transformations 8
1.3.3 Linear FEM Discretization 9
1.3.4 Partial Derivatives 10
1.3.5 Assembling the Element Matrix 12
1.4 Error Estimation 12
1.4.1 Interpolation Error with Piecewise Linear Functions 12
1.4.2 A Priori Error Estimation 18
1.4.3 A Posterior Error Estimation and Adaptive Mesh-Refinement Techniques 19
1.5 Finite Element Approximation for Singular Functions 21
1.5.1 Lagrange Interpolation 22
1.5.2 Singular Element 23
2 Multigrid Method 26
2.1 Introduction of Multigrid Method 26
2.2 Relaxation Process 27
2.2.1 Jacobi Method 28
2.2.2 Gauss-Seidel Method 29
2.2.3 Successive Overrelaxation Method 30
2.3 Inter-grid Interpolation : Restriction and Prolongation 32
2.4 Multigrid Algorithm 34
2.5 Complexity 37
2.6 Numerical Experiments 37
3 Research Method 43
4 Numerical Results 49
References 55

[1]Kellogg RB.Singularities in interface problems .In Symposium on Numerical Solutions of Partial Differential Equations,vol.II. Acadamic Press: New York,1971;351-400

[2]Grisvard P.Singularities in Boundary Value Problems. Masson: Paris, Springer: Berlin, 1992.

[3]S.C.Brenner ,Multigrid methods for the computation of singular solutions and stress intensity factors I: Corner singularities,preprint ,1996.

[4]M.Dauge,Elliptic Boundary Value Problems on Corner Domains,Lecture Notes in Mathematics 1341,Springer-Verlag,Berlin-Heidelberg,1988.

[5]P.Grisvard,Elliptic Problrms in Non Smooth Domains,Pitman,Boston,1985.

[6]V.Kondratiev,Boundary value problems for elliptic equations in domains with conical or angular points, Tran.Moscow Math.Soc.,16(1967),pp.277-313.

[7]S.A.Nazarov and B.A.Plamenevsky,Elliptic Problems in Domains with Piecewise Smooth Boundaries,de Gruyter,Expositions in Mathematics ,13,Berlin,New York,1994.

[8]Satya N.Atluri, Computational Methods for Plane Problems of Fracture,Georgia Institue of Technology,U.S.A

[9]S.C.Brenner and L.-Y.Sung ,Multigrid methods for the computation of singular solutions and stress intensity factors II: Crack singularities,1997.

[10]S.C.Brenner,Overcoming corner singularuties by multigrid methods ,preprint,1996.

[11]I.Babuska and A. Miller,The post-processing approach in the finite element method part 2 : The calculation of stress intensity factors,Int.J.Numer.Methods Engrg.,20(1984),pp.1111-1129.

[12]M.S.Birman and G.E.Skvorcov,On the square integrability of the highest derivatives of the solution of the Dirichlet problem on domain with piecewise smooth boundaries(in Russian),Izv.Vyssh.Uchebn.Zaved.Mat., Vol (1962),pp.136-155.

[13]H.Blum and M.Dobrowolski, On finite element methods for elliptic equations on domains with corners,Computing,28 (1982),pp.53-63.

[14]M.Dauge, M.-S. Lubuma, and S. Nicaise, Coefficient des singularities pour le probleme de Dirichlet sur un polygone,C.R.Acad. Sci.Paris Ser. I Math., 304 (1987),pp.483-486.

[15]M.Dauge, S. Nicaise, M.Bourlard, and M.-S. Lubuma ,Coefficients des singularities pour des problemes aux limites elliptiques sur un domaine a points coniques I: resultats generaux pour le probleme de Dirichlet, M2AN,24 (1990),pp.27-52.

[16]M.Dauge, S. Nicaise, M.Bourlard, and M.-S. Lubuma ,Coefficients des singularities pour des problemes aux limites elliptiques sur un domaine a points coniques II: quelques operateurs particuliers, M2AN,24 (1990),pp.343-367.

[17]R.Verfurth,A posteriori error estimation and adaptive mesh-refinement techniques, Journal of Computational and Applied Mathematics 50 (1994) 67-83.

[18]Claes Johnson, "Numerical solution of partial differential equations by the finite element method",Cambridge University Press, 1988

[19]William L.Briggs,"A Multigrid Tutorial",Department of Mathematics University of Colorado at Denver Denver, Colorado.

[20]Richard Barrett,Michael Berry,Tony F.Chan,James Demmel ,June M. Donato,Jack Dongarra,Victor Eijkhout,Roldan Pozo,Charles Romine,and Henk Van der Vorst"Templates for the Solution of Linear Systems :Building Blocks for Iterative Methods".

[20]I.Babuska, R.B. kellogg, J,Pitkaranta, Direct and Inverse Error Estinates for Finite Elements with Mesh Refinements,1972.

[21]I.Babuska, Michael B. Rosenzweig, A finite element scheme for domains with corners,1971.
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