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

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