裂縫問題為一具奇異之橢圓邊值問題，經常被拿來當作模型問題測試數值方法之用。
我們計畫求出其最高精度之解，總務差達到O(10-100)，此解便可視為理論解來使用。
另外我們也計畫改變此問題的邊界條件，得到許多相近之裂縫模型，並計算出其解。
將這些模型之解與奇異問題的經典”莫茲問題”之解比較，我們可得知各模型之優劣與最佳之數值測試模型。

Cracked beam problem is an elliptic boundary value problem with singularity. It is often used as a testing model for numerical methods.
We use numerical and symbolic boundary approximation methods and boundary collocation method to compute its extremely high accurate solution with global error $O(10^{-100})$.
This solution then can be regarded as the exact solution. On the other hand, we vary the boundary conditions of this problem to obtain several related models.
Their numerical solutions are compared to those of cracked beam and Motz problems, the prototypes of singularity problems.
From the comparison we can conclude the advantage of each model and decide the best testing model for numerical methods.

Chapter1: Cracked Beam Problem.........2
1.1 Introduction......................2
1.2 Cracked Beam Problem..............5
1.3 Symbolic BAM......................12
1.4 Numerical BAM.....................22
1.5 Gauss-Chebyshev Collocation.......35
1.6 Numerical Comparison..............42
1.7 Cracked-Beam vs Motz Problem......48

Chapter2: Related Singularity Models...55
2.1 Related Models....................55
2.2 Numerical Solutions...............57
2.3 Conclusion........................138

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