Laplace方程式的特別解和這些解對應的singularities是基本的知識對於學習數值偏微分方程的演算法和誤差收斂分析。我們最先回顧Laplace方程式的特別解在扇形上及邊界有Dirichlet和Neumann的條件。 Harmonic函數清楚地顯示那些特別解在頂點的regularity/singularity。所以我們可以藉此分析Laplace的特別解有singularities時在多邊形上當不同的頂點有不同邊界條件的情況下。使用這些知識，我們可以創造出許多我們要的不同的singularities的模型，例如像discontinuous和mild singularities，除此之外，還有一般singularity 模型，Motz’s和cracked beam problems。
我們使用邊界近似法 (boundary approximation method 簡稱BAM)，換句話說，在工程的文獻上叫做collocation Trefftz method。去解上述測試模型屬於Laplace邊界值問題在多邊形，當沒有singularity在所有頂點，這方法所產生的誤差有exponential收斂，但是它的收斂速度將退化成polynomial假如某些頂點有singularity。從實驗數據來觀察，我們有三種收斂形式：exponential、polynomial、和他們混合的形式。我們將學習這些收斂的行為和產生。最後，我們將發現介於收斂的order和頂點singularity的關係。

The particular solutions of the Laplace equations and their singularities are fundamental
to numerical partial di erential equations in both algorithms and error analysis. We
first review the explicit solutions of Laplace’s equations on sectors with the Dirichlet
and the Neumann boundary conditions. These harmonic functions clearly expose the
solution’s regularity/singularity at the vertex. So we can analyze the singularity of
the Laplace’s solutions on polygons at di erent domain corners and for various boundary
conditions. By using this knowledge we can designed many new testing models
with di erent kind of singularities, like discontinuous and mild singularities, beside the
popular singularity models, Motz’s and the cracked beam problems,
We use the boundary approximation method, i.e. the collocation Tre tz method
in engineering literatures, to solve the above testing models of Laplace boundary value
problems on polygons. Suppose the uniform particular solutions are chosen in the entire
domain. When there is no singularity on all corners, this method has the exponential
convergence. However, its rate of convergence will deteriorate to polynomial if there
exist some corner singularities. From experimental data, we even have three type of
convergence, i.e. exponential, polynomial or their mixed types. We will study these
convergent behaviors and their causes. Finally, we will uncover the relation between
the order of convergence and the intensity of corner singularities.

1 Introduction 2
2 Solutions of Laplace Equation 4
2.1 Solution of D-D type . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Solution of D-N type . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.3 Solution of N-D type . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.4 Solution of N-N type . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.5 Singularity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3 Boundary Approximation Method 16
3.1 NBAM and SBAM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.2 Error Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
4 Convergence Types 21
4.1 Three Convergence Types . . . . . . . . . . . . . . . . . . . . . . . . . 21
4.2 Comparison of SBAM and NBAM . . . . . . . . . . . . . . . . . . . . . 25
5 Singularity at Rectangles 28
5.1 Singularity at Near Corners . . . . . . . . . . . . . . . . . . . . . . . . 28
5.2 Singularity at Far Corners . . . . . . . . . . . . . . . . . . . . . . . . . 33
5.3 Distances and Singularity . . . . . . . . . . . . . . . . . . . . . . . . . 37
6 Singularity on Polygons 44
6.1 Isosceles Triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
6.2 Other Triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
6.3 More on Polygons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
6.3.1 Angle of 90◦ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
6.3.2 Angle of 60◦ and 75◦ . . . . . . . . . . . . . . . . . . . . . . . . 54
7 Conclusion 60

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