當我們在同心圓邊界上的簡單環形域裡做誤差分析時，我們可以取得它的誤差範圍，並且得到它的最佳收斂速度。對於在圓形邊界上的圓型域，我們可以經由嚴格的證明去得到同樣的收斂速度。
此外，零場法及其守恆型的方案可以適用於非常小的圓洞，這是很難被其他的數值方法來處理。無論是零場法和Trefftz方法都是用於非常小的圓孔上，而Trefftz方法在精度和穩定性上優於零場法。

The error analysis is made for the simple annular domain with the circular boundaries having the same origin. The error bounds are derived, and the optimal convergence rates can be archived. For circular domains with circular boundaries, the same convergence rates can be achieved by strict proof. Moreover, the NFM algorithms and its conservative schemes can be applied to very small holes, which are difficult for other numerical methods to handle. Both the NFM and the collocation Trefftz method(CTM) are used for very small circular holes, the CTM is superior to the NFM in accuracy and stability.

Contents
Abstract i
1 Explicit Algorithms of Null Field Methods 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Explicit Algorithms and Their Conservative Schemes of Null Field Methods 3
1.2.1 Basic Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2.2 Explicit Algebraic Equations . . . . . . . . . . . . . . . . . . . . 5
1.2.3 Explicit Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2.4 Conservative Schemes . . . . . . . . . . . . . . . . . . . . . . . . 8
1.3 The First Kind Boundary Integration Equations . . . . . . . . . . . . . . 10
1.3.1 The First Kind Boundary Integration Equations . . . . . . . . . . 10
1.3.2 Relations of TM with IFM and The First Kind BIE . . . . . . . . 11
2 Further Investigation on Null and Interior Field Methods 15
2.1 Interior and Exterior Problems . . . . . . . . . . . . . . . . . . . . . . . 15
2.1.1 Interior Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.1.2 Exterior Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.2 Circular Boundaries Having The Same Origin . . . . . . . . . . . . . . . 18
2.2.1 Exact Coefficients from NFM . . . . . . . . . . . . . . . . . . . . 18
2.2.2 Interior Solutions from NFM . . . . . . . . . . . . . . . . . . . . . 23
2.3 Error Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.3.1 Error Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.3.2 Properties of Sobolev Norms . . . . . . . . . . . . . . . . . . . . . 30
2.3.3 The Case with Very Small R1 . . . . . . . . . . . . . . . . . . . . 34
2.4 True Solutions for Eccentric Cable in Circular Cable . . . . . . . . . . . . 35
3 Numerical Experiments and Conclusion 41
3.1 Numerical Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.1.1 Model Problems by NFM . . . . . . . . . . . . . . . . . . . . . . 41
3.1.2 Conservative Schemes . . . . . . . . . . . . . . . . . . . . . . . . 42
3.1.3 Numerical Results for Very Small R1 by IFM . . . . . . . . . . . 44
3.1.4 Model Problems by CTM . . . . . . . . . . . . . . . . . . . . . . 44
3.1.5 Numerical Results by CTM and BIE . . . . . . . . . . . . . . . . 54
3.1.6 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . 55

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