在過去的文獻中，使用邊界近似法一般都具有幾何 ( 指數 ) 的收斂速度。最近有許多學者發現在某些情況下，使用譜方法可以大到比幾何收斂速度更快的超幾何收斂速度。邊界近似法也是譜方法的一種，我們猜想在某些情況下使用邊界近似法可能也具有超幾何收斂速度。在這篇論文中，我們制訂了一種方法去判斷一組誤差數據是何種收斂速度。終於，我們在使用邊界近似法解決Helmholtz邊界值問題的數值結果中，獲得了超幾何收斂。

In literature Trefftz method normally has geometric (exponential) convergence. Recently many scholars have found that spectral method in some cases can converge faster than exponential, which is called super-geometric convergence. Since Trefftz method can be regarded as a kind of spectral method, we expect it might possess super-geometric convergence too. In this thesis, we classify all types of super-geometric convergence and compare their speeds. We develop a method to decide the convergent type of given error data. Finally we can observe in many numerical experiments the super-geometric convergence of Trefftz method to solve Helmholtz boundary value problems.

1 Introduction 1
2 The Basis of Numerical Solution 2
2.1 Bessel function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
2.2 Solutions of Helmholtz equation . . . . . . . . . . . . . . . . . . . . . 3
2.3 The particular solution of Helmholtz equation . . . . . . . . . . . . . 4
2.3.1 D-D type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.3.2 D-N type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.3.3 N-D type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.3.4 N-N type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.4 Singularity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3 Testing Models 17
3.1 Singular Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.2 Analytic Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
4 The Tre_tz Method 19
4.1 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
4.2 Exponential rates of convergence . . . . . . . . . . . . . . . . . . . . 21
5 Rate of convergence 21
6 Numerical Experiment 30
References 56

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