在這篇碩士論文，我們以數學上的角度探討橢圓算子特徵模組問題，進而可以獲得 – 當形狀參數趨近於零，使用標準徑向基底函數之方法會收斂至使用趨近平坦之徑向基底函數所得到之內差多項式的方法。首先，我們使用特殊的轉換方式，讓我們可以使用徑向基底函數把原始問題轉化成一個有限維的矩陣特徵值問題。接下來也轉換使用趨近平坦之徑向基底函數所得到之內差多項式的方法，再將兩者做比較，歸納其收斂性，而此項結果將使用理論以及數值結果作證明。

Although an elliptic operator eigenmode problem can solve easily by using lots of method. In this thesis, we want to show the significance of mathematics that for eigenmodes problem using the RBF collocation method converges to that of using increasingly flat radial basis functions when ε goes to 0 in 2D. First, we use radial basis functions (RBFs) to solve eigenmodes problem for elliptic operator by converting the eigenmodes problem to an eigenpairs problem of a finite dimensional matrix. And then formulate RBF interpolation polynomials as eigenfunctions, and proved this result converges to the solution obtained by using increasingly flat RBFs. And conclude that two approaches merge when RBFs are getting flatter. The results are supported by numerical examples.

[Thesis Approval Sheet + i]
[摘要 +Š ii]
[Abstract + iii]
[List of Figures + vi]
[List of Tables + viii]
[1 Introduction + 1]
[2 Definition + 3]
[2.1 Multi-index Notation + 3]
[2.2 Polynomial Spaces and Unisolvency + 4]
[3 Radial Basis Function + 7]
[3.1 Introduction to Radial Basis Function + 7]
[3.2 Generate the Interpolation Polynomial of the Increasingly Flat Radial Basis Function + 9]
[3.3 Interpolation + 13]
[4 Algorithm for Computing Eigenmodes + 17]
[4.1 Solving Two-Dimensional Eigenmodes Problem for Laplace Operator Using RBF Polynomial Interpolation on Type 1 Point Sets +17]
[4.2 Solving Two-Dimensional Eigenmodes Problem for Laplace Operator Using RBF Polynomial Interpolation on Type 2 Point Sets +19]
[4.3 Solving Two-Dimensional Eigenmodes Problem for Laplace Operator Using RBF +19]
[5 Numerical Results +23]
[6 Conclusion +50]
[References +51]

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