許多種類的徑向基底函數(RBFs)，如multiquadrics，包含一個我們稱為shape factor 的自由參數，它可以控制RBFs 的平坦程度。在一維內插問題中，Fornberg 等人證明在一些簡單的條件下，使用趨近平坦的徑向基底函數所得到的解會等價於Lagrange 內插法。在本篇論文中，我們學習這項特性並將它延申到一維Poisson equation RBFs direct solver，且觀察到解會收斂到譜方法(Spectral Collocation Method using Polynomial)。 在二維內插問題，Fornberg 等人觀察到內插子的極限無法在高度規律結構的格點下收斂；我們也在二維PDE 問題測試這個現象，並觀察到一些不同於內插問題的結果。

Many types of radial basis functions, such as multiquadrics, contain a free parameter called shape factor, which controls the flatness of RBFs. In the 1-D problems, Fornberg et al. [2] proved that with simple conditions on the increasingly flat radial basis function, the solutions converge to the Lagrange interpolating. In this report, we study and extend it to the 1-D Poisson equation RBFs direct solver, and observed that the interpolants converge to the Spectral Collocation Method using Polynomial. In 2-D, however, Fornberg et al. [2] observed that limit of interpolants fails to exist in cases of highly regular grid layouts. We also test this in the PDEs solver and found the error behavior is different from interpolating problem.

1 Introduction 5
1.1 Radial Basis Functions 5
1.2 Solving Interpolation Problems 6
1.3 Solving Elliptic PDE Problems 7
1.4 The conditioning of RBF systems in 1-D interpolations 7
1.5 Error Measures and Arbitrary Precision Computation 9

2 1-D interpolations with increasingly flat RBFs 10
2.1 Determine the limiting approximations for two points in 1-D 12
2.2 A generalized algorithm of limiting approximations in 1-D 14

3 An extension of limiting approximations to Poisson Equation in 1-D 17
3.1 1-D Spectral Collocation Method using Polynomial 18
3.1.1 Example 1 19
3.1.2 Example 2 19
3.2 Numerical Verifications when N≥3 20
3.3 Asymptotical εrms 24

4 Error behaviors v.s. layout of collocation points in 2-D 25
4.1 For interpolations 25
4.2 For PDE solvers 25
4.3 Numerical results of interpolations 27
4.3.1 Example 1: f1 = cos(2πx) 27
4.3.2 Example 2: f2 = cos(2πy) + sin(2πx) 29
4.3.3 Example 3: f3 = cos(2πx) sin(2πx) 31
4.3.4 Example 04: f4 = cos(2πy) sin(2πx) 33
4.3.5 Example 5: f5 = sin(πx/6)sin(7πx/4)sin(3πx/4)sin(5πx/4) 36
4.3.6 Example 6: f6 = exp(10x) 39
4.3.7 Example 7: f7 = exp(10√(x^2 + y^2)) 41
4.4 Numerical results of Poisson equations 44
4.4.1 Example 1: u1(x, y) = cos(2πx) 44
4.4.2 Example 2: u2 = cos(2πy) + sin(2πx) 46
4.4.3 Example 3: u3 = cos(2πx) sin(2πx) 48
4.4.4 Example 4: u4 = cos(2πy) sin(2πx) 50
4.4.5 Example 5: u5 =sin(πx/6)sin(7πx/4)sin(3πx/4)sin(5πx/4) 52
4.4.6 Example 6: u6 = exp(10x) 54
4.4.7 Example 7: u7 = exp(√(x^2 + y^2)) 56
4.5 Disturbing the layout grid of collocation points 57

5 Conclusions 67

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[2] Tobin A. Driscoll & Bengt Fornberg. Interpolation in the limit of increasingly flat radial basis functions, Computers Math. Applic. 43 (3V5) 413V422 (2002).

[3] R. L. Hardy. Theory and applications of the multiquadric-biharmonic method - 20 years of discovery, Comput Math Appl 1990;Vol. 19,No. 8/9, pp. 163-208.

[4] C.-S. Huang. Error Estimate, Optimal Shape Factor, and High Precision Computation of Multiquadric Collocation Method, Engineering Analysis with Boundary Elements Volume 31, Issue 7, 614-623(2007).

[5] Elisabeth Larsson & Bengt Fornberg. A Numerical Study of some Radial Basis Function based Solution Methods for Elliptic PDEs, Comp. Math. Appl. v46. 891-902.

[6] Elisabeth Larsson & Bengt Fornberg. Theoretical and computational aspects of multivariate interpolation with increasingly °at radial basis functions, Comp. Math. Appl. v49. 103-130.