本論文分為兩部分，第一部分為介紹二變數的伯克霍夫插值法之定義，並且描述其在交面問題上之應用。接下來展示如何用伯克霍夫類的緊緻插值法進行插值並且探討行列式之性質以幫助我們計算行列式。
在第二部分，我們將會展示二階，三階及四階逼近的數據和可行的圖表。

This thesis consists of two parts. In the first part, we will introduce the definition of bivariate Birkhoff interpolation and describe its application
on interface problem. Next, we will show an example that how we interpolates a function by compact Birkhoff type interpolation and discuss the property of determinant to help us compute the determinant.

In the second part, we will show the data and workable figures of the quadratic, cubic and fourth degree approximation.

Contents
1 Introduction and Definition of Bivariate Birkhoff Interpolation ....... 1
1.1 Definition of bivariate Birkhoff interpolation . . . . . . . . . . . . . 1
1.2 Approximation to a smooth function in u+f case. . . . . . . . . . 2
1.3 The determinant ............................ 7

2 Simulations and Figures .................8
2.1 The quadratic approximation ..................... 9
2.2 The cubic approximation........................ 15
2.3 The fourth degree approximation ................... 21
Reference ..............28

[1] Rudolph A. Lorentz, Multivariate Birkhoff Interpolation.
[2] Richard L. Burden & J. Douglas Faires, Numerical Analysis.
[3] W. J. Ying, C. S. Henriquez, A kernel-free boundary integral method for elliptic boundary value problems, J. Comp. Phy. 227 (2) (2007) 1046–1074.