在本篇論文中，我們藉由定義不同的離散型梯度，改進之前原盤上的離散型散度定理及格林等式，讓這些定理有更漂亮的結果；接著我們再將這樣的結果推廣到三度空間，建立三度空間球上的離散型散度定理及格林等式。

In this thesis, we used different definitions for the discrete gradient so that the discrete divergence theorems are expressed in a great way. We adopt our definition of discrete gradient to improve the discrete divergence theorems and Green’s identities on a disk. And then, we establish the discrete divergence theorems and the Green’s identities on a mesh ball in R3.

1.Introduction..................................................1
2.Discrete Divergence Theorem on a Disk..........3
3.Discrete Green’s identities on a Disk............11
4.Discrete Divergence Theorem on a 3-Ball......16
5.Discrete Green’s Identities on a 3-Ball..........30
6.Conclusion.................................................38
References....................................................39

References:
[1] D.Gilbarg and N.S.Trudinger, Elliptic partial
differential equations of second order, 2nd ed.,
Springer-Verlag, Berlin, Heidelberg, New York, and
Tokyo, 1983.
[2] Bi-Jin Huang, Some Discrete Green Identities on a Disk,
theses of master degree, National Chung-Hsing
University, 2005.
[3] Pei-Yu Jhong, On Sharper Forms of the Discrete
Divergence Theorem and Green’s Identities in a Disk,
theses of master degree, National Chung-Hsing
University, 2007.