三維橢圓介面問題的數值研究

Abstract

在本篇論文中,系使用內嵌介面法把一個二維橢圓介面問題推廣到三維。此方法為調整過的有限差分法，其修正項透過介面上的不連續條件而得。藉由由法向量方向所展開的泰勒展開式，在介面附近的網格點，離散差分方程會被調整。因為在介面上不連續條件型態的關係， 在解橢圓介面問題的過程中，必須要解一個線性系統。首先用重新初始化水平集方法來找網格點在界面上的正交投影，接下來我們運用內嵌介面法，廣義最小殘量方法，和最小平方法來解橢圓介面問題。

In this thesis, we extend the immersed interface method for a 2D elliptic interface problem to 3D. The numerical method is a finite difference method modified with some correction terms from the jump conditions. By using Taylor's expansions along the normal direction, the discrete difference equation is modified at the gird point close to interface. Because of the types of the jump condition, we have to solve a linear system in the process of solving the elliptic interface problem. We first use the reinitialization of level set method to find the orthogonal projection of the grid point and perform it to check its accuracy. Then, we solve the elliptic interface problem by the immersed interface method, GMRES and least squares method, and make some numerical tests to check the rate of convergence.