多重網格法解一些不可分離的橢圓方程式

Abstract

這篇論文主要之目的是使用多重網格法來解一些不可分離的橢圓方程式有著Dirichlet條件在矩形的區域上(當然這種方法也可應用在其他的邊界條件下)。首先，我們會學習基本的多重網格網法。再來，我們簡要地介紹預加條件共軛梯度法和Concus and Golub法。最後，我們會給一些例子並且列出數值結果其中包含了達到判停條件所需的計算時間和迭代次數，然後做出結論。

The primary objective of this thesis is to use multigrid method (MG) for solving nonseparable elliptic equations with Dirichlet boundary condition on a rectangle. (Of course, this method can be applied with any boundary conditions.) First, we study elements of multigrid method. Next, we introduce roughly the preconditioned conjugate gradient (PCG) method and Concus and Golub's method to compare with MG. Finally, we give some examples and show numerical results including CPU time and the number of necessary iterations to achieve stopping criterion, and the conclusion follows.