求解偏微分方程，經過各種數值方法 (有限元素、有限差分)，皆得到線性聯立方程組:Ax=b。如果x有n 個未知數,用高斯消去法來求解 ，需要n的三次方的計算量，使用傳統迭代法來求解 ，如 Jacobi , Gauss-Seidel ， 對於稀疏矩陣 ，每次需n 的計算量。
但是迭代法僅能很快地將高頻誤差抹去，也就是為抹去低頻誤差，需要相當多的迭代次數。為解決此問題，我們有(幾何型)多重網格法：選取細網格的一半間距作為粗網格，先在粗網格點上得到微分方程的近似解，再線性外差到細網格，作為迭代法的初始值。在求解Poisson方程中，這個方法有效地將為達收斂所需時間，由只用迭代法的n的平方降到n。
求解另一個方程：-e*ddu/ddx-ddu/ddy=f ，e~0 ，我們發現真正相關的未知數在y 方向。我們必須調整外差時的係數，或是調整粗網格點的位置來達到比較好的收斂效果。代數型多重網格法從另一個角度來解決此問題：由矩陣直接構造出合適的粗網格、相對應的外差矩陣，和粗網格上的聯立方程組。
我們套用代數型多重網格法來解介面問題：
-a*ddu/ddx-c*ddu/ddy+b*ddu/dxdy=f ，a,b,c 在四區域各不相同，和biharmonic方程，都得到收斂。收斂快慢則和原矩陣A的好壞(diagonal dominate與否)有關。

In this paper, how multigrid method accelerate the convergence rate of simple iterative methods, such as Jacobi and Gauss-Seidel, is discussed.
We also use Krylov subspace methods, such as PCG and BiCGSTAB, as smoothers, and compare the results with those related to classical smoothers.
Since geometric multigrid is powerful in solving ordinary elliptic P.D.E., we illustrate some examples geometric one cannot work, or performs poorly, to show the merit of Algebraic Multigrid (AMG).

1. Introduction
2. Preliminary
2.1. Construction of Multigrid
2.1.1. Coarse Grid Correction Scheme
2.1.2. Elements of Multigrid
2.1.3. V-cycle And W-cycle
2.2. Why Multigrid
2.3. How Multigrid Works
3. Algebraic Multigrid (AMG)
3.1. Concepts of AMG
3.2. The Basic AMG Algorithm
4. Differences between AMG AND GMG
5. Numerical Results
6. Summary And Conclusions

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