Multigrid Methods: Managing Massive Meshes

In our last homework assignment, we investigated iterative methods for solving large, sparse, linear systems of equations. we saw that the Gauss-Seidel (GS) method was intolerably slow, but various forms of preconditioned conjugate gradient (CG) algorithms gave us reasonable results. The test problems we used were discretizations of elliptic partial differential equations, but for these problems, we can use a faster class of methods called multigrid algorithms. Surprisingly, the GS method (or some variant) is one of the two main ingredients in these algorithms!