TABLE OF CONTENTS The convergenKrylov subspace methods for solving systems arising from discretized partial differential equations (PDEs) tends to slow down considerably as these systems become larger. This deterioration in the convergence rate, compounded with the increased operation count per step due to the sheer problem size, results in a severe loss of efficiency. In contrast, the class of methods to be described in this chapter is capable of achieving convergence rates that are, in theory, independent of the mesh size. One significant difference with the preconditioned Krylov subspace approach is that multigrid (MG) methods were initially designed specifically for the solution of discretized elliptic PDEs. The method was later extended in different ways to handle other PDE problems, including nonlinear ones, as well as problems not modeled by PDEs. Because these methods exploit more information on the problem than do standard preconditioned Krylov subspace methods, their performance can be vastly superior. On the other hand, they may require implementations that are specific to the physical problem at hand, in contrast with preconditioned Krylov subspace methods, which attempt to be general purpose.
MG techniquesons with different mesh sizes of a given problem to obtain optimal convergence from relaxation techniques. At the foundation of these techniques is the basic and powerful principle of divide and conquer. Though most relaxation-type iterative processes, such as Gauss Seidel, may converge slowly for typical problems, it can be noticed that the components of the errors (or residuals) in the directions of the eigenvectors of the iteration matrix...
