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Twelve years have passed since the publication of the first edition of A Multigrid Tutorial. During those years, the field of multigrid and multilevel methods has expanded at a tremendous rate, reflecting progress in the development and analysis of algorithms and in the evolution of computing environments. Because of these changes, the first edition of the book has become increasingly outdated and the need for a new edition has become quite apparent.
With the overwhelming growth in the subject, an area in which I have never done serious research, I felt remarkably unqualified to attempt a new edition. Realizing that I needed some help, I recruited two experts to assist with the project. Steve McCormick (Department of Applied Mathematics, University of Colorado at Boulder) is one of the original researchers in the field of multigrid methods and the real instigator of the first edition. There could be no better collaborator on the subject. Van Emden Henson (Center for Applied Scientific Computing, Lawrence Livermore National Laboratory) has specialized in applications of multigrid methods, with a particular emphasis on algebraic multigrid methods. Our collaboration on a previous SIAM monograph made him an obvious choice as a co-author.
With the team in place, we began deliberating on the content of the new edition. It was agreed that the first edition should remain largely intact with little more than some necessary updating. Our aim was to add a roughly equal amount of new material that reflects important core developments in the field. A topic that probably should have been in the first edition comprises Chapter 6: FAS (Full Approximation Scheme),which is used for nonlinear problems. Chapter 7 is A collection of methods for four special situations that arise frequently in solving boundary value problems: Neumann boundary conditions, anisotropic problems, variable-mesh problems, and variable-coefficient problems. One of the chief motivations for writing a second edition was the recent surge of interest in algebraic multigrid methods, which is the subject of Chapter
8. In Chapter 9, we attempt to explain the complex subject of adaptive grid methods, as it appears in the FAC Fast Adaptive Composite) Grid Method. Finally, in Chapter 10, we depart from the predominantly finite difference approach of the book and show how finite element formulations arise. This chapter provides a natural closing because it ties a knot in the thread of variational principles that runs through much of the book.
There is no question that the new material in the second half of this edition is more advanced than that presented in the first edition. However, we have tried to create a safe passage between the two halves, to present many motivating examples, and to maintain a tutorial spirit in much of the discourse. While the first half of the book remains highly sequential, the order of topics in the second half is largely arbitrary.
The FAC examples in Chapter 9weredeveloped by Bobby Philip and Dan Quinlan, of the Center for Applied Scientific Computing at Lawrence Livermore National laboratory, using AMR   within the Overture framework. Overture is a parallel object-oriented framework for the solution of PDEs in complex and moving geometries. More information on Overture can be found at http://www.llnl.gov/casc/ Overture.
Model Problems
Basic Iterative Methods
Elements of Multigrid
Implementation
Some Theory
Nonlinear Problems
Selected Applications
Algebraic Multigrid (AMG)
Multilevel Adaptive Methods
Finite Elements