Preface to the Second Edition
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changes, the first edition of the book has become increasingly outdated and the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
The FAC examples in Chapter 9 were developed 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/
We thank Irad Yavneh for a thorough reading of the book, for his technical
insight, and for his suggestion that we enlarge Chapter 4. We are also grateful
to John Ruge who gave Chapter 8 a careful reading in light of his considerable
knowledge of AMG. Their suggestions led to many improvements in the book.
Deborah Poulson, Lisa Briggeman, Donna Witzleben, Mary Rose Muccie, Kelly
Thomas, Lois Sellers, and Vickie Kearn of the editorial staff at SIAM deserve
thanks for coaxing us to write a second edition and for supporting the project from
beginning to end. Finally, I am grateful for the willingness of my co-authors to
collaborate on this book. They should be credited with improvements in the book
and held responsible for none of its shortcomings.
November 15, 1999