542 BOOK REVIEWS

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M. TAYLOR, Reflection ofsingularities ofso-lutions to wave equations, Comm. PureAppl. Math., 28 (1975), pp. 457-478.

Pseudodifferential Operators,Princeton University Press, Princeton,NJ, 1981.

E TREVES, Introduction to PseudodifferentialOperators and Fourier Integral Opera-tors, Plenum, New York, 1980.

MICHAEL TAYLORUniversity ofNorth Carolina

Multigrid Methods for Finite Elements. ByV. V. Shaidurov. Kluwer Academic Publishers,Norwell, MA, 1995. $148.50. 331 pp., hard-cover. ISBN 0-7923-3290-3.

This is a book mainly on the convergence the-ory for multigrid methods. In contrast with otherexisting multigrid books (such as Hackbusch [2]and Wesseling [3]), this book is more theoret-ical and mainly about finite element discretiza-tions. Its contents may be compared with thoseof Bramble’s [1], but this book is not as con-cise as and it contains more material on prob-lems such as nonsymmetric/indefinite systemsand nonlinear equations. The focus of the bookis on the convergence properties of V- and W-cycle multigrid methods applied to finite elementdiscretized systems for elliptic boundary valueproblems.

The theoretical nature of the book makes ita good reference for theoreticians on the multi-grid methodology. The book is also mostly selfcontained with detailed presentations; hence itshould be useful to most researchers or graduatestudents who want to have a mathematical under-standing of why multigrid methods work. Thebook should be considered more as a researchmonograph than an introductory book on multi-grid methods; thus readers should be warned thatit is not very concerned with practical aspects ofmultigrid methods or details on some basic com-ponents such as smoothers, prolongations, andrestrictions.

The first three chapters contain backgroundmaterial on Sobolev spaces, boundary value prob-lems, and finite element discretizations. The pre-sentations in these chapters are descriptive in na-ture and most results are listed without proofs.Chapter (36 pages) gives a brief description ofseveral model elliptic boundary value problems(including linear elasticity and biharmonic and

Stokes equations) and their variational formula-tions with finite element approximations (includ-ing mixed formulations). This chapter is basi-cally to introduce notation and list the main re-suits for Sobolev spaces. Chapter 2 (38 pages)gives some basic descriptions of some standardfinite element spaces and brief descriptions ofsome triangulation techniques. Chapter 3 (42pages) concerns the approximation properties offinite element solutions.

The main topic of the book is treated in thelast three chapters. Chapter 4 (89 pages) givesa general description of multigrid algorithms inan abstract setting for both symmetric positivedefinite problems and nonsymmetric or indef-inite problems. Mixed finite element methodsand eigenvalue problems are also discussed. Ap-plications of the theories in Chapter 4 are givenin Chapter 5 (41 pages) to second-order ellip-tic boundary value problems. Special discus-sions are also given here regarding the treatmentof curved boundaries and corner singularities.Chapter 6 is on nonlinear problems and systemsof equations. Quasilinear second-order ellipticboundary value problems are discussed in de-tail. The biharmonic problem is also discussed,mainly in a mixed formulation. As examples forsystems, the linear elasticity and Stokes equationsare also studied.

The book, however, does not mention morerecent multigrid convergence theories that do notmake explicit use of elliptic regularity and that inparticular prove the V-cycle still converges uni-formly "in the nonregular case" and also for cer-tain locally refined meshes (cf. Bramble or Xu[4]). The analysis of a locally refined (graded)mesh (around p. 229) lacks enough details andhence looks questionable.

The bibliography contains 200 publications.Although it is not a very complete list of ref-erences on the multigrid theory, it does con-tain much Russian literature, which should bevery valuable to many researchers outside ofRussia.

In summary, recommend this book to re-searchers who are interested in the theoretical as-pects of multigrid methods. It is a good referencebook.

REFERENCES

[1] J. BRAMBLE, Multigrid Methods, Pitman, Boston,MA, Notes on Mathematics, 1994.

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BOOK REVIEWS 543

[2] W. HACKBUSCH, Multigrid Methods and Applica-tions, Vol. 4 of Computational Mathematics,Springer-Verlag, Berlin, 1985.

[3] P. WESSELING, An Introduction to Multigrid Meth-ods, John Wiley and Sons, Chichester, UK,1992.

[4] J. Xu, Iterative methods by space decomposi-tion and subspace correction, SIAM Rev., 34(1992), pp. 581-613.

JINCHAO XUPenn State University

Multigrid Methods for Process Simulation. ByW Joppich and S. Mijalkovic. Springer-Verlag,Berlin, New York, 1993. $129.00. 309 pp., hard-cover. ISBN 3-211-82404-9. Springer-VerlagWien New York ISBN 0-387-82404-9 Springer-Verlag New York Wien.

The multigrid method has proven to be one ofthe most powerful methods in many applicationsand the topic of this book is on how the multi-grid method can also be effectively applied toa very special field, namely, process simulation.The book was, as stated in its preface, a prod-uct of collaboration between two authors fromtwo different backgrounds, one from a multigridresearch group and the other from an engineer-ing discipline. The intention of this book is "Togive the descriptions of the ingredients which areneeded for successful multigrid algorithms andto give the compact information (of) why andhow multigrid works in a realistic process sim-ulation environment." The book indeed offersmany mathematical insights, but the style of itspresentation appears to be more attuned to engi-neering.

The book is exclusively on finite differencemethods, although finite element methods arehighly recommended in its introduction (Chapter1). Indeed, for the model problems studied in thisbook, the finite element approach seemed to beadvantageous in many respects, such as its flexi-bility for complicated geometries and free surface(with moving triangular meshes, for example), itsnatural treatment of the natural boundary condi-tions, and its easy application of adaptive grids. Itis a pity that the book does not have any materialon finite element discretizations.

The contents of the book are briefly summa-rized below.

Chapter 2, which takes about half of the wholebook, gives an introduction to the basic ideas

and general principles of multigrid methods. Thestated aim of this chapter is "to explain why andhow multigrid works." Most of the basic ele-ments ofmultigrid methods are discussed to someextent in this chapter.

Chapter 3 discusses, in a very general setting,adaptive time step control techniques for generalinitial boundary value problems which are onlydiscretized in their temporal variable. Some briefdescriptions of adaptive multigrid methods are

given, but the discussions are mostly philosophi-cal and few technical details are given. An exam-

ple one dimensional linear problem is studied.Chapter 4 is devoted to a specific diffusion

model problem for process simulations, namely,a free-surface nonlinear parabolic problem. Aftera brief introduction of the model problem, somerather detailed discussions are given on how themodel problem can be discretized by finite dif-ference schemes and how multigrid methods canbe applied to solve the discretized systems.

Chapter 5 describes many multigrid proce-dures using techniques discussed in previouschapters with an emphasis on spatial local adap-tive grid refinement. The grid refinement is basedon strictly regular grids on a rectangular domain.The discussions in this chapter are quite formal,and no technical details are given on how the lo-cal refinement is obtained. Several pictures ofnumerical simulations are shown in this chapter.

Process simulation is a relatively less-developed area of research encompassing math-ematical modeling to numerical analysis. Multi-grid methods, as demonstrated in the book, area very powerful technique for problems in thisfield. This book should be a very useful referencefor researchers in the related fields, especially forthose who use finite difference discretizations.

JINCHAO XUPenn State University

Mathematical Go Chilling Gets the Last Point.By Elwyn Berlekamp andDavid Wolfe. A. K. Pe-ters, MA, 1994. $34.95. 234 pp., hardback.ISBN 1-56881-032-6.

Compared with chess, computer programs for Goare still in a primitive age. This is illustratedby the fact that the best chess program can rankamong at least the top 100 human players in theworld, whereas the best Go program still needs totake more than a 13 stone handicap from human

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