Free and Moving Boundary Problems. by John CrankReview by: C. M. ElliottSIAM Review, Vol. 30, No. 4 (Dec., 1988), pp. 668-669Published by: Society for Industrial and Applied MathematicsStable URL: http://www.jstor.org/stable/2030583 .

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668 BOOK REVIEWS

Applications, CBMS-NSF Regional Confer- ence Series in Applied Mathematics 26, So- ciety for Industrial and Applied Mathematics, Philadelphia, 1977.

[3] R. G. VOIGT, D. GOTTLIEB, AND M. Y. HUSSAINI, EDS., Spectral Methods for Partial Diferential Equations, Society for Industrial and Applied Mathematics, Philadelphia, 1984.

[4] B. A. FINLAYSON, The Method of Weighted Residuals and Variational Principles, Aca- demic Press, New York, 1972.

[5] R. PEYRET AND T. D. TAYLOR, Computa- tional Methods for Fluid Flow, Springer- Verlag. Berlin. New York. 1983.

JOHN P. BOYD Laboratory for Scientific Computation University of Michigan

Free and Moving Boundary Problems. By John Crank. Clarendon Press, The Oxford University Press, Oxford, UK, 1987. x + 425 pp. $39.95, paper. ISBN 0-19-853370-5. An Oxford Science Publication.

In many important applications, partial differential equations are to be solved in domains that must be determined as part of the solution. Since part (or all) of the do- main's boundary is unknown, it is common practice to use the termsfree-boundary prob- lems for steady-state situations, and moving- boundary problems for evolutionary situa- tions with an unknown time-dependent boundary. The book under review is a 1987 paper edition of a volume first published in 1984. It is a compendium and survey of free- and moving-boundary problems and numer- ical methods for their solution. As such, it is aimed at practitioners in the applied sciences and engineering, together with applied math- ematicians and numerical analysts. The main virtues of the book are its accessibility and its comprehensive nature. It is similar in style to the author's previous bestseller enti- tled The Mathematics of Diffusion (Claren- don Press, Oxford, 1975).

The contents of the book are as follows:

Chapter 1. Moving-Boundary Prob- lems: Formulation. The idea of moving- boundary problems for parabolic equations is described, together with examples from heat conduction with change of phase (the

Stefan problem), flow in a porous medium, and diffusion.

Chapter 2. Free-Boundary Problems: Formulation. Free-boundary problems for elliptic equations are described with the main application area being steady-state flow in a porous medium. Other applications consid- ered include electrochemical machining and lubrication.

Chapter 3. Analytical Solutions. In general it is not possible to find explicit for- mulae for the solution of unknown boundary problems. A list of some solutions of the Stefan problem is given together with some approximate analytical methods.

Chapter 4. Front-Tracking Methods. These are discretization methods for mov- ing-boundary problems that explicitly ap- proximate the partial differential equation in a time-dependent domain.

Chapter 5. Front-Fixing Methods. Transforming to new space coordinates can fix the moving front at the expense of intro- ducing complications into the differential equation.

Chapter 6. Fixed-Domain Methods. Frequently the unknown boundary is a level surface of a dependent variable that solves a conservation equation or a variational prob- lem. Natural discretizations of the resulting formulation lead to nonlinear algebraic problems without explicit reference to the unknown boundary, which is then recovered after the computation by seeking the level surface of the discrete approximation. This leads to the "enthalpy" method for the Stefan problem and the practical application of var- iational inequalities.

Chapter 7. Analytical Solution of Seepage Problems. This is an account of analytical methods (hodograph, conformal transformation) for the solution of free- boundary problems for Laplace's equation arising in porous flow.

Chapter 8. Numerical Solution of Free Boundary Problems. The trial free-boundary method, boundary singularities, method of variable interchange, variational inequalities, and integral equation methods are described. Numerical results are quoted for a variety of applications in porous flow and electrochem- ical machining.

The author gives clear expositions of methods and examples of their use. There

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BOOK REVIEWS 669

are numerous figures and tables, taken from original papers, illustrating the results of numerical experiments.

Previous books on this subject matter have specialized in the pure mathematics or numerical analysis of a particular method- ology, namely that of variational inequalities and weak solutions. Thus a gap in the liter- ature is filled by this book, which should be the one a practitioner, without any prior knowledge and desperate for help, would first turn to.

The book succeeds in its aim of pre- senting a broad but detailed account of math- ematical and numerical methods for free- and moving-boundary problems that will be accessible to researchers both in the applied sciences and in applied mathematics.

C. M. ELLIOTT University of Sussex

Computational Geometry: An Introduction. By Franco P. Preparata and Michael Ian Shamos. Springer-Verlag, New York, 1985. xii + 390 pp. ISBN 0-387-96131-3. Texts and Monographs in Computer Science series.

The term computational geometry was first used by Robin Forrest to describe computer-aided design and by Minsky and Papert to describe geometric algorithms us- ing perceptrons. This book refers to a third meaning, the recasting of classical geometry into a form amenable to efficient execution on a computer. This use of computational geometry was begun by Shamos in his Ph.D. thesis; this book is a very heavily revised and extended version of his thesis.

The field of computational geometry is a source of inspiration for theoretical com- puter science since it consists of so many useful problems, which are difficult but solv- able. This is the first book in the field and is a necessity for the serious researcher. It cov- ers the field quite extensively. This reviewer has found it suitable as a text for a graduate course, and there are many exercises in each chapter. Unfortunately, there are assorted errors, but they are minor. The chapter notes are a very desirable part of each chapter also.

The introduction has a good summary of the history of geometry and presents some preliminary algorithms and data structures.

The chapter on geometric searching shows some of the possible time-complexity tradeoffs as illustrated by a sequence of algorithms for two-dimensional range searching with preprocessing time 0(N' +) for arbitrarily small e > 0.

The chapters on convex hulls present many worst-case efficient and expected-case efficient algorithms, and present applications to statistics, such as minimum diameter clus- tering.

In the chapters on proximity we learn about Voronoi diagrams that solve closest point, farthest point, Euclidean minimum spanning tree, triangulation of a point set, and many other problems. The (two- dimensional, closest-point) Voronoi diagram is a partition of the plane determined by N points. It consists of a polygon around each point that contains the part of the plane closer to that point than to any other point. We also see the reduction of a k-d Voronoi diagram problem to a (k+ 1)-d convex hull problem, which can be reduced to a (k+ 1)-d Voronoi diagram problem. Voronoi diagrams are intrinsically harder in higher dimensions; for N points in E', with d even, there may be (d/2)!N"'(2) Voronoi vertices.

The section on intersections starts with convex polygons, which can be intersected in 0(N) time for input polygons with N sides, and covers assorted generalizations such as visibility problems. The last section, The Geometry of Rectangles, discusses material appropriate to efficient VLSI algorithms.

Since this is a more theoretical book, it does not cover implementation considera- tions such as special cases, which can account for half the total lines of code. The numerical stability of the algorithms is not considered. (Roundoff errors may cause a numerically unstable algorithm to self-destruct when given degenerate input. For example, some Voronoi diagram implementations cannot handle four concentric points.) An imple- mentor must also consider whether an algo- rithm with the minimum asymptotic rate of growth is needed, or whether one with a smaller constant factor on the time, and which can be implemented in a reasonable amount of time, is preferable. This is relevant since many computational geometry algo- rithms have never been implemented. These considerations might bias a practical user

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