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Multivariate splines
192
Book reviews
Singluar perturbations are one of the central topics in the asymptotic analysis, which has been rapidly developing during the last decade. They play a special rule in mathematically describing several important physical phenomena such as, propagation of waves in media in the presence of small energy dissipations or dispersion, appearance of boundary or interior layers in fluid and gas dynamics, and also in the elasticity theory, semiclassical asymptotic approximation in quantum mechanics, and others, The topic of this volume is to study the linear singular perturbation theory of smooth manifolds without boundary and its applications, considered as equicontinuous linear mappings between corresponding families of Sobolev-Slobodetski's type spaces of vectorial order. Contents:
fluid dynamics which is one of the focal areas of applied scientific supercomputing, several papers deal with advanced features such as vectorization and parallelization, multigrid techniques and mesh adaptivity. In the past fifteen years, improved computers have reduced the cost of computation by a factor ofabout 100. Over the same period, better algorithms have reduced the cost of computation on a given computer by a factor of almost 1000. This emphasizes the need for a continuous exchange of experience between engineers, applied mathematicians and computer scientists involved in numerical modelling in CFD. This book is a useful source of information on recent advances in this field.
• Chapter 1: manifolds, functional analysis, distributions. This chapter provides the necessary background. • Chapter 2: Sobolev spaces of vectorial orders. It is devoted to the study of such spaces and their finite difference versions.
Multivariate Splines. Charles K. Chui, Published by Society for Industrial and Applied Mathmatics (SIAM),
• Chapter 3: Singular perturbations on smooth manifolds. It contains the core of the book and deals essentially with elliptic and hyperbolic singular
Spline functions in one variable have found numerous
perturbation and their difference counterparts, This book is available from the Amsterdam address, or the USA/Canada from Elsevier Science Publishing, Co. Inc., P.O. Box 882, Madison Square Station, New York, USA. A. Charafi
Computational Fluid Dynamics. Edited by G. de Vahl Davis & C. Fletcher, Elsevier, Amsterdam, 1988. pp. 810, US $131.50/Dfl. 250.00. ISBN: 044470430.2 This volume contains invited and contributed papers presented at the International Symposium on Computational Fluid Dynamics, which took place in August 1987 in Sydney, Australia. There is a total of 74 papers covering the following research areas: boundary layer flow, combustion and chemically reacting flows, free surface flows, geophysical flows, inviscid flows, meteorological flows, non-Newtonian flow, numerical methods and analysis, porous media, separated flow, shallow water problems, shocks wave interactions, stability and transition, supercomputers, supersonic and transonic flow, thermal convection, turbulent flows and modelling, viscous flow and vortex flow. Following the trend of recent CFDconferences, many of the papers are devoted to efficient algorithms for solution of advection-dominated flows, including high Reynolds number viscous flow, and to mathematical and numerical aspects of turbulence modelling. Also, being
L.C. Wrobel
Philadelphia, PA, USA, 1988. pp. 189. ISBN: 0 89871 226 2
applications in many different areas of engineering and applied mathematics. However, to represent a mathematical model described by several parameters and to interpret a higher-dimensional data, functions of two or more variables are often required. Charles Chui, in his book, presents the family of piecewise polynominal functions in more than one variable which are usually called multivariate splines. The subject is introduced in a parallel way to the theory and development of univariate sptine analysis. Chapter 4 reviews univariate B-splines theory with topics related to species, basis and basic properties and Chapter 2 generalizes it to the multivariate settings. Chapters 3 and 4 are devoted by bivariate splines with emphasis on dimension, locally supported splines, minimal supports, basis approximation orders and super spline subspace. Chapter 5 discusses Bezier's representation and smoothing techniques and Chapter 6 presents vertex splines with a prescribed continuity degree, useful for applications in finite element methods. Chapter 7 provides some computational algorithms of approximation schemes. Chapter 8 is devoted to quasiinterpolation schemes and Chapter 9 presents multivariate interpolation. Chapter 10 develops some topics related to shapepreserving approximation and also contains some interesting applications. The book is useful for applied mathematicians and engineers who work and research in CAD/CAM, computer graphics, mathematical modelling, numerical methods such as FEM and BEM and some related areas. J.J.S.P. Cabral
Book reviews
Singluar perturbations are one of the central topics in the asymptotic analysis, which has been rapidly developing during the last decade. They play a special rule in mathematically describing several important physical phenomena such as, propagation of waves in media in the presence of small energy dissipations or dispersion, appearance of boundary or interior layers in fluid and gas dynamics, and also in the elasticity theory, semiclassical asymptotic approximation in quantum mechanics, and others, The topic of this volume is to study the linear singular perturbation theory of smooth manifolds without boundary and its applications, considered as equicontinuous linear mappings between corresponding families of Sobolev-Slobodetski's type spaces of vectorial order. Contents:
fluid dynamics which is one of the focal areas of applied scientific supercomputing, several papers deal with advanced features such as vectorization and parallelization, multigrid techniques and mesh adaptivity. In the past fifteen years, improved computers have reduced the cost of computation by a factor ofabout 100. Over the same period, better algorithms have reduced the cost of computation on a given computer by a factor of almost 1000. This emphasizes the need for a continuous exchange of experience between engineers, applied mathematicians and computer scientists involved in numerical modelling in CFD. This book is a useful source of information on recent advances in this field.
• Chapter 1: manifolds, functional analysis, distributions. This chapter provides the necessary background. • Chapter 2: Sobolev spaces of vectorial orders. It is devoted to the study of such spaces and their finite difference versions.
Multivariate Splines. Charles K. Chui, Published by Society for Industrial and Applied Mathmatics (SIAM),
• Chapter 3: Singular perturbations on smooth manifolds. It contains the core of the book and deals essentially with elliptic and hyperbolic singular
Spline functions in one variable have found numerous
perturbation and their difference counterparts, This book is available from the Amsterdam address, or the USA/Canada from Elsevier Science Publishing, Co. Inc., P.O. Box 882, Madison Square Station, New York, USA. A. Charafi
Computational Fluid Dynamics. Edited by G. de Vahl Davis & C. Fletcher, Elsevier, Amsterdam, 1988. pp. 810, US $131.50/Dfl. 250.00. ISBN: 044470430.2 This volume contains invited and contributed papers presented at the International Symposium on Computational Fluid Dynamics, which took place in August 1987 in Sydney, Australia. There is a total of 74 papers covering the following research areas: boundary layer flow, combustion and chemically reacting flows, free surface flows, geophysical flows, inviscid flows, meteorological flows, non-Newtonian flow, numerical methods and analysis, porous media, separated flow, shallow water problems, shocks wave interactions, stability and transition, supercomputers, supersonic and transonic flow, thermal convection, turbulent flows and modelling, viscous flow and vortex flow. Following the trend of recent CFDconferences, many of the papers are devoted to efficient algorithms for solution of advection-dominated flows, including high Reynolds number viscous flow, and to mathematical and numerical aspects of turbulence modelling. Also, being
L.C. Wrobel
Philadelphia, PA, USA, 1988. pp. 189. ISBN: 0 89871 226 2
applications in many different areas of engineering and applied mathematics. However, to represent a mathematical model described by several parameters and to interpret a higher-dimensional data, functions of two or more variables are often required. Charles Chui, in his book, presents the family of piecewise polynominal functions in more than one variable which are usually called multivariate splines. The subject is introduced in a parallel way to the theory and development of univariate sptine analysis. Chapter 4 reviews univariate B-splines theory with topics related to species, basis and basic properties and Chapter 2 generalizes it to the multivariate settings. Chapters 3 and 4 are devoted by bivariate splines with emphasis on dimension, locally supported splines, minimal supports, basis approximation orders and super spline subspace. Chapter 5 discusses Bezier's representation and smoothing techniques and Chapter 6 presents vertex splines with a prescribed continuity degree, useful for applications in finite element methods. Chapter 7 provides some computational algorithms of approximation schemes. Chapter 8 is devoted to quasiinterpolation schemes and Chapter 9 presents multivariate interpolation. Chapter 10 develops some topics related to shapepreserving approximation and also contains some interesting applications. The book is useful for applied mathematicians and engineers who work and research in CAD/CAM, computer graphics, mathematical modelling, numerical methods such as FEM and BEM and some related areas. J.J.S.P. Cabral