11R23. Variational Inequality Approach to Free Boundary Problems with Applications in Mould Filling. ISNM, Vol 136. - J Steinbach (Gartnerstr 8, Augsburg, 86153, Germany). Birkhauser Verlag AG, Basel, Switzerland. 2002. 294 pp. ISBN 3-7643-6582-X. $149.00.

Reviewed by L Mishnaevsky Jr (MPA, Univ of Stuttgart, Pfaffenwaldring 32, Stuttgart, D-70569, Germany).

The purpose of this monograph is to study the evolutionary variational inequality approach to a moving free boundary problem with respect to both the mathematical analysis and to the numerical treatment. The author is successful in his aim to provide a detailed systematic treatment of the conceptual, mathematical, and numerical aspects of the approach, including the problems of existence, uniqueness, regularity, and time evolution of solutions, the numerical (finite element and finite volume) approximations of the problem, and practical applications of the approach.

The book is primarily addressed to applied mathematicians working in the field of nonlinear differential equations as well as to scientists from the application areas of engineering and physics. The layout is pleasant, the figures are original, and a well-done subject index is available. The lists of symbols and figures are included in the book as well.

The book is logically divided into three major parts: mathematical analysis (Chs 2 and 3), numerical treatment (Chs 4 and 5), and the applications of the approach to the numerical analysis of the injection and compression moulding process (Ch 6). The short history of the problem, its place in the general theory of nonlinear differential equations, as well as the general outline of the following chapters are given in the Introduction.

In Chapter 2, Derivation of the Evolutionary Variational Inequality Approach, an evolutionary variational inequality is derived as a fixed domain formulation for a general moving free boundary problem. The relation to another fixed domain formulation (weak formulation) is discussed, and the formulations are compared.

In Chapter 3, Properties of the Variational Inequality Solution, the analytical aspects of the evolutionary variational inequality formulation and properties of the solution of the problem are studied. The existence of a unique solution for the problem as a continuous mapping to the Sobolev space is proved. The regularity of the solution with respect to time as well as the spatial regularity is studied on the basis of the investigation of the associated penalty problem.

Chapter 4, Finite Volume Approximations for Elliptic Variational Inequalities, provides a systematic treatment of the finite volume method for the numerical solution of interior and boundary obstacle problems of elliptic type with mixed boundary conditions in two and three dimensions. A study of the finite volume approximation, which includes the solvability, stability, error analysis, and maximum principle, is performed. The comparison with the piecewise linear finite element approximation is carried out. The results of the numerical analysis of the finite volume schemes applied to elliptic variational inequalities, described in this chapter, present basic tools for the numerical analysis of the evolutionary inequalities, given in the next chapter.

In Chapter 5, Numerical Analysis of the Evolutionary Variational Inequalities, the finite element and finite volume schemes applied to the solution of the evolutionary variational inequalities with mixed boundary conditions are studied. The solvability and stability of discrete problems, maximum principle, error estimation, and different penalization methods of the approximations are considered.

In Chapter 6, Injection and Compression Moulding as Application Problems, the mathematical models of the injection and compression molding process based both on the generalized Hele-Shaw flow (which includes non-isothermal and possible non-Newtonian effects) and on three-dimensional non-Newtonian Navier-Stokes equations without the Hele-Shaw simplifications are presented. The numerical implementations of the models (finite volume and finite elements) are discussed in this chapter. The influence of geometrical and operating conditions of the injection-compression molding process on the flow front movement, pressure distribution, and the appearance of air traps in molding is studied numerically on the basis of the evolutionary variational inequality approach.

The big merit of the book is that it gives a systematic, detailed, and comprehensive treatment of the evolutionary variational inequality approach to a degenerate moving free boundary problem, including mathematical, numerical, and application aspects. This approach is very efficient and can be useful in solving many problems of considerable importance for practical applications. The book is well structured, and all the concepts, ideas and solutions are presented taking into account the history of the problem and the state-of-the-art of corresponding areas of applied mathematics and engineering.

In general, Variational Inequality Approach to Free Boundary Problems is highly recommended to libraries and to specialists working in the areas of nonlinear differential equations and their applications to engineering and physical problems.