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I. Farago, J. Karatson "Numerical Solution of Nonlinear Elliptic Problems Via Preconditioning Operators: Theory and Applications"

For researchers in numerical analysis and other readers interested in any step in the process from the mathematical modelling to the computer realization of real-life problems, Faragó and Karátson develop the framework of preconditioning operators for discretized nonlinear elliptic problems, which means that the proposed preconditioning matrices are the discretization of suitable linear elliptic operators. They assume readers to be familiar with the basic level of the finite element method and functional analysis.

In the present book, the authors successfully take up the challenge to develop in a systematic way a functional analytic framework for construction of preconditioners.

Thereby they give a proper general background on how to understand and improve preconditioners for elliptic boundary value problems, both in continuous function space and in the finite dimensional spaces arising after proper discretization of the differential operators. Their approach provides a natural way to prove spectral equivalence between the pairs of precondtitioning and given operators, i.e. for which the spectral condition number is bounded uniformly with respect to the discretization (mesh) parameter. Doing so they also can prove mesh independent convergence of the methods, such as the Newton method, in a general way.

The monograph gives a connection between Sobolev space theory and iterative solution methods required for the actual numerical solution of nonlinear boundary value problems. In particular, a general presentation of how to understand the behaviour of preconditioners and to improve them is given. Both nonlinear operators and nonlinear boundary conditions are analyzed. Furthermore both theory and applications are presented.

The book consists of three parts: Part I gives motivation for the subject in different respects, Part II contains the required theoretical background, and the main ideas of the book are presented in Part III. The chapters of the book are devoted to the following topics. In Part I the model problems of Chapter 1 both show the wide scope and represent typical kinds of nonlinear elliptic equations. This is followed by a summary on linear equations, involving algebraic systems and elliptic problems in Chapters 2 and Chapter 3, respectively, with focus on preconditioning. Chapter 2 includes the discussion of computer realization. In Chapter 3 the Sobolev space background also plays a central role, and the motivating results on preconditioning operators are summarized here. Chapter 4 contains a brief summary on iterations for nonlinear algebraic systems, and, as a conclusion, demonstrates the importance of finding good preconditioners. Part II starts with Chapter 5 which summarizes the underlying Hilbert space theory involving monotone and potential operators. The framework of variable preconditioning is also developed here. Chapter 6 deals with nonlinear elliptic problems from the aspect of solvability: the discussion of existence and uniqueness, based on the corresponding results of the previous chapter, both helps the understanding of the nature of these equations and gives a starting point for the study of convergent iterations. Part III consists of Chapters 7-10: Chapter 7 summarizes iterative methods in Sobolev space: a detailed presentation is given for simple and Newton-like iterations, and some other iterative methods are sketched. The focus is on preconditioning, which is even used to give a common framework for these two types of method via the Sobolev gradient idea. Chapter 8 first summarizes briefly some general properties of the derived preconditioning methods for the discretized problems, then a list of various preconditioners is presented using the preconditioning operator background. The latter makes this chapter a central part of the book. The given preconditioners basically rely on spectral equivalence. Chapter 9 gives algorithmic realization and convergence results for the numerical methods based on preconditioning operators. Special attention is devoted to FEM realization, which is the most natural choice owing to the Sobolev space background. Finally, in Chapter 10, the authors return to the model problems of Chapter 1 and, as an illustration, they give algorithms for some of them using the previous results. A brief appendix in Chapter 11 gives some background information.

This monograph can be expected to be important both for researchers involved in the analysis and development of solution methods for nonlinear boundary value problems as well as for practitioners, who will use the methods for their particular applications.

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