Numerical Solution of Nonlinear Elliptic Problems Via Preconditioning Operators [Repost]

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.