When Herb Keller suggested, more than two years ago, that we update our
lectures held at the Tata Institute of Fundamental Research in 1977, and then
have it published in the collection Springer Series in Computational Physics,
we thought, at first, that it would be an easy task. Actually, we realized very
quickly that it would be more complicated than what it seemed at first glance,
for several reasons:
1. The first version of Numerical Methods for Nonlinear Variational
Problems was, in fact, part of a set of monographs on numerical
mathematics published, in a short span of time, by the Tata Institute of
Fundamental Research in its well-known series Lectures on Mathematics and
Physics; as might be expected, the first version systematically used the
material of the above monographs, this being particularly true for
Lectures on the Finite Element Method by P. G. Ciarlet and Lectures on
Optimization—Theory and Algorithms by J. Cea. This second version
had to be more self-contained. This necessity led to some minor additions
in Chapters I-IV of the original version, and to the introduction of a
chapter (namely, Chapter V of this book) on relaxation methods, since
these methods play an important role in various parts of this book. For
the same reasons we decided to add an appendix (Appendix I) introducing
linear variational problems and their approximation, since many of the
methods discussed in this book try to reduce the solution of a nonlinear
problem to a succession of linear ones (this is true for Newton's method,
but also for the augmented Lagrangian, preconditioned conjugate
gradient, alternating-direction methods, etc., discussed in several parts
of this book).