In the early eighties, when Jacques-Louis Lions and I considered the idea of a Handbook of Numerical Analysis, we carefully laid out specific objectives, outlined in the following excerpts from the "General Preface" which has appeared at the beginning of each of the volumes published so far:
During the past decades, giant needs for ever more sophisticated mathematical models and increasingly complex and extensive computer simulations have arisen, hi this fashion, two mdissociable activities, mathematical modeling and computer simulation, have gained a major status m all aspects of science, technology and industry.
hi order that these two sciences be established on the safest possible grounds, mathematical rigor is indispensable. For this reason, two companion sciences. Numerical Analysis and Scientific Software, have emerged as essential steps for validating the mathematical models and the computer simulations that are based on them.
Numerical Analysis is here understood as the part of Mathematics that describes and analyzes all the numerical schemes that are used on computers; its objective consists in obtaining a clear, precise, and faithful, representation of all the ''information" contained in a mathematical model; as such, it is the natural extension of more classical tools, such as analytic solutions, special transforms, functional analysis, as well as stability and asymptotic analysis.
The various volumes comprising the Handbook of Numerical Analysis will thoroughly cover all the major aspects of Numerical Analysis, by presenting accessible and in-depth surveys, which include the most recent trends.
More precisely, the Handbook will cover the basic methods of Numerical Analysis, gathered under the following general headings:
- Solution of Equations in R".
- Finite Difference Methods.
- Finite Element Methods,
- Techniques of Scientific Computing.
It will also cover the numerical solution of actual problems of contemporary interest in Applied Mathematics, gathered under the following general headmss:
- Numerical Methods for Fluids,
- Numerical Methods for Solids.
hi retrospect, it can be safely asserted that Volumes I to IX. which were edited by both of us. fulfilled most of these objectives, thanks to the eminence of the authors and the quality of their contributioiis.
After Jacques-Louis Lions* tragic loss in 2001. it became clear that Volume IX would be the last one of the type published so far. i.e., edited by both of us and devoted to some of the general headings defined above. It was then decided, in consultation with the publisher, that each future volume will instead be devoted to a single "specific application" and called for tins reason a "Special Volume". "Specific applications" will include Mathematical Fmance, Meteorology, Celestial Mechanics, Computational Chemistry. Living Systems. Electromagnetism. Computational Mathematics etc. It is worth noting that the inclusion of such "specific applications" in the Handbook of Numerical Analysis was part of our initial project.
To ensure the continuity of this enterprise, I will continue to act as Editor of each Special Volume, whose conception will be jointly coordinated and supervised by a Guest Editor.