Partial Differential Equations - Modelling and ... - ResearchGate

VIII
Preface
computers has been at the origin of an explosive development of numerical
mathematics, leading itself to applications of size and complexity unthinkable
a not so long time ago.
There has been simultaneity in the progress achieved on both the theory
and the numerics of partial differential equations, each feeding the other one:
indeed, methods for proving the existence of solutions have lead to numerical
methods for the actual computation of these solutions; on the other hand,
conjectures on mathematical properties of solutions have been verified first
computationally providing thus a justification for further analytical investigations.
Applications of partial differential equations are essentially everywhere
since to the areas mentioned above we have to add bio and health sciences,
finance, image processing. (It is worth mentioning that today the term partial
differential equations has to be taken in a broader sense than let say fifty years
ago in order to include partial differential inequalities, which are of fundamental
importance in, for example, the modeling of non-smooth phenomena.)
From the above comments, it is quite obvious that the “world of partial
differential equations” is a very large and complex one, and, therefore, quite
difficult to explore. Not surprisingly, the many aspects of partial differential
equations (theory, modeling and computation) have motivated a huge number
of publications (books, articles, conference proceedings, websites). Concerning
books, most of them are necessarily specialized (unless elementary) with topics
such as elliptic equations, parabolic equations, Navier–Stokes equations,
Maxwell equations, to name some of the most popular ones. We think thus
that there is a need for books on partial differential equations addressing at a
reasonably advanced level a variety of topics. From a practical point of view,
the diversity we mentioned above implies that such books have to be necessarily
multi-authors. We think that the present volume is an answer to such a
need since it contains the contributions of experts of international reputation
on a quite diverse selection of topics all partial differential equation related,
ranging from well-established ones in mechanics and physics to very recent
ones in micro-electronics and finance. In all these contributions the emphasis
has been on the modeling and computational aspects.
This volume is structured as follows: In Part I, discontinuous Galerkin and
mixed finite element methods are applied to a variety of linear and nonlinear
problems, including the Stokes problem from fluid mechanics and fully nonlinear
elliptic equations of the Monge-Ampère type. Part II is dedicated to the
numerical solution of linear and nonlinear hyperbolic problems. In Part III one
discusses the solution by domain decomposition methods of scattering problems
for wave models and of electronic structure related nonlinear variational
problems. Part IV is devoted to various issues concerning the modeling and
simulation of fluid mechanics phenomena involving free surfaces and moving
boundaries. The finite difference solution of a problem from spectral geometry
has also been included in this part. Part V is dedicated to inverse problems.
Finally, in Part VI one addresses the parabolic variational inequalities based
modeling and simulation of finance related processes.