Complexity in industrial problems some remarks

Abstract

When working on the numerical approximation of the solution of the state equations modeling industrial (complex) systems, one meets a number of problems which are rather universal.

We present here an introduction to some of these problems and to some of the methods which can be used, some of these methods being already quite old (we give then a short introduction to them). Others are on the contrary in development stage, and we then present them with more details.

We begin with stiff problems, i.e. models where one has several orders of magnitude of difference in the size of the coefficients. We introduce in such situations stiff asymptotic expansions. Questions of this type are met very often, for instance in electromagnetism.

In Section 3 we consider multi scales problems, where the “multi scales” refer to time scales and to space scales as well. In these cases, associated with multi physics with intrinsic mixing properties of the system under study, one can use expansions based on slow and fast variables, in time and, or in space. The main ideas are presented for rapidly rotating fluids and for flows in porous media, where the expansion leads to Darcy’s law.

Section 4 deals with the “universal” difficulty of the complexity of the geometry of the system under study. A classical method used to tackle this type of question is the DDM (domain decomposition method).

An introduction is presented in Section 4, an introduction to the decomposition of the energy space being presented in Section 5. The Section 4 and 5 are based on the virtual control method, introduced by O. PIRONNEAU and the A. in several conferences in 1998 and in the notes, referred to in Section 4. It contains as a particular case artificial or fictitions domains methods. Many applications and extensions are now under progress, in joint papers by O. PIRONNEAU and the A. The Decomposition of the Energy space has been introduced in a note of R. GLOWINSKI, O. PIRONNEAU and the A. (CRAS, 1999), developments being under development with T.W. PAN Others are in progress with J.P. PERIAUX

The method of virtual control applies in a very efficient manner to problems where there is an effective control (a real control !). Extensive developments are then needed. They will be presented elsewhere, a short introduction being given in the second note of O. PIRONNEAU and the A. referred to above.