Applications of Asymptotic Analysis

Abstract

The workshop Applications of Asymptotic Analysis, organised by Rupert Klein (Potsdam), Evariste Sanchez-Palencia (Paris), Jan Sokolowski (Nancy) and Barbara Wagner (Berlin) was held June 18th–June 24th, 2006. This meeting was well attended with 46 participants with a broad geographic representation. This workshop was a nice blend of young and senior researchers with various mathematical backgrounds.

The objective of this workshop was to present the new developments of multiple scale asymptotics as they are developed for problems in various fields of application. It brought together experts working in different areas of asymptotic analysis and application fields and initiated exchange of new ideas and discussions of parallel developments.

On the whole the atmosphere of this workshop was very cheerful and characterized by the mutual interest into each others expertise and approach.

The themes of the workshop included:

• Applications in shape optimization, where we discussed new tools like the internal and external topological derivatives and topological variations and their close relationship to basic research in asymptotic analysis of elliptic problems under singular perturbations of boundaries.
• Also, new asymptotic problems that arise in thin shell theory were discussed.
• A major field of application of asymptotic analysis in this workshop turned out to be numerical analysis. Examples within this workshop included new procedures for problems that involve multiple time and spatial scales, where adaptive tools alone are not robust. For boundary layer problems anisotropic finite element methods and the method of asymptotic decomposition of domains point to promising directions to tackle these complex problems. Furthermore, application of asymptotic analysis to finite difference schemes, where the grid spacing is the small parameter, could be shown to be very useful for studying consistency, stability and long-time behavior.
• Naturally, new emerging singular perturbation methods were another focus of discussions. These included methods such as Gevrey asymptotics to treat phenomena of boundary layer resonance and logarithmic switchback. Other new developments that require asymptotics beyond all orders analysis were demonstrated for problems that exhibit the Stokes phenomenon, such as for the finger selection problem for Hele–Shaw flow with kinetic undercooling.

Apart from these themes, various other lectures on applications of asymptotic analysis were given, ranging from applications in biology, solid mechanics to fluid mechanics, in particular thin liquid films.

Most of the lectures were given in the morning, with one hour overview lectures, followed by shorter half-hour talks. In the afternoon ample time was left for discussion sessions. Wherever possible the lectures and discussion sessions were grouped according to specific topics, which concerned the interplay of asymptotics and shape optimization (Monday), numerical analysis and asymptotics (Tuesday), mathematical theory and asymptotic methods (Wednesday, Thursday) and some new developments in homogenization theory (Friday).