by Steffen Härting
by Moritz Mercker
by Moritz Mercker
by Steffen Härting

Methods of Multiscale Analysis

Prof. Dr. Anna Marciniak-Czochra
Master or PhD Students in Mathematics/Scientific Computing/Physics
4 (Block sessions possible)
The seminar is devoted to mathematical methods of multiscale and asymptotic analysis with application to differential equations. The methods are motivated by problems of model derivation, upscaling and reduction. Real systems can be modeled at various levels of resolution. Effective behavior of the system depends on parameters and, ideally, by changing these parameters we should be able to move between microscopic, mesoscopic and macroscopic level depending on whether we are interested in the accuracy of the description and disregard the cost, or conversely. The goal of this seminar is to explore different methods for derivation of effective equations and reduction of model complexity. The program includes:
1. Singular perturbation methods:
(ii) Tikhonov Theorem with application to ODEs (quasi-stationary approximation):
(iii) Singular perturbation methods for reaction-diffusion equations (shadow systems, SLEP method):
(iii) Renormalization Group method applied to ODE and PDE problems of singular perturbation (quasi-stationary approximation; shadow limit, etc):
2. Methods of homogenization:
(i) Heuristic approach and two-scale expansions. Examples of diffusion and filtration in porous medium.:
(ii) Techniques to study convergence of multi-scale expansions: two-scale convergence, Tartar's energy method, Gamma-convergence, periodic unfolding. Applications to reactive flows in multiscale systems, including heterogeneous surface reactions.
Suggested Literature
[1] J.Banasiak and M.Lachowicz, Methods of small parameter in mathematical biology. Birkhäuser, Boston 2014.
[2] L. Y. Chen, N .Goldenfeld, Y. Oono, Renormalization group and singular perturbations : Multiple scales, boundary layers, and reductive perturbation theory, Phys. Rev. E 54, (1996), 376-394.
[3] U. Hornung, Homogenization and Porous Media, Springer 1997.
[4] A. Marciniak-Czochra and A.Mikelic, Multiscale Methods with Applications in Sciences, chapters from the book in preparation, 2014.
[5] I.Takagi, Mathematical Analysis of Biological Pattern Formation - Singular Perturbation Methods, Script, 2011.
[6] V.V. Zhikov, S.M. Kozlov, O. A. Oleinik, Homogenization of differential operators and integral functionals. Springer-Verlag, 1994.