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This course is run by MIGSAA. It is open to everyone, including non-MIGSAA students, as an SMSTC supplementary module.
Homogenization theory I: multiscale modelling and analysis of physical and biological processes
The aim of homogenization theory is to determine the macroscopic behaviour of a system comprising microscopic heterogeneities, e.g. transport processes in a porous medium, signalling processes in a cell tissue, deformations of composite materials. This means that the mathematical model defined in a heterogeneous medium is replaced by equations posed in a homogeneous one, which provide a good approximation of properties of the original microscopic system.
In this course we will learn the main methods of homogenization theory, which are used to prove that solutions of microscopic problem, depending on a small parameter, converge to a solution of the corresponding macroscopic problem, as the small parameter (determined by the characteristic size of the microscopic structure) goes to zero.
Cioranescu D., Donato P. An introduction to homogenization, Oxford University Press, 1999
Bensoussan A., Lions J.L., Papanicolaou G. Asymptotic analysis for periodic structures, North-Holland, 1978
Hornung U. Homogenization and Porous Media, Springer, 1997
Braides A. Gamma-Convergence for beginners, Oxford University Press, 2002
Braides A. A handbook of Gamma-convergence, online
Pavliotis G.A., Stuart A. Multiscale methods: averaging and homogenization, Springer, 2008
Allaire G. Shape optimization by the homogenization method, Springer, 2002.
Assessment: (20 hrs, 15 credits)
7 credits for active participation
8 credits for solving tutorial/homework questions (under guidance of instructor)
Recommended prerequisites: some knowledge of Sobolev spaces and PDEs.