Homogenization, Shape Optimization, Variational Techniques for Elliptic PDE

Dr. T. Muthukumar, Department of Mathematics & Statistics

Homogenization is a technique to understand the macroscopic behaviour of a composite medium through its microscopic properties. The technique is commonly used for PDE with highly oscillating coefficients. The idea is to replace a given heterogeneous medium by a fictitious homogeneous one (the 'homogenized' material) for numerical computations. The technique is also known as "Multi scale analysis". The known and unknown quantities in the study of physical or mechanical processes in a medium with micro structure depend on a small parameter ε. The study of the limit as ε→0, is the aim of the mathematical theory of homogenization. The notion of G-convergence, H-convergence, two-scale convergence, periodic unfolding method are some examples of the techniques employed for specific cases. The variational characterization of the technique for problems in calculus of variations is given by γ-convergence.