Lecture in summer term 2014:

V5E1 – Numerical Homogenization of Partial Differential Equations

Prof. Dr. Daniel Peterseim

Many physical processes in microheterogeneous media such as modern composite and functional materials are described by partial differential equations (PDEs) with rough coefficients or domains with a complex microstructure. Given the complexity of these processes, the key to reliably simulate some relevant classes of such processes involves the construction of appropriate macroscopic (homogenized or effective) models.

Numerical homogenization is a multiscale method for the derivation of meaningful macroscopic models. This lecture reviews the state-of-the-art techniques for numerical homogenization (analytically and experimentally). Recent results of numerical analysis strongly support the added value of numerical homogenization when compared with classical analytical homogenization techniques, i.e., its applicability, reliability, and accuracy in the absence of strong (unrealistic) assumptions such as periodicity and scale separation.

Since it is crucial for the feasibility and the efficiency of numerical homogenization that local problems on microscopic scales can be solved with minimal complexity, this lecture also addresses numerical techniques for this purpose, e.g., generalized and extended finite elements or network methods.