"Modeling and Numerical Methods 
for Multiscale Problems"

• Albert Cohen : "Adaptive multiscale approximation and applications to PDE's"

• Tom Hou : "Numerical Approximations to Multiscale Solutions in Partial     Differential Equations"
  

    Many problems of fundamental and practical importance have multiple scale solutions. The direct numerical solution of multiple scale problems is difficult to obtain even with modern supercomputers. The major difficulty of direct solutions is the scale of computation. The ratio between the largest scale and the smallest scale could be very large in each space dimension. From an engineering perspective, it is often sufficient to predict the macroscopic properties of the multiple-scale systems, such as the effective conductivity, elastic moduli, permeability, and eddy diffusivity. Therefore, it is desirable to develop a method that captures the small scale effect on the large scales, but does not require resolving all the small scale features. In my lectures, I will review some of the recent advances in developing systematic multiscale methods such as homogenization, multiscale analysis with many or continuous spectrum of scales, numerical samplings, multiscale finite element methods, variational multiscale methods, wavelets based homogenization,efficient numerical methods for nonlinear stochastic PDEs. Applications of these multiscale methods to transport through heterogeneous porous media and incompressible flows will be discussed. These lectures are not intended to be a detailed survey and the discussion is limited by both the taste and expertise of the author.

• John Osborn : "Generalized Finite Element Methods, Meshfree Methods, and Applications"

Topics will include :

1. Generalized Finite Element Methods; Basic theory of generalized Finite Element Methods; Survey of basic results.
2. Meshfree Methods; A unified approach to Meshless Methods; survey of basic results.
3. Selection of shape functions for the Generalized Finite Element Method and for Meshfree Methods. Principles for the selection of shape functions; optimal and nearly optimal shape functions; analysis based on the notion of n-width.
4. Applications 
• Applications to problems with rough coefficients
• The work of Babuska, Strouboulis, and Copps
• Other applications

• Felix Otto : "Multiscale Analysis in Micromagnetis"

    Domains and walls in ferromagnets are a paradigm for pattern formation in materials science. Domains are subregions of the sample $\Omega$ in which the magnetization $m$ is nearly constant; the transition layers separating domains are called walls. We will focus on the technologically important ferromagnetic films.
    Mathematically speaking, the micromagnetic model is a non--convex, non--local variational problem for the magnetization $m$. It is characterized by several length scales: On one end, there are the scales given by the sample geometry (film thickness and film diameter) and on the other end, there are the scales which depend only on the material. This set--up drives the pattern formation on intermediate scales.
    In the lectures, we shall try to explain specific experimental observations on walls and domains in ferromagnetic films starting from the micromagnetic model. First, we shall try to understand domain formation neglecting wall energy. Then, we'll take wall energy into account and will discover that there are different modes of walls. Finally, we'll have to take wall interaction into account. We will use a mixture of heuristic and rigorous arguments and shall present some numerical simulations.