# Functional Analysis

## Discrete PDEs

Winter term 2013/2014

Graduate seminar on PDEs S4B2

M. Helmers, M. Zaal

### Synopsis

A discrete PDE is, loosely speaking, a set of finite or countably many equations whose structure resembles a partial differential equation. A simple example is the discretisation of a well-known PDE such as the heat equation; other, more interesting, examples are models for coarsening processes or crystal lattices.

In the seminar we aim to study some of these discrete PDEs and their properties. The talks will we chosen from the following topics:

- lattice models for friction mechanisms on the atomistic scale [1],
- large particle systems [5],
- coarsening processes [3], droplets in thin films [4],
- approximation of ill-posed diffusion equations, discrete diffusion [2,3].

### Prerequisites

Knowledge about PDEs (classical or weak solutions) and ordinary differential equations.

### Organisation

The seminar meetings will be Wednesday 2--4pm, room N0.003.

### Literature sample

- S. Aubry and P. Y. le Daeron. The discrete Frenkel-Kontorova model and its extensions: I. Exact results for the ground-states.
*Physica D*8(3), 381–422, 1983. - G. Bellettini, C. Geldhauser and M. Novaga. Convergence of a semidiscrete scheme for a forward-backward parabolic equation.
*Adv. Differential Equations 18(5/6), 495*–522, 2013. - S. Esedoglu and J. B. Greer. Upper bounds on the coarsening rate of discrete, ill-posed nonlinear diffusion equations.
*Comm. Pure Appl. Math. 62(1), 57*–81, 2009. - K. B. Glasner et al. Ostwald ripening of droplets: The role of migration.
*European J. Appl. Math. 20(1), 1*–67, 2009. - C Kipnis, S. Olla and S. Varadhan. Hydrodynamics and large deviations for simple exclusion processes.
*Comm. Pure Appl. Math. 42(2),*115–137, 1989.