Topics
The central topics of this workshop are continuous and time dependent diffusion operators and their discrete counterparts on smooth as well as on highly irregular domains. The idea of the meeting is to bring together researchers who work on corresponding discretization problems from two complementing perspectives.
One aspect is the theory of Numerical Dynamics that has emerged over the last 20 years. Its principal goal is to analyze the effects of time discretizations on trajectories, invariant manifolds and more general dynamic invariants such as Lyapunov exponents. While the theory for ODE's is fairly well developed there are only partial results for parabolic systems on smooth domains which cover discretization in time as well as in space. The main theme here is to understand in detail the transition from infinite dimensional to finite dimensional dynamical systems. For example, it can happen that replacing smooth domains by regular lattices creates new phenomena that are of truly discrete nature.
The other perspective occurs in potential theory for highly irregular domains such as fractals. On their finite approximations (pre-fractals) one knows how to define diffusions. The main problem then is to construct limiting diffusion operators exhibiting the self-similarities of the underlying fractal. Classical and fractal operators can be treated in a joint framework but the latter show non-classical effects in terms of scaling and fluctuations. Moreover, one would like to look for corresponding effects in time-dependent nonlinear reaction diffusion systems. The existence and uniqueness of several equations has been established on finitely ramified fractals as for example the Sierpinski gasket.