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Piotr Gwiazda
Institute of Applied Mathematics and Mechanics
University of Warsaw
Banacha 2,
02-097 Warsaw
phone: +48 22 5544551
fax: +48 22 5544700

Diffusion-driven dynamic phase separation in binary systems over complex geometries: a numerical study
Marek Niezgódka (ICM University of Warsaw) and Willi Jäger (IWR, University of Heidelberg)
The project is devoted to numerical analysis of the processes described in Project 3.

Among the challenging mathematical questions to be addressed within the project it is to mention a numerical study of:

(1) the remote large-time effects of local in space activations at distant locations, including a characterization of the resulting steady-state equilibria,
(2) the large-time developments for specific representative classes of domain geometries and topologies.

Until recently none of the above questions was explored in detail, the only available results refer to rather simple special cases of the domain shapes and the driving mechanisms [1], numerical aspects studied in a number of papers by [2].

As the problems to be studied in particular arise from mathematical modelling of processes in a range of biosystems, an essential part of the project will address the questions of data assimilation into the model and predictive quantitative feasibility of the numerical approaches. The Ph.D. project will primarily address fundamental questions of the numerical analysis of such nonlinear systems, with special focus on the challenges due to:

(1) multiscale nature of the nonlinear models, both in space and, especially, in time (the latter imposing hard challenges on the extreme efficiency of the computational algorithms and their adaptive parallel implementations),
(2) the role of nonlinear dynamic couplings that play a decisive role for the structural stability of the dynamical systems,
(3) an impact of the nonlinearity of the media and that of the variable domain geometry and topology.

[1] M. Gokieli and N. Varchon, (2005) The proceedings of Third Polish-Japanese Days on Mathematical Approaches to Nonlinear Phenomena: Modelling, Analysis and Simulations, 5: 55-72.
[2] C. M Elliott and D. A. French, (1987) IMA Journal of Applied Mathematics, 38: 97-128.