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The aim of this project is the multi-scale analysis of porous media equations. We shall start from a mesoscopic description of two phase flow in porous media and consider a wide range of heterogeneities. We plan to study

1. problems with heterogeneities at a large scale such that subdomains with homogeneous properties can be introduced at the cost of interface conditions between the subdomains;
2. problems where the scales can not be separated but a proper multi-scale discretization technique can be exploited in order to obtain a macro-scale solution enriched by meso-scale information;
3. problems where the meso-scale solution is the target and multi-scale preconditioning techniques can serve as a basis for robust and efficient solution methods.

In case (1) with a clear scale separation, we want to adapt and apply homogenization techniques in order to justify and analyze effective models and efficient approximation schemes. We want to develop numerical methods that exploit the multi-scale character of the problem and we wish to accompany the practical implementation with a rigorous analysis. As an extension of the basic model, we will consider interface conditions between different porous materials; this can lead to either coupling conditions between one- and two-phase flow equations, to effective Dirichlet conditions or to outflow conditions.
Furthermore, in case (2) of more complex heterogeneities, when the scales can not be separated but it is still too expensive or impossible to solve a full meso-scale problem, we want to further develop and analyze multi-scale discretization techniques, such as the heterogeneous multiscale method, HMM, and the multiscale finite volume method, MSFV.
When, as in (3), the full meso-scale solution is the target, we plan to develop proper multi-scale preconditioners.

There are three main groups working on this project. Alongside

1. our group ,

these are the groups of

1. Prof. Dr. Oleg Iliev

and

1. Prof. Dr. Ben Schweizer .

While, in this project, our group is mainly concerned with Two-Phase flow in porous media with rapidly oscillating coefficients, the group of Prof. Dr. Ben Schweizer is manly working on Two-phase flow in porous media with internal interfaces and the group of Prof. Dr. Oleg Iliev is mainly concerned with General multi-scale media.

This project is partly supported by the Deutsche Forschungsgemeinschaft
Fraunhofer Institut für Techno- und Wirtschaftsmathematik		http://www.itwm.fraunhofer.de/
Technische Universität Dortmund

Please contact Prof. Dr. M. Ohlberger or F. Albrecht for further information.

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