We consider problems characterized by the existence and interaction of nonuniform materials (multi-physics, multi-domain), and modelled by partial differential equations (PDEs) involving discontinuous coefficients at the interfaces between different materials. These discontinuities result in discontinuities of the solution functions and/or their derivatives.

Examples of such problems are the spreading of cancer tumor (glioma) in the brain (white and gray matters of the brain), and the contamination of fresh ground water by saline waters, due to overpumping. Both problems are of strategic importance and their effective solution affects the decisions of governments and institutions, as well as the society.

For the effective solution of such problems we plan to:

- develop and study new numerical and analytical methods,
- adapt known numerical methods to the above described complex problems,
- implement the methods on modern computer architectures,
- test, evaluate and compare the methods and software, and validate them on the above two practical applications.

Large scale and complexity simulations, such as those needed for the solution of the above described complex problems, require the appropriate integration of physical concepts and laws, mathematical/numerical solution methods, discretization methods, algebraic system solvers, software and computer architectures, in a flexible and user-friendly manner.

The target of this project is the recognition and deeper understanding of the special characteristics of multi-physics, multi-domain problems, and the development of a platform for their realistic modelling and efficient solution, that will lead to:

- easier mathematical modelling,
- more effective and re-usable software,
- flexibility in choosing the most appropriate mathematical model, discretization method and algebraic solver for each part of the problem domain,
- effective exploitation of several modern/emerging computer architectures.