Development of adaptive methods for the efficient resolution of Navier-Stokes equations and hyperbolic systems with source terms

The purpose of the project is the mathematical and numerical survey of non linear complex systems derived from problems linked to natural management resources, in particular water management. We will put the emphasis on working out efficient numerical methods that summarize as follows: - The use of self-adaptive methods in finite elements or finite volumes methods , through working out a posteriori error estimations for nonlinear systems derived from conservation laws. - The use of these estimations for automatic adaptation of meshes in an optimal way, by setting "in a better way" the degrees of freedom and developing new strategies of refinement in two or three dimensions. - Working out optimal and efficient solvents, by developing preconditioned methods allowing an efficient resolution at low cost of the large systems obtained after discretization . The numerical schemes should be appropriate for use in engineering, geophysical as well as biomechanical problems. We want to deal with the following types of problems: the Euler and Navier-Stokes equations with source terms, the shallow-water equations with source terms, which are often used in meteorological or geophysical modeling, conservation laws with nonlinear diffusion, which describe, for example, motion of oils, composites, polymers or blood (hemodynamics). In principal, providing usable results is the purpose of a numerical computation for realistic complex flows, but it is eminently desirable to be able to estimate its validity. Hence we will seek to check the efficiency and reliability of computer solution,starting from the computed solution it self and eventually from an auxiliary simple calculation. The a posteriori error estimations constitute a major tool to achieve this task in every numerical simulation and in every adaptable methods. One hould keep in mind that beyond the knowledge of the accuracy of our calculations, we also seek to minimize the cost in order to obtain the precision. A further important component for three-dimensional problems is the development of an iterative solution technique together with appropriate preconditioning. Fundamental to achieving success in any of these aspects are the choice of formulation and the approximation them selves. It is proposed that all these aspects be considered in this project: there are strong relationships and obvious interactions. Stabilized formulations, precondioned iteration and a posteriori error estimation and adaptive gird refinement are all areas which are internationally at the cutting edge research.