Préconditionneurs hiérarchiques pour les problèmes de convection-diffusion

Numerical simulation is a strategic tool since it is indispensable for research and many industrial applications that require multiphase flow resolution. To carry out these numerical simulations, many challenges must be met in terms of mathematical formulation and numerical analysis, related to the couplings of thermal transfers by conduction, convection, and mechanical deformation of the porous media. Given these complexities, the accuracy of simulations for better predictions depends in part on the effectiveness of numerical algorithms. To achieve this accuracy and allow simulations in realistic times, it is crucial to have efficient, robust, and low CPU time-consuming linear solvers. Over the past ten years, new methods of resolution called H-matrix methods have emerged to reduce the algorithmic cost of elementary algebraic operations applied to dense matrices (for example: vector-matrix product, inversion, LU factorization). 
The purpose of this thesis will be the development of an effective preconditioner based on the H-matrix method (characterized by a hierarchical representation of matrices) coupled with matrix compression techniques, in order to efficiently exploit all the zeros of the sparse matrices derived from our applications while preserving a global hierarchical structure. The purpose of this thesis will also be to explore the relevance of the compression techniques initially introduced for purely elliptic problems, to the problems of convection-diffusion in a dominant convection regime where the linear systems produced are sparse.
 

Keywords : Préconditionement, matrices hiérarchiques, convection-diffusion, factorisation LU