Fast solution of sparse linear systems with adaptive choice of preconditioners. (Résolution rapide de systèmes linéaires creux avec choix adaptatif de préconditionneurs)

This thesis analyzes the use of adaptive preconditioned Krylov methods in applications which can be modeled by partial differential equations. Preconditioning is generally essential for efficiently solving large sparse nonlinear systems of equations. However, the optimality of the available preconditioners is not guaranteed for all uses due to the changing nature of the linearized operator. This thesis explores some types of preconditioners and solve procedures that can adapt to the complexity of linear systems using information from a posteriori error estimates. First, we propose global and local adaptive strategies based on a posteriori error estimation and a hybrid block-jacobi and ILU(0) preconditioner. Second, the a posteriori error estimation is used to partition the matrix, and a Schur complement-based approach is used for the preconditioning of the block with a high error. Then, we introduce a variant of this latter approach which replaces the costly exact factorizations by low-rank approximations. We also define an adaptive preconditioner based on a posteriori error estimation that allows to control a local algebraic error norm. Finally, we prove the efficiency of our adaptive strategies on two-dimensional reservoir simulation examples for heterogeneous porous media.