| In this thesis, we consider the solution of linear systems arising from the finite-element discretization of incompressible and non miscible flows. The location of the interface between the fluids is a variable of the problem. Our approach is to use a Newton-Krylov scheme with adaptive stopping tolerance on the mixed formulation. We introduce a new class of iterative methods specifically tailored for the structure of the linear system encountered at each Newton iteration. Those methods draw from the projected conjugate gradient method, often used in quadratic programming, and implicitly work in the nullspace of the divergence operator. They only require matrix-vector products with the convection-diffusion matrix and a one-time symmetric indefinite factorization of a projection matrix. This results in important gains in terms of computational cost and memory requirements. We first validate our approach on test problems with one and two fluids. In a second stage, we present three applications that can be solved with our new class of iterative methods: non isothermal fluid flows, viscoelastic fluids and flows with moving boundaries using the fictitious domains method. This last method has the advantage that the discretization of the boundaries is independent of the finite-element mesh. The mesh is thus only generated once. |
