A preconditioner for linearized Navier-Stokes problem in exterior domains

Projet SIMPAFWe aim to approach the solution of the stationary incompressible Navier-Stokes equations in a three-dimensional exterior domain. Therefore, we cut the exterior domain by a sphere of radius $R$ and we impose some suitable approximate boundary conditions (ABC) to the truncation boundary of the computational domain: the minimal requirement of these conditions is to ensure the solvability of the truncated system and the decay of the truncation error if $R$ grows. We associate to the truncated problem a mesh made of homothetic layers, called exponential mesh, such that the number of degrees of freedom only grows logarithmically with $R$ and such that the optimal error estimate holds. In order to reduce the storage, we are interested in discretizations by equal-order velocity-pressure finite elements with additional stabilization terms. Therefore, the linearisation inside a quasi-Newton or fixed-point method leads to a generalized saddle-npoint problem, that may be solved by a Krylov method applied on the preconditioned complete system matrix. We introduce a bloc-triangular preconditioner such that the decay rate of the Krylov method is independent of the mesh size $h$ and we give an estimate for this rate in function of the truncation radius and of the Reynolds number. Some three-dimensional numerical results well confirm the theory and show the robustness of our method

Interior penalty finite element approximation of Navier-Stokes equations and application to free surface flows

In the present work, we investigate mathematical and numerical aspects of interior penalty finite element methods for free surface flows. We consider the incompressible Navier-Stokes equations with variable density and viscosity, combined with a front capturing model using the level set method. We formulate interior penalty finite element methods for both the Navier-Stokes equations and the level set advection equation. For the two-fluid Stokes equations, we propose and analyze an unfitted finite element scheme with interior penalty. Optimal a priori error estimates for the velocity and the pressure are proved in the energy norm. A preconditioning strategy with adaptive reuse of incomplete factorizations as preconditioners for Krylov subspace methods is introduced and applied for solving the linear systems. Different and complementary solutions for reducing the matrix assembly time and the memory consumption are proposed and tested, each of which is applicable in general in the context of either multiphase flow or interior penalty stabilization. As level set reinitialization method, we apply a combination of the interface local projection and a fast marching scheme. We provide for the latter a reformulation of the distance computation algorithm on unstructured simplicial meshes in any spatial dimension, allowing for both an efficient implementation and geometric insight. We present and discuss numerical solutions of reference problems for the one-fluid Navier-Stokes equations and for the level set advection problem. Solutions of benchmark problems in two and three dimensions involving one or two fluids are then approximated, and the results are compared to literature values. Finally, we describe software design techniques and abstractions for the efficient and general implementation of the applied methods