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