LU factorizations and ILU preconditioning for stabilized discretizations of incompressible Navier-Stokes equations

The paper studies numerical properties of LU and incomplete LU factorizations applied to the discrete linearized incompressible Navier-Stokes problem also known as the Oseen problem. A commonly used stabilized Petrov-Galerkin finite element method for the Oseen problem leads to the system of algebraic equations having a 2x2-block structure. While enforcing better stability of the finite element solution, the Petrov-Galerkin method perturbs the saddle-point structure of the matrix and may lead to less favorable algebraic properties of the system. The paper analyzes the stability of the LU factorization. This analysis quantifies the effect of the streamline upwind Petrov-Galerkin stabilization in terms of the perturbation made to a nonstabilized system. The further analysis shows how the perturbation depends on the particular finite element method, the choice of stabilization parameters, and flow problem parameters. The analysis of LU factorization and its stability helps to understand the properties of threshold ILU factorization preconditioners for the system. Numerical experiments for a model problem of blood flow in a coronary artery illustrate the performance of the threshold ILU factorization as a preconditioner. The dependence of the preconditioner properties on the stabilization parameters of the finite element method is also studied numerically.