ILU Preconditioners for Nonsymmetric Saddle-Point Matrices with Application to the Incompressible Navier-Stokes Equations

Motivated by the numerical solution of the linearized incompressible Navier–Stokes equations, we study threshold incomplete LU factorizations for nonsymmetric saddle-point matrices. The resulting preconditioners are used to accelerate the convergence of a Krylov subspace method applied to finite element discretizations of fluid dynamics problems in three space dimensions. The paper presents and examines an extension for nonsymmetric matrices of the Tismenetsky–Kaporin incomplete factorization. It is shown that in numerically challenging cases of higher Reynolds number flows one benefits from using this two-parameter modification of a standard threshold ILU preconditioner. The performance of the ILU preconditioners is studied numerically for a wide range of flow and discretization parameters, and the efficiency of the approach is shown if threshold parameters are chosen suitably. The practical utility of the method is further demonstrated for the haemodynamic problem of simulating blood flow in a right coronary artery of a real patient.