Preconditioned Nullspace Method for the Two-Dimensional Oseen Problem

Abstract

The Oseen problem, which arises in the simulation of the time-dependent Navier-Stokes equations for incompressible fluid flow, leads to indefinite, nonsymmetric and possibly ill-conditioned linear systems of equations. This paper presents a method for obtaining a reduced linear system from the original system, which is then solved by the preconditioned BiCGStab method. The system reduction is obtained through an efficient implicit representation of a basis of discretely divergence-free functions, also known as the nullspace method. We will show a close relationship between this preconditioned nullspace method and the BFBt-preconditioned pressure Schur complement problem arising in typical block preconditioners. We will use this relationship to analyze the spectrum of the preconditioned reduced system. We will also present numerical tests for the two-dimensional Oseen problem to illustrate the performance of this method, which is similar to the performance of the BFBt-preconditioned pressure Schur complement solver. In particular, it is robust w.r.t. the strength of the viscosity for simple constant wind but not for more complicated flows, and it shows only a slight dependence on the mesh size.