Recent research has shown that in some practically relevant situations like multi-physics flows [11] divergence-free mixed finite elements may have a significantly smaller discretization error than standard nondivergence-free mixed finite elements. In order to judge the overall performance of divergence-free mixed finite elements, we investigate linear solvers for the saddle point linear systems arising in ((Pk)d; Pdisc k-1 )) Scott-Vogelius finite element implementations of the incompressible Navier-Stokes equations. We investigate both direct and iterative solver methods. Due to discontinuous pressure elements in the case of Scott-Vogelius elements, considerably more solver strategies seem to deliver promising results than in the case of standard mixed finite elements like Taylor-Hood elements. For direct methods, we extend recent preliminary work using sparse banded solvers on the penalty method formulation to finer meshes, and discuss extensions. For iterative methods, we test augmented Lagrangian and H