Two-and multilevel truncated Newton finite element discretizations are presently a very promising approach for approximating the (nonlinear) Navier-Stokes equations describing the equilibrium flow of a viscous, incompressible fluid. Their combination with mesh adaptivity is considered in this article. Specifically, locally calculable c1 posteriori error estimators are derived, with full mathematical support, for the basic two-level discretization of the Navier-Stokes equations. 0 1996