  We propose and study numerically the implicit approximation in time of the
Navier-Stokes equations by a Galerkin-collocation method in time combined with
inf-sup stable finite element methods in space. The conceptual basis of the
Galerkin-collocation approach is the establishment of a direct connection
between the Galerkin method and the classical collocation methods, with the
perspective of achieving the accuracy of the former with reduced computational
costs in terms of less complex algebraic systems of the latter. Regularity of
higher order in time of the discrete solution is ensured further. As an
additional ingredient, we employ Nitsche's method to impose all boundary
conditions in weak form with the perspective that evolving domains become
feasible in the future. We carefully compare the performance poroperties of the
Galerkin-collocation approach with a standard continuous Galerkin-Petrov method
using piecewise linear polynomials in time, that is algebraically equivalent to
the popular Crank-Nicholson scheme. The condition number of the arising linear
systems after Newton linearization as well as the reliable approximation of the
drag and lift coefficient for laminar flow around a cylinder (DFG flow
benchmark with $Re=100$) are investigated. The superiority of the
Galerkin-collocation approach over the linear in time, continuous
Galerkin-Petrov method is demonstrated therein.
