  In this paper we consider the r\^ole that numerical computations -- in
particular Galerkin approximations -- can play in problems modelled by the 3d
Navier-Stokes equations, for which no rigorous proof of the existence of unique
solutions is currently available. We prove a robustness theorem for strong
solutions, from which we derive an {\it a posteriori} check that can be applied
to a numerical solution to guarantee the existence of a strong solution of the
corresponding exact problem.
  We then consider Galerkin approximations, and show that {\it if} a strong
solution exists the Galerkin approximations will converge to it; thus if one is
prepared to assume that the Navier-Stokes equations are regular one can justify
this particular numerical method rigorously.
  Combining these two results we show that if a strong solution of the exact
problem exists then this can be verified numerically using an algorithm that
can be guaranteed to terminate in a finite time.
  We thus introduce the possibility of rigorous computations of the solutions
of the 3d Navier-Stokes equations (despite the lack of rigorous existence and
uniqueness results), and demonstrate that numerical investigation can be used
to rule out the occurrence of possible singularities in particular examples.
