AbstractIt is generally accepted that mixed and penalty finite element methods can routinely solve the incompressible Navier–Stokes equations. This paper shows by means of simple examples that problems can arise even for the simpler Stokes equations. The causes of the problem fall in either of two categories: round‐off and ill conditioning, or a poor choice of pressure discretization. Nonsensical solutions can be obtained. Computation of the discrete divergence of the flow field is a simple and powerful tool to diagnose such conditions. In the first part of the paper several simple techniques for minimizing the effect of round‐off are reviewed. In the second part it is shown that, for coupled flow problems, care must be exercised in the choice of the pressure approximation. A unified treatment of various observations by different workers is presented. This should prove useful for general users of the finite element m