  We show that, using the Crouzeix-Raviart scheme, a cheap algebraic
transformation, applied to the coupled velocity-pressure linear systems issued
from the transient or steady Stokes or Navier-Stokes problems, leads to a
linear system only involving as many auxiliary variables as the velocity
components. This linear system, which is symmetric positive definite in the
case of the transient Stokes problem and symmetric invertible in the case of
the steady Stokes problem, with the same stencil as that of the velocity
matrix, provides the exact solution of the initial coupled linear system.
Numerical results show the increase of performance when applying direct or
iterative solvers to the resolution of these linear systems.
