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

Exact pressure elimination for the Crouzeix-Raviart scheme applied to the Stokes and Navier-Stokes problems