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We provide optimal order pressure error estimates for the Crank-Nicolsonsemidiscretization of the incompressible Navier-Stokes equations. Second orderestimates for the velocity error are long known, we prove that the pressureerror is of the same order if considered at interval midpoints, confirmingprevious numerical evidence. For simplicity we first give a proof under highregularity assumptions that include nonlocal compatibility conditions for theinitial data, then use smoothing techniques for a proof under reducedassumptions based on standard local conditions only.

Second order pressure estimates for the Crank-Nicolson discretization of the incompressible Navier-Stokes Equations