Second Order Pressure Estimates for the Crank--Nicolson Discretization of the Incompressible Navier--Stokes Equations

Abstract

We provide optimal order pressure error estimates for the Crank--Nicolson semidiscretization of the incompressible Navier--Stokes equations. Second order estimates for the velocity error are long known; we prove that the pressure error is of the same order if considered at interval midpoints, confirming previous numerical evidence. For simplicity we first give a proof under high regularity assumptions that include nonlocal compatibility conditions for the initial data, then use smoothing techniques for a proof under reduced assumptions based on standard local conditions only.