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

TL;DR

This paper establishes optimal second-order pressure error estimates for the Crank-Nicolson scheme applied to the incompressible Navier-Stokes equations, confirming numerical evidence and extending theoretical understanding.

Contribution

It provides the first rigorous proof of second-order pressure error estimates at midpoints for the Crank-Nicolson discretization of Navier-Stokes equations, under both high and reduced regularity assumptions.

Findings

Pressure error is of the same order as velocity error at midpoints.

Optimal second-order pressure estimates are proven under high regularity.

Smoothing techniques extend results to less regular initial data.

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.