MATHEMATICS OF COMPUTATION
Volume 73, Number 248, Pages 1719–1737
Article electronically published on December 19, 2003
ON THE ERROR ESTIMATES
FOR THE ROTATIONAL PRESSURE-CORRECTION
J. L. GUERMOND AND JIE SHEN
In this paper we study the rotational form of the pressure-correc-
tion method that was proposed by Timmermans, Minev, and Van De Vosse.
We show that the rotational form of the algorithm provides better accuracy in
terms of the
-norm of the velocity and of the
-norm of the pressure than
the standard form.
There are numerous way to discretize the unsteady incompressible Navier-Stokes
equations in time. Undoubtedly, the most popular one consists of using projection
methods. This class of techniques has been introduced by Chorin and Temam
[2, 3, 17]. They are time marching algorithms based on a fractional step technique
that may be viewed as a predictor-corrector strategy aiming at uncoupling viscous
diﬀusion and incompressibility eﬀects. The method proposed originally, although
simple, is not satisfactory since its convergence rate is irreducibly limited to
This limitation comes from the fact that the method is basically an artiﬁcial com-
pressibility technique as shown in  and . To cure these problems, numerous
modiﬁcations have been proposed, among which are pressure-correction methods
(see [6, 20]) and splitting techniques based on extrapolated pressure boundary con-
ditions (see [10, 9]).
Pressure-correction methods are widely used and have been extensively ana-
lyzed. These schemes are composed of two substeps by time step: the pressure is
made explicit in the ﬁrst substep and is corrected in the second one by projecting
the provisional velocity onto the space of incompressible vector ﬁelds. Rigorous
second-order error estimates for the velocity have been proved by E and Liu  and
Shen  in the semi-discrete case and by Guermond  and E and Liu  in the
fully discrete case. We refer also to  and  for diﬀerent proofs based on normal
mode analysis in the half plane and in a periodic channel, respectively.
It is well known that standard pressure-correction schemes still suﬀer from the
nonphysical pressure boundary condition which induces a numerical boundary layer
Received by the editor February 11, 2002 and, in revised form, March 2, 2003.
Mathematics Subject Classiﬁcation.
Primary 65M12, 35Q30, 76D05.
Key words and phrases.
Navier-Stokes equations, projection methods, fractional step methods,
incompressibility, ﬁnite elements, spectral approximations.
The work of the second author is partially supported by NFS grants DMS-0074283 and DMS-
0311915. Part of the work was completed while this author was a CNRS “Poste Rouge” visitor
2003 American Mathematical Society