VELOCITYCORRECTION PROJECTION METHODS FOR
INCOMPRESSIBLE FLOWS
∗
J. L. GUERMOND
†
AND
JIE SHEN
‡
SIAM J. N
UMER.
A
NAL
.
c
°
2003 Society for Industrial and Applied Mathematics
Vol. 41, No. 1, pp. 112–134
Abstract.
We introduce and study a new class of projection methods—namely, the velocity
correction methods in standard form and in rotational form—for solving the unsteady incompressible
Navier–Stokes equations. We show that the rotational form provides improved error estimates in
terms of the
H
1
norm for the velocity and of the
L
2
norm for the pressure. We also show that
the class of fractionalstep methods introduced in [S. A. Orsag, M. Israeli, and M. Deville,
J. Sci.
Comput
., 1 (1986), pp. 75–111] and [K. E. Karniadakis, M. Israeli, and S. A. Orsag,
J. Comput.
Phys
., 97 (1991), pp. 414–443] can be interpreted as the rotational form of our velocitycorrection
methods. Thus, to the best of our knowledge, our results provide the Frst rigorous proof of stability
and convergence of the methods in those papers. We also emphasize that, contrary to those of the
above groups, our formulations are set in the standard
L
2
setting, and consequently they can be
easily implemented by means of any variational approximation techniques, in particular the Fnite
element methods.
Key words.
Navier–Stokes equations, projection methods, fractionalstep methods, incom
pressibility, Fnite elements, spectral approximations
AMS subject classifcations.
65M12, 35Q30, 35J05, 76D05
PII.
S0036142901395400
1. Introduction.
We consider in this paper the time discretization of the un
steady incompressible Navier–Stokes equations in primitive variables. For a given
body force
f
(
t
) and an initial solenoidal vector ±eld
v
0
,welookfor
u
and
p
such that
∂
t
u
−
ν
∇
2
u
+
u
·∇
u
+
∇
p
=
f
in Ω
×
[0
,T
]
,
∇·
u
= 0
in Ω
×
[0
]
,
u

Γ
=0
,
u

t
=0
=
v
0
in Ω
.
(1.1)
The boundary condition on the velocity is set to zero for the sake of simplicity. The
ﬂuid domain Ω is open and bounded in
R
d
(
d
= 2 or 3 in practical situations). The
domain boundary Γ is assumed to be smooth; e.g., Γ is Lipschitzian and Ω is locally
on one side of its boundary.
The goal of this paper is to present a new class of fractionalstep projection meth
ods. The original projection method was introduced by Chorin [3] and Temam [15]
in the late 60s. An important class of projection methods is the socalled pressure
correction methods introduced in [5, 8]. These schemes consist of two substeps per
time step: the pressure is treated explicitly in the ±rst substep and corrected in the sec
ond substep by projecting the intermediate velocity onto the space of divergencefree
∗
Received by the editors September 20, 2001; accepted for publication (in revised form) August
12, 2002; published electronically March 13, 2003.
http://www.siam.org/journals/sinum/411/39540.html
†
LIMSI (CNRSUPR 3152), BP 133, 91403 Orsay, ±rance ([email protected]). Part of this
work was completed while this author was visiting Texas Institute of Computational and Applied
Mathematics, Austin, TX, during the period of August, 2001, to July, 2002, under a TICAM Visiting
±aculty ±ellowship.
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 Spring '09
 Applied Mathematics, Finite Element Method, Ω, Projection methods, J. L. GUERMOND, JIE SHEN

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