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ANALYSIS OF CONTINUOUS H –1
LEAST-SQUARES METHODS FOR THE STEADY
NAVIER-STOKES SYSTEM
Jerôme Lemoine, Arnaud Munch, Pablo Pedregal
To cite this version:
Jerôme Lemoine, Arnaud Munch, Pablo Pedregal. ANALYSIS OF CONTINUOUS H –1 LEAST-
SQUARES METHODS FOR THE STEADY NAVIER-STOKES SYSTEM. Applied Mathematics
and Optimization, Springer Verlag (Germany), In press, 83 (1), pp.461-488. �hal-01774607v2�
ANALYSIS OF CONTINUOUS H
−1
-LEAST-SQUARES METHODS FOR THE
STEADY NAVIER-STOKES SYSTEM
J
´
EROME LEMOINE, ARNAUD M
¨
UNCH, AND PABLO PEDREGAL
Abstract. We analyze two H
−1
-least-squares methods for the steady Navier-Stokes system
of incompressible viscous fluids. Precisely, we show the convergence of minimizing sequences
for the least-squares functional toward solutions. Numerical experiments support our analysis.
Key Words. Steady Navier-Stokes system, Least-squares approach, Gradient method.
1. Introduction
Let Ω ⊂ R
N
, N = 2 or N = 3 be a bounded connected open set whose boundary ∂Ω is
Lipschitz. We denote by n = n(x) the outward unit normal to Ω at any point x ∈ ∂Ω. Bold
letters and symbols denote vector-valued functions and spaces; for instance L
2
(Ω) is the Hilbert
space of the functions v = (v
1
, . . . , v
N
) with v
i
∈ L
2
(Ω) for all i.
This work is concerned with the (numerical) approximation for the steady Navier-Stokes
system
(1.1)
(
− ν∆y + (y · ∇)y + ∇π = f , ∇ · y = 0 in Ω,
y = 0 on ∂Ω,
which describes a viscous incompressible fluid flow in the bounded domain Ω, submitted to the
external force f. Our strategy is to use a least-squares approach, much in the spirit of [2], [3],
[7], but in a systematic way as in [15], having in mind some applications to control problem as
described in [12, 13] for the Stokes system. From a purely analytical perspective, the following
is a well-known existence theorem.
Theorem 1.1 ([17]). For any f ∈ H
−1
(Ω), there exists at least (y, π) ∈ H
1
0
(Ω) ×L
2
0
(Ω) solution
of (1.1). Moreover, if ν
−2
kfk
H
−1
(Ω)
is small enough, then the couple (y, π) is unique.
We put L
2
0
(Ω) for the space of functions in L
2
(Ω) with zero mean. Assuming f ∈ H
−1
(Ω),
the solution of (1.1) may be investigated by considering the following functional
(y, π) → k − ν∆y + (y ·∇)y + ∇π −fk
2
H
−1
(Ω)
+ k∇ · yk
2
L
2
(Ω)
over the space H
1
0
(Ω)×L
2
0
(Ω). The minimization of this functional leads to a so-called continuous
H
−1
-least-squares type method, following the terminology of [5] and later use in [3].
Least-squares methods to solve non linear boundary value problems have been the subject
of intensive developments in the last decades, as they present several advantages, notably on
computational and stability viewpoints. We refer to the book [2]. The main reason of this work
is to show that, under the assumption of Theorem 1.1, minimizing sequences for this so-called
Date: 23-04-2018.
Laboratoire de Math´ematiques, Universit´e Clermont Auvergne, UMR CNRS 6620, Campus des C´ezeaux,
63177 Aubi`ere, France. e-mail: jerome.lemoine@uca.fr.
Laboratoire de Math´ematiques, Universit´e Clermont Auvergne, UMR CNRS 6620, Campus des C´ezeaux,
63177 Aubi`ere, France. e-mail: arnaud.munch@uca.fr.
INEI. Universidad de Castilla La Mancha. Campus de Ciudad Real (Spain). e-mail: pablo.pedregal@uclm.es.
Research supported by MTM2017-83740-P, by PEII-2014-010-P of the Conserjer´ıa de Cultura (JCCM), and by
grant GI20152919 of UCLM.
1
2 J
´
EROME LEMOINE, ARNAUD M
¨
UNCH, AND PABLO PEDREGAL
error functional do actually converge strongly to the solution of (1.1). We first consider in Section
2 the case where minimizing sequences live in V × L
2
0
(Ω) with V defined below by (2.1), a set
of divergence free fields. Then, in Section 3, we discuss the general case where the field y is
not a priori assumed to be divergence free. In the two cases, we provide a sufficient condition
of the convergence of the values of the error functional E in terms of the convergence of the
values of its derivative. Then, in Section 4, we show that gradient methods, under hypothesis of
Theorem 1.1, produce converging sequences to the unique solution of (1.1). Section 5 describes
the conjugate gradient algorithm associated to the error functional E while section 6 discusses
numerically the celebrated exemple of the 2D channel with a backward facing step.
2. Steady case under the div-free constraint
As indicated in the Introduction, in order to solve the boundary value problem (1.1), we
use a least-squares type approach. If we insist in keeping explicitly the div-free constraint for
fields, then the pressure field does not play a specific role, so that we can eliminate it from the
formulation.
We consider the Hilbert space
(2.1) V := H
1
0,div
(Ω) = {y ∈ H
1
0
(Ω) : ∇ · y = 0 in Ω},
endowed with the norm of the gradient k∇yk
L
2
(Ω)
and define the functional E : V → R
+
by
putting
(2.2) E(y) =
1
2
Z
Ω
|∇v|
2
dx
where the corrector v is the unique minimizer in V for the variational problem
Minimize in v ∈ V :
Z
Ω
1
2
|∇v|
2
+ (ν∇y −y ⊗y) · ∇v
dx − hf, vi
H
−1
(Ω),H
1
0
(Ω)
.
Notice that the differential constrained imposed on V, leads to the existence of a multiplier π,
the pressure, such that
(2.3)
(
− ∆v + ∇π + (−ν∆y + div(y ⊗ y) − f ) = 0, ∇· v = 0 in Ω,
v = 0 on ∂Ω,
with div(y ⊗ y) = (y.∇)y + y∇ · y = (y.∇)y since y is here divergence free.
The existence of a unique solution for this quadratic, constrained problem in v is standard.
Optimality conditions lead directly to the weak form of (2.3).
The functional E is a so-called error functional which measures, through the corrector variable
v, the deviation of the pair (y, π) from being a solution of the underlying system (1.1). We then
consider the following extremal problem
(2.4) inf
y∈V
E(y).
Note that the error functional E is differentiable as a functional defined on the Hilbert space V,
because the operator y 7→ v taking each y ∈ V into its associated corrector v, as stated above,
is a differentiable operation. Indeed, E
0
(y) can always be identified with an element of V itself.
This computation is performed below in the proof of Proposition 2.4.
From Theorem 1.1, the infimum is equal to zero, and is reached by a solution of (1.1). Beyond
this statement, we would like to argue why we believe it is a good idea to use a (minimization)
least-squares approach to approximate the solution of (1.1) by solving (2.4).
The least-squares problem (2.2)-(2.3)-(2.4) has been actually introduced in [3], section 4.2 (and
numerically discussed in [4]), in order to solve one step in time of an implicit Euler scheme, time
approximation for the unsteady Navier-Stokes system. However, the analysis of the convergence
of the method was not given there. In that direction, our main theorem is the following.
H
−1
-LEAST-SQUARES METHODS FOR NAVIER-STOKES 3
Theorem 2.1. There is a positive constant C
2
, such that if {y
j
}
j>0
is a sequence in
B := {y ∈ V : ky k
H
1
0
(Ω)
≤ C
2
}
with E
0
(y
j
) → 0 as j → ∞, then the whole sequence y
j
converges strongly as j → ∞ in H
1
0
(Ω)
to the unique solution y
0
of (1.1) guaranteed by Theorem 1.1, if ν
−2
kfk
H
−1
(Ω)
is small enough.
We divide the proof in two main steps.
(1) First, we use a typical a priori bound to show that leading the error functional E down
to zero implies strong convergence to the unique solution of (1.1).
(2) Next, we will show that taking the derivative E
0
to zero actually suffices to take E to
zero.
Proposition 2.2. Assume that ν
−2
kfk
H
−1
(Ω)
is small enough and let y
0
∈ V be the unique
solution of (1.1), mentioned in Theorem 1.1. If dimension n ≤ 4, there is a positive constant
C ≡ C(ν, n, kf k
H
−1
(Ω)
) such that for every y ∈ V, we have
ky − y
0
k
2
H
1
0
(Ω)
≤ CE(y).
This proposition very clearly establishes that as we take down the error E to zero, we get closer,
in the strong norm, to the solution of the problem, and so, it justifies why a promising strategy
to find good approximations of the solution of problem (1.1) is to look for global minimizers of
(2.4).
The proof of this proposition basically amounts to a typical a priori estimate which is essen-
tially the same that the proof of uniqueness in page 112 in [17]. Recall the following basic fact
which will be utilized several times in what follows.
Lemma 2.3. For all u, v ∈ V, we have
Z
Ω
(v ⊗ v : ∇u + u ⊗ v : ∇v) dx =
Z
Ω
u ⊗ v : ∇u dx = 0.
Proof. (of Proposition 2.2) Bear in mind (2.3)
−∆v + (−ν∆y + div(y ⊗ y) − f ) + ∇π = 0, in Ω,
in addition to the boundary condition v = 0 on ∂Ω. Since (y
0
, π
0
) is the unique solution of
problem (1.1), we can write
−∆v + (−ν∆y + div(y ⊗ y)) − (−ν∆y
0
+ div(y
0
⊗ y
0
)) + ∇(π −π
0
) = 0 in Ω.
If we put Y ≡ y
0
− y, reorganizing terms it is also true that
(2.5) ∆v − ν∆Y + ∇(π
0
− π) − div(y ⊗ y −y
0
⊗ y
0
) = 0 in Ω.
In a linear situation, we would immediately achieve the bound in the statement. But the pres-
ence of the non-linear (quadratic) term that is so essential to Navier-Stokes requires a bit more
analysis.
Let us focus on the difference
Div ≡ div(y ⊗ y −y
0
⊗ y
0
).
We first put
−Div = div(y
0
⊗ Y) + div(Y ⊗ y).
If we take this identity back to (2.5), multiply by Y, and integrate by parts, taking into account
boundary conditions and bearing in mind that divY = 0, we are led to
−
Z
Ω
∇v · ∇Y dx + ν
Z
Ω
|∇Y|
2
dx −
Z
Ω
y
0
⊗ Y : ∇Y dx −
Z
Ω
Y ⊗y : ∇Y dx = 0.
4 J
´
EROME LEMOINE, ARNAUD M
¨
UNCH, AND PABLO PEDREGAL
By the second part of Lemma 2.3, the last term above vanishes, while for the third term, the
first part of the same lemma leads to
Z
Ω
y
0
⊗ Y : ∇Y dx = −
Z
Ω
Y ⊗Y : ∇y
0
dx ≤ C
2
(n)kYk
2
H
1
0
(Ω)
ky
0
k
H
1
0
(Ω)
,
where C(n) is the constant of the Sobolev embedding of H
1
(Ω) into L
4
(Ω) provided n ≤ 4 (see
Lemmas 1.1 and 1.2 in pages 108 and 109, respectively, of [17]). By putting together all of this
information, we have
ν
Z
Ω
|∇Y|
2
dx =
Z
Ω
y
0
⊗ Y : ∇Y dx +
Z
Ω
∇v · ∇Y dx,
and then
(2.6) νkYk
2
H
1
0
(Ω)
≤ C
2
(n)kYk
2
H
1
0
(Ω)
ky
0
k
H
1
0
(Ω)
+ kvk
H
1
0
(Ω)
kYk
H
1
0
(Ω)
.
But y
0
is precisely the solution of problem (1.1), and using it as a test function in its own system,
it is again immediate to check that
(2.7) ky
0
k
H
1
0
(Ω)
≤
C
ν
kfk
H
−1
(Ω)
for some C = C(Ω) > 0. Altogether, we find that
ν
1 − C C
2
(n) ν
−2
kfk
H
−1
(Ω)
kYk
H
1
0
(Ω)
≤ kvk
H
1
0
(Ω)
,
and under our hypotheses and the fact that 2E(y) = kvk
H
1
0
(Ω)
, this is the statement in the
proposition.
A practical way of taking a functional to its minimum is through some (clever) use of descent
directions, i.e. the use of its derivative. In doing so, the presence of local minima is always
something that may dramatically spoil the whole scheme. The unique structural property that
discards this possibility is the strict convexity of the functional. However, for non-linear equations
like (1.1), one cannot expect this property to hold for the functional E in (2.2). Nevertheless, we
insist in that for a descent strategy applied to our extremal problem (2.4), numerical procedures
cannot converge except to a global minimizer leading E down to zero. In doing so, thanks to
Proposition 3.4, we are establishing the strong convergence of approximations to the unique
solution of (1.1).
Indeed, we would like to show that the only critical points for E correspond to solutions of
(1.1). In such a case, the search for an element y solution of (1.1) is reduced to the minimization
of E, as indicated in the preceding paragraph. Precisely, we would like to prove the following
proposition, in the spirit of [16]. Our computations here follow closely those in [15].
Before proceeding to our second step for a full proof of Theorem 2.1, we stress that the error
functional E(y) is coercive in the sense
E(y) → ∞ if kyk
H
1
0
(Ω)
→ ∞.
Indeed, from (2.3), and using y itself as a test function, it is elementary to arrive, using
Z
Ω
y ⊗ y : ∇y dx = 0
according to Lemma 2.3, that
(2.8) νkyk
H
1
0
(Ω)
≤ C(n, Ω)
kfk
H
−1
(Ω)
+ kvk
H
1
(Ω)
for some constant C(n, Ω) > 0, leading to the coercivity of E.