An L2-stable approximation of the Navier–Stokes convection
operator for low-order non-conforming finite elements
G. Ansanay-Alex, F. Babik, J.C. Latché∗ , D. Vola
Institut de Radioprotection et de Sûreté Nucléaire (IRSN)
BP3 - 13115 Saint Paul-lez-Durance CEDEX, France.
email: [guillaume.ansanay-alex, fabrice.babik, jean-claude.latche, didier.vola]@irsn.fr
SUMMARY
We develop in this paper a discretization for the convection term in variable density
unstationary Navier-Stokes equations, which applies to low-order non-conforming finite
element approximations (the so-called Crouzeix-Raviart or Rannacher-Turek elements). This
discretization is built by a finite volume technique based on a dual mesh. It is shown to enjoy
an L2 stability property, which may be seen as a discrete counterpart of the kinetic energy
conservation identity. In addition, numerical experiments confirm the robustness and the accuracy
of this approximation; in particular, in L2 norm, second order space convergence for the velocity
and first order space convergence for the pressure are observed.
KEY WORDS: Stability, kinetic energy theorem, Rannacher-Turek finite element, CrouzeixRaviart finite element, low Mach number flows, incompressible flows
1. INTRODUCTION
We address in this paper the numerical solution of the variable density unstationary NavierStokes equations:
∂t ̺ + div(̺u) = 0
(1a)
∂t (̺u) + div(̺u ⊗ u) − divτ + ∇p = f
(1b)
where ∂t stands for the derivation operator with respect to time, u for the fluid velocity,
p for the pressure, f for a known volume forcing term and ̺ for the fluid density, which
is supposed to be a known positive quantity. The quantity ̺u ⊗ u is a tensor of Rd×d of
entries (̺u ⊗ u)i,j = ̺ui uj and the divergence of a tensor T is a vector quantity defined by
Pd
(divT )i = j=1 ∂j T i,j , where ∂j stands for the derivative with respect to the j th coordinate.
The tensor τ is the viscous part of the stress tensor, given by the following expression:
2
τ (u) = µ ∇u + ∇t u − (divu)I
3
(2)
where the viscosity µ is a positive constant real number and the gradient of the vector field
u is the tensor defined by (∇u)i,j = ∂j ui . The problem is posed over Ω, an open bounded
connected subset of Rd with d = 2 or d = 3. An initial condition for u must be provided,
and we suppose that the boundary conditions are of mixed type: the boundary ∂Ω of Ω is
split into ∂Ω = ∂ΩD ∪ ∂ΩN , the velocity is prescribed over the part ∂ΩD of positive (d − 1)dimensional measure while, on ∂ΩN , we suppose that the external forces are given:
∗ Correspondence
on ∂ΩD ,
u = uD
(3a)
on ∂ΩN ,
(τ (u) − pI) n = g
(3b)
to: J.C. Latché, IRSN ([email protected])
2
G. ANSANAY-ALEX ET AL.
where uD is a known velocity field, n is the outward normal vector to ∂Ω and g is a known
surface forcing term.
System (1) is a building block for many physical problems. For instance, supplementing
it by an energy balance and supposing ̺ given by the equation of state of perfect
gases evaluated at a fixed pressure yields the asymptotic governing equations for natural
convection flows in the low Mach number limit. Considering chemical reactions in the flow,
radiative transfer and turbulence phenomena, one obtains a model for the simulation of
fires, which is the aim of the ISIS free software, developed in France at the Institut de
Radioprotection et de Sûreté Nucléaire (IRSN).
The numerical scheme developed in this paper is based on low-order non-conforming finite
elements, namely the Crouzeix-Raviart [9] elements for simplicial meshes or the RannacherTurek element [32] for quadrilaterals and hexahedra. These pairs of finite element spaces
are the lowest order ones that fulfill the discrete version of the so-called Babuska–Brezzi
stability condition. They are thus well suited for a coupling with other balance equations
discretized by a finite volume technique for monotonicity reasons, as encountered in the
above-mentioned context.
For vanishing external forces f and g and prescribed velocity uD , if one supposes that the
unknown functions u, p and ̺ are regular and that the normal component of the velocity u
is prescribed to zero at the boundary, multiplying Equation (1b) by u and integrating over
Ω yields the following a priori estimate:
Z
Z
Z
1 d
2
p divu dx
(4)
τ (u) : ∇u dx =
̺|u| dx +
2 dt Ω
Ω
Ω
Since the second term at the left hand side is non-negative, this relation provides an estimate
for the velocity u, provided that an adequate treatment is possible for the right hand side,
which is the case, for instance, for incompressible or for compressible barotropic flows. The
key ingredient to establish (4) is the following identity:
Z
Z
1 d
∂t (̺u) + div(̺u ⊗ u) · u dx =
̺|u|2 dx
(5)
2 dt Ω
Ω
which holds provided that the mass balance equation (1a) is satisfied. Unfortunately, if
standard finite element techniques are used to discretize the terms ∂t (̺u) + div(̺u ⊗ u),
this identity does not hold at the discrete level with the chosen (non-conforming) finite
element spaces, essentially because some terms involving jumps across the interfaces of the
elements appear when performing the integration by parts necessary to its derivation. As a
consequence, some blow up of the solution was observed in our simulations of moderate-tohigh Reynolds number flows, especially with rather coarse meshes, as often used in real life
applications.
Some techniques to stabilize the discretization of convection-dominant flows suitable for
the Crouzeix-Raviart or Rannacher-Turek elements have been presented in the literature.
In [25], the authors introduce some upwinding a posteriori, i.e. after the assembling of
the discrete operators, by directly modifying the obtained algebraic system. An edge
stabilization method consisting in adding penalization terms involving the jumps of the
gradient of the velocity is proposed in [6] (see also [30] for a review); at the present time, this
technique seems to be analyzed only for Oseen equations (i.e. considering the convection
field as a given function), and has the drawback of enlarging the stencil of the discrete
convection operator. Another approach, which consists in discretizing the convection term
by a finite volume technique based on a dual mesh, has been proposed in the literature.
It first has been applied for the Crouzeix-Raviart element to a scalar linear [29] and
nonlinear [1, 10, 13] convection diffusion equation, with the goal to obtain a monotone
approximation of the convection operator and a general discretization of the diffusion (in
particular, suitable for anisotropic diffusion). Then this technique has been extended to
incompressible stationary flows, for the Crouzeix-Raviart element [34, 35], and further for
the Rannacher-Turek element [38, section 3.1.4, pp. 115–134]. A common point of these
AN L2 -STABLE APPROXIMATION OF THE NAVIER–STOKES CONVECTION OPERATOR
3
works is to introduce some upwinding, either for the sake of monotonicity [29, 1, 10, 13] or
stability [34, 35, 38].
In this paper, we follow the finite-volume route, and extend it to the case of variable
density flows. In addition, essentially because we address the unstationary case, and thus
we may rely on the stability induced by the conservation of the kinetic energy (4) for the
control of the solution, we develop a centered approximation.
This paper is structured as follows. We first describe the space discretization (Section 2).
Then the proposed approximation for the unstationary and convection terms is given and
its stability is proven in Section 3. A fractional step algorithm for the Problem (1) is then
built (Section 4) on this basis. We finally compute the solution to different analytical and
benchmark tests (Section 5) in order to assert the potentialities of the scheme.
The discretization presented here enters as an ingredient in already published (entropy
preserving) schemes for compressible flows [16, 18, 19]; compared to these works, we make
here a much more involved description, with a detailed proof of the essential stability result,
deal with general boundary conditions, extend the discussion to implementation aspects and
report an in-depth numerical study.
2. MESHES AND DISCRETIZATION SPACES
Let T be a decomposition of the domain Ω either in simplices (triangles in 2D or tetrahedra
in 3D) or, in the case where the shape of Ω allows it, in rectangles or rectangular
parallellepipeds. This decomposition T is assumed to be regular in the usual sense of the
finite element literature (eg. [8]). By E(K), we denote the set of the faces σ of the element
K ∈ T . The set of all faces of the mesh is denoted by E ; the set of faces included in the
boundary of Ω is denoted by Eext and the set of internal faces (i.e. E \ Eext ) is denoted by Eint .
The set Eext itself decomposes into the set of external faces included in ∂ΩD , denoted by ED ,
and the set of external faces included in ∂ΩN , denoted by EN . The face σ ∈ Eint separating
the cells K, L ∈ T is denoted by K|L and an external face σ ∈ Eext of the cell K ∈ T is
denoted by K|ext. By |K| and |σ|, we denote the measure, respectively, of an element K
and of a face σ . For σ ∈ E(K), nK,σ stands for the unit normal vector to σ outward to K .
The finite elements used in this paper are the Crouzeix-Raviart element for simplicial
meshes (see [9] for the seminal paper and, for instance, [11, p. 199–201] for a synthetic
presentation), and the so-called ”rotated bilinear element” introduced by Rannacher and
Turek for quadrilateral or hexahedric meshes [32].
For the discretization of the velocity components, the reference element for the CrouzeixRaviart is the unit d-simplex and the discrete functional space is the space P1 of affine
b for the rotated bilinear element is the unit d-cube
polynomials. The reference element K
b is Q̃1 (K)
b :
and the discrete functional space on K
b = span 1, (xi )i=1,...,d , (x2 − x2 )i=1,...,d−1
(6)
Q̃1 (K)
i
i+1
For both elements used here, the degrees of freedom are determined by the following set of
nodal functionals {mσ , σ ∈ E(K)} with:
Z
1
v dγ
(7)
mσ (v) =
|σ| σ
The mapping from the reference element to the actual discretization cell is the standard
affine mapping for the Crouzeix-Raviart element, and the standard Q1 mapping for the
Rannacher-Turek element. Finally, in both cases, the continuity of the average value mσ (v)
(i)
of a discrete function v across each face of the mesh is required, thus the discrete space VT
for the ith component of the velocity is defined as follows:
n
VT(i) =
v ∈ L2 (Ω) : v|K ∈ W (K), ∀K ∈ T ;
o
(8)
(i)
mσ (v|K ) = mσ (v|L ), ∀σ = K|L ∈ Eint ; mσ (v) = mσ (uD ), ∀σ ∈ ED
4
G. ANSANAY-ALEX ET AL.
where the space W (K) is thus the space of affine functions over K for the Crouzeix-Raviart
b through the Q1 mapping for the
element and the space of functions obtained from Q̃1 (K)
Rannacher-Turek element.
Remark 1
The above description of the Rannacher-Turek element is rectricted to its so-called
parametric version, which is sufficient here because we work, at least for the theory, with (2D
or 3D) rectangular meshes. Note however that this element is known to loose its accuracy
for ”far from parallelogram” cells, and that this phenomenon can be cured by using a nonparametric version of the element, i.e. a version where the basis functions which span the
discrete space are defined by an analog of (6) as a function of the coordinates in the actual
cell, without referring to a reference element [38, Section 3.1.2].
Since only the continuity of the integral over the faces of the mesh is imposed, the velocity
is discontinuous through each face; the discretization is thus non-conforming in H 1 (Ω)d .
From the definition (7), each velocity degree of freedom can be associated to a face of
an element. Hence, the velocity degrees of freedom may be indexed by the number of the
component and the associated face, and the set of velocity degrees of freedom reads:
{v σ,i , σ ∈ E \ ED , 1 ≤ i ≤ d}
P
We define v σ = di=1 v σ,i e(i) where e(i) is the ith vector of the canonical basis of Rd . We
(i)
denote by ϕσ the vector shape function associated to v σ,i , which, by the definition of the
Crouzeix-Raviart and Rannacher-Turek finite elements, reads:
(i)
ϕ(i)
σ = ϕσ e
where ϕσ is the scalar basis function.
For both the Crouzeix-Raviart and the Rannacher-Turek discretizations, the
approximation space for the pressure is the space of piecewise constant functions. In
addition, since we want to design an algorithm suitable for coupling with possible other
balance equations the unknown of which governs the value of the density, we suppose that
̺ is approximated by a discrete function, belonging to the same space than the pressure.
The degrees of freedom for the pressure thus are {pK , K ∈ T } and the density is defined
by {̺K , K ∈ T }.
In the definition of the scheme, we also need a dual mesh, which is defined as follows. For
any K ∈ T and any face σ ∈ E(K), let DK,σ be the cone of basis σ and of opposite vertex
the mass center of K . The volume DK,σ is referred to as the half-diamond mesh associated
to K and σ . We now define the diamond mesh Dσ associated to σ as follows: if σ ∈ Eint ,
σ = K|L, Dσ = DK,σ ∪ DL,σ ; if σ ∈ Eext , σ = K|ext, Dσ = DK,σ . The set of volumes (Dσ )σ∈E
provides the dual mesh T̄ of Ω. We denote by Ē(Dσ ) the set of faces of any Dσ ∈ T̄ , and
by ǫ = Dσ |Dσ′ the face separating two diamond meshes Dσ and Dσ′ (see Figure 1). Note
that, for a diamond cell Dσ adjacent to a boundary of the domain (i.e. a diamond cell Dσ
associated to a face σ ∈ Eext ) the external face is also a face of the primal mesh; we denote
such a face by Dσ |ext. The unit vector normal to ǫ ∈ Ē(Dσ ) outward to Dσ is denoted by
nσ,ǫ .
Since the velocity is prescribed on the faces of ED , the associated diamond cells (Dσ )σ∈ED
do not play any role in the definition of the scheme, whereas their internal faces play a
special role. We thus decide to remove these diamond cells from the dual mesh, and to
define the set ĒD as their internal faces (see Figure 2). On the part ∂ΩN of the boundary,
the faces of the primal and dual mesh are the same, and we define ĒN by E¯N = EN .
3. A L2 -STABLE CONVECTION OPERATOR
The goal of this section is to present the proposed discretization for the convection operator.
This presentation does not need to be linked to any specific time-marching algorithm, and is
AN L2 -STABLE APPROXIMATION OF THE NAVIER–STOKES CONVECTION OPERATOR
5
|σ |
ǫ=
Dσ |
Dσ ′
|L
σ=K
Dσ
L
|M
σ′ = K
K
Dσ ′
M
Figure 1. Notations for control volumes and diamond cells.
∪ǫ∈ĒN ǫ
∪σ∈EN σ
∂ΩN
∪ǫ∈ĒD ǫ
∪σ∈ED σ
∂ΩD
Figure 2. Notations for the boundaries of the computational domain, of the primal mesh and of
the dual mesh.
thus disconnected from the algorithm used for the solution of the complete system: we just
suppose that we know a density field, at the end and beginning of the time step (respectively
denoted by ̺ and ̺∗ ), the beginning-of-step velocity (u∗ ) and a convection field such that a
finite-volume-like mass balance on the primal mesh is satisfied (Equation (16) below), and
build on this basis the desired discrete convection operator. This form of the discrete mass
balance is of course the form which can be expected from the discretization of (1a) by the
considered finite elements.
This construction is made in two steps: first we suppose that a similar mass balance
relation holds for each diamond cell (section 3.1), then show how to get it from a discrete
velocity field satisfying the mass balance on the primal mesh (section 3.2). The general form
(i.e. without specifying the time marching algorithm) of the discrete momentum balance is
then given (section 3.3), and we conclude by some general remarks (section 3.4).
3.1. General form of the convection operator and stability analysis
Let us address in this section a sub-problem of the discretization of (1) which consists in
building a discrete convection operator ∂t (̺u) + div(uq) for a scalar unknown u satisfying
Dirichlet and Neumann boundary conditions on ∂ΩD and ∂ΩN respectively, supposing that
the momentum field q is known and such that the mass balance holds. More specifically, we
6
G. ANSANAY-ALEX ET AL.
assume that q and ̺ are such that:
(9)
∂t ̺ + divq = 0
and that this yields a discrete mass balance of the form:
|Dσ |
(̺σ − ̺∗σ ) +
δt
∀σ ∈ E \ ED ,
X
(10)
Fσ,ǫ = 0
ǫ∈Ē(Dσ )
In this relation, ̺σ and ̺∗σ stand for an approximation of the density in the diamond cell
Dσ at the current and previous time step respectively, and Fσ,ǫ is an approximation of the
outward mass flux through the face ǫ of Dσ , associated to the momentum q :
Z
Fσ,ǫ = q · nσ,ǫ dγ
ǫ
The families of real numbers (̺σ )σ∈E\ED and (̺∗σ )σ∈E\ED are supposed to be positive. The
mass flux is assumed to be outward on ∂ΩN , which means that, for any ǫ ∈ ĒN , Fσ,ǫ ≥ 0.
Let u and u∗ be two (scalar) Crouzeix-Raviart or Rannacher-Turek functions, the
prescribed value for u on ∂ΩD being uD . For the discretization of the convection operator
u 7→ ∂t (̺u) + div(uq) (which acts in (1b) on each component of the velocity), we propose a
finite volume operator C based on the dual mesh and defined as follows:
∀σ ∈ E \ ED ,
Cu
σ
=
1
1
(̺σ uσ − ̺∗σ u∗σ ) +
δt
|Dσ |
X
Fσ,ǫ uǫ
(11)
ǫ∈Ē(Dσ )
where uǫ is given by:
for ǫ ∈ Ēint ∪ E¯D , ǫ = Dσ |Dσ′ ,
for ǫ ∈ ĒN , ǫ ∈ Ē(Dσ ),
1
(uσ + uσ′ )
2
uǫ = uσ
uǫ =
(12)
For internal and Dirichlet faces, the choice thus corresponds to a centered one (remember
that the faces of ĒD indeed separate two diamond-cells, one of which is associated to a
primal Dirichlet face where the value of the unknown is given by the boundary data (8)).
For faces located on ∂ΩN , since the mass flux is outward, this choice is the upwind one, and
also seems to be the only reasonable one, since no natural value is readily available for the
external side of the boundary.
At the continuous level, supposing that (9) holds and that the functions appearing in the
following relations are regular, we have:
Z
Z
Z
1 ̺ ∂t u2 + q · ∇u2 dx
̺ ∂t u + q · ∇u u dx =
∂t (̺u) + div(uq) u dx =
2 Ω
Ω
Ω
Integrating by parts and using once again (9), we get:
Z
Z
Z
1 1
∂t (̺u) + div(uq) u dx =
̺ ∂t u2 − u2 divq dx +
u2 q · n dγ
2
2
Ω
∂Ω
ZΩ
Z
1
1
2
2
∂t (̺ u ) dx +
u q · n dγ
=
2 Ω
2 ∂Ω
and thus, finally:
Z
Z
Z
Z
1
1
1
u2 q · n dγ +
u2 q · n dγ (13)
∂t (̺ u2 ) dx +
∂t (̺u) + div(uq) u dx =
2 Ω
2 ∂ΩN
2 ∂ΩD D
Ω
which, since q · n is supposed to be non-negative over ∂ΩN , may be used, by an integration
with respect to the time, to obtain a stability estimate for u (namely a generalization to
general boundary conditions of Relation (4) of the introduction) The following theorem
AN L2 -STABLE APPROXIMATION OF THE NAVIER–STOKES CONVECTION OPERATOR
7
states that the operator C satisfies a discrete analogue of this relation in the case where
uD = 0. It adapts to the case of a discretization based on a dual mesh and generalizes to
mixed boundary conditions a similar stability result which can be found in [16] (theorem
3.1, p. 317). An example of how this result may be used to obtain a discrete energy estimate
is given in Section 4 (derivation of Inequality (36)).
Theorem 3.1
Provided that the discrete mass balance (10) holds and that uD = 0, the convection operator
defined by (11)-(12) satisfies the following stability result:
X
X
1
1 X |Dσ | ̺σ u2σ − ̺∗σ (u∗σ )2 +
Fσ,ǫ u2σ
(14)
|Dσ | (Cu)σ uσ ≥
2
δt
2
σ∈E\ED
σ∈E\ED
Proof
We have:
X
ǫ=Dσ |ext ∈ĒN
|Dσ | (Cu)σ uσ = T1 + T2
σ∈E\ED
with:
T1 =
X
σ∈E\ED
|Dσ | ̺σ uσ − ̺∗σ u∗σ uσ ,
δt
T2 =
X
uσ
σ∈E\ED
X
Fσ,ǫ uǫ
ǫ∈Ē(Dσ )
For the first term, we get T1 = T1,1 + T1,2 + T1,3 with:
X |Dσ |
T1,1 =
(̺σ − ̺∗σ ) u2σ
δt
σ∈E\ED
1 X |Dσ | ∗ 2
̺σ uσ − (u∗σ )2
T1,2 =
2
δt
σ∈E\ED
2
1 X |Dσ | ∗ T1,3 =
̺σ uσ − u∗σ
2
δt
σ∈E\ED
The term T1,3 is always positive and can be seen as a dissipation associated to the backward
Euler time discretization. Turning now to T2 , we get T2 = T2,1 + T2,2 with:
X
X
X
X
Fσ,ǫ (uǫ − uσ )uσ
Fσ,ǫ ,
T2,2 =
T2,1 =
u2σ
σ∈E\ED
σ∈E\ED ǫ∈Ē(Dσ )
ǫ∈Ē(Dσ )
From the discrete mass balance (10), the term T2,1 cancels with T1,1 . Using the identity
2a (a − b) = a2 + (a − b)2 − b2 valid for any real number a and b, we get:
X
1 X
T2,2 = −
Fσ,ǫ u2σ + (uσ − uǫ )2 − (uǫ )2
2
σ∈E\ED ǫ∈Ē(Dσ )
So T2,2 = T2,2,1 + T2,2,2 with:
1 X 2
uσ
T2,2,1 = −
2
σ∈E\ED
X
Fσ,ǫ =
ǫ∈Ē(Dσ )
1
2
X
σ∈E\ED
|Dσ |
(̺σ − ̺∗σ ) u2σ
δt
and, introducing the notation (u2 )σ,ǫ = (uǫ )2 − (uσ − uǫ )2 :
X
1 X
T2,2,2 =
Fσ,ǫ (u2 )σ,ǫ
2
σ∈E\ED ǫ∈Ē(Dσ )
Reordering the summations and using the fact that, for any ǫ = σ|σ ′ , Fσ,ǫ = −Fσ′ ,ǫ , we get:
X
1
Fσ,ǫ (u2 )σ,ǫ − (u2 )σ′ ,ǫ
T2,2,2 =
2
ǫ=Dσ |Dσ′ ∈Ēint
(15)
X
X
1
1
Fσ,ǫ (u2 )σ,ǫ +
Fσ,ǫ (u2 )σ,ǫ
+
2
2
ǫ=Dσ |ext ∈ĒD
ǫ=Dσ |ext ∈ĒN
8
G. ANSANAY-ALEX ET AL.
For ǫ = Dσ |Dσ′ ∈ Ēint , from (12), we have uǫ = (uσ + uσ′ )/2 and so (u2 )σ,ǫ = uσ uσ′ and
finally (u2 )σ,ǫ − (u2 )σ′ ,ǫ = 0. For ǫ = Dσ |ext ∈ E¯D , supposing that the prescribed value is
uD = 0, we have uǫ = uσ /2 and so (u2 )σ,ǫ = 0. Finally, for ǫ = Dσ |ext ∈ E¯N , uǫ = uσ and
(u2 )σ,ǫ = u2σ . Gathering all the terms, we thus have:
X
|Dσ | (Cu)σ uσ ≥ T1,2 + T2,2,1 +
σ∈E\ED
1
2
X
Fσ,ǫ u2σ
ǫ=Dσ |ext ∈ĒN
which concludes the proof.
Remark 2 (Upwind scheme)
An upwind choice for the value of the unknowns at the dual faces is also possible, and its
effect is to introduce an artificial dissipation.
Indeed, the only changes induced by this choice in the preceding proof lie in the evaluation
of the terms appearing in T2,2,2 in Relation (15). For an internal face ǫ = Dσ |Dσ′ , supposing
without loss of generality that the chosen orientation for the face is such that Fσ,ǫ ≥ 0,
2
2
2
we would have uǫ = uσ and thus
2 (u )σ,ǫ −2 (u )σ′ ,ǫ = (uσ′ − uσ ) . The corresponding energy
flux in (15), which reads Fσ,ǫ (u )σ,ǫ − (u )σ′ ,ǫ is thus always positive. Let us compare this
term with the dissipation which would be induced by a diffusion term. For a two-point flux
finite volume scheme, this latter takes the form λ |ǫ| (uσ′ − uσ )2 /hǫ where λ is the diffusion
coefficient and hǫ is a geometric quantity associated to the face ǫ and of the same magnitude
as its diameter. We thus see that the ”numerical diffusion” through the dual face ǫ is given
by hǫ q ǫ · n, where q ǫ stands for the mean value of q over ǫ.
For faces of ĒD , by a similar computation, we easily check that the energy flux is either
zero or positive. For faces of ĒN , the computation is unchanged, since the performed choice
was already the upwind one.
Remark 3 (Non-homogeneous Dirichlet boundary conditions)
When the prescribed value uD is not zero, the ”energy flux” over ǫ = Dσ |Dσ′ ∈ ĒD (i.e. the
quantity appearing in the second summation of Relation (15)) reads:
Z
1
Fσ,ǫ (u2 )σ,ǫ = Fσ,ǫ uσ uσ′ = Fσ,ǫ uσ ′
uD dγ
|ǫ | ǫ′
which is consistent with the inlet flux term appearing at the continuous level (i.e. the last
term of Equation (13)), but with the following two differences:
(i) the persistence of the unknown uσ makes that the identity (14) does not yield, at least
in its present form, a control on the solution;
(ii) the momentum flux Fσ,ǫ is not exactly the one entering the domain. This slight
difference only has a weak impact if the momentum field q is given, but, for NavierStokes equations, this field is itself a function of the unknown u (which, in this context,
will be a component of the velocity), and, once again, the solution reappears in an a
priori uncontrolled term.
Note however that this difficulty associated to boundary conditions seems to have no impact
in practice, since no uncontrolled growth of the kinetic energy was ever observed in our
computations with non-homogeneous Dirichlet boundary conditions.
3.2. Interpolating the mass flowrates
We now turn to the case where the field q = ̺ u is itself obtained from a discretization of the
complete problem (1). With the chosen finite elements and a backward Euler discretization
with respect to time, the pressure and the density being piecewise constant per primal cell,
the discrete mass balance takes the form:
∀K ∈ T ,
X
|K|
FK,σ = 0
̺K − ̺∗K ) +
δt
σ∈E(K)
(16)
AN L2 -STABLE APPROXIMATION OF THE NAVIER–STOKES CONVECTION OPERATOR
9
In this relation, ̺K and ̺∗K stand for an approximation of the density in the primal cell K at
the end and beginning of the time step, and FK,σ is an approximation of the outward mass
flux through the face σ of K . Since the density is discontinuous through σ , the expression
of FK,σ must be written as:
FK,σ = |σ| ̺m
(17)
σ uσ · nK,σ
where any reasonable approximation for ̺m
σ seems to be suitable in the present context,
because the density is supposed to be given and positive. Throughout this paper, the
centered choice is performed for the internal faces, a value is computed from the data
on ∂ΩD and the upwind choice is made on ∂ΩN :
1
(̺K + ̺L )
2
Z
1
=
̺ dγ
|σ| σ
= ̺K
for σ ∈ Eint , σ = K|L,
̺m
σ =
for σ ∈ ED ,
̺m
σ
for σ ∈ EN , σ ∈ E(K),
̺m
σ
(18)
Remark 4
In a more general context, choosing for ̺m
σ an upwind discretization may be a convenient
way to ensure the positivity of the density (see [16, 17, 12, 19] for works exploiting this
argument in the context of compressible flows).
For the Crouzeix-Raviart element and, with the specific meshes (i.e. rectangles or
rectangular parallellepipeds) considered here, for the Rannacher-Turek element, we have
Z
ϕσ dx = |DK,σ |
K
where |DK,σ | is the measure of the half diamond cell DK,σ . Thus, still with a backward Euler
time discretization, applying a mass lumping technique to the finite element discretization
of the unstationary term ∂t (̺u) in Equation (1b) yields, in the discrete equation associated
(i)
with σ (i.e. obtained with the test functions ϕσ , 1 ≤ i ≤ d):
|Dσ |
̺σ uσ − ̺∗σ u∗σ + other terms = 0
δt
(19)
where u and u∗ stand for the velocity at the current time step and the previous time step
respectively, and ̺σ is defined by:
for σ ∈ Eint , σ = K|L,
|Dσ | ̺σ = |DK,σ | ̺K + |DL,σ | ̺L
for σ ∈ EN , σ ∈ E(K),
̺σ = ̺K
(20)
No value for ̺σ needs to be specified for σ ∈ ED , since the velocity is prescribed on this
boundary, and no equation is consequently written for the associated degrees of freedom.
To be in position to apply the theory developed in the previous section, the task we
have to complete is thus the following one: obtain a discretization of the term div(̺u ⊗ u)
such that, associated to the time derivative term of Equation (19) with the density defined
by (20), the structure of Equation (11) is recovered. In fact, this problem reduces to the
definition of momentum fluxes through the faces of the diamond cells such that a discrete
mass balance over the diamond cell (i.e. Equation (10)) holds, starting from the mass balance
over the primal cells (16). The construction of these fluxes is the goal of the remainder of
this section. We first give an argument which provides a general technique for this purpose,
then successively address the case of the Crouzeix-Raviart elements, the Rannacher-Turek
elements in two and three dimensions and, finally, in axisymmetrical coordinates, for a
specific type of mesh.
3.2.1. A general argument The approach adopted here is based on the following elementary
result.
10
G. ANSANAY-ALEX ET AL.
Lemma 3.2 (Mass balance in a sub-volume of a mesh)
Let K ∈ T , let ̺K and ̺∗K be two real numbers, and consider a family of real numbers
(FK,σ )σ∈E(K) such that (16) holds. Let wK be a momentum field on K , such that divwK is
constant over K and satisfying:
Z
∀σ ∈ E(K),
wK · nK,σ dγ = FK,σ
(21)
σ
Let D be a subset of K with boundary ∂D, and n∂D be the normal vector to ∂D outward
D. Then the following property holds:
|D|
(̺K − ̺∗K ) +
δt
Z
wK · n∂D dγ = 0
∂D
Proof
Using the fact that the divergence of w is constant over K , then Relation (16), we have:
Z
∂D
w · n∂D dγ =
Z
divw dx =
D
|D|
|K|
Z
divw dx = −
K
|D| |K|
(̺K − ̺∗K )
|K| δt
Suppose now that we are able to build for any K ∈ T a constant divergence field w such
that (21) holds, and that we evaluate the fluxes at each face of a half-diamond cell DK,σ
by integration of w K · n over the face. The set of the faces of DK,σ , denoted by Ẽ(DK,σ ), is
the union of σ and a set of faces of the dual mesh. By definition of w K , we thus get a flux
on σ which is FK,σ , and a family of additional fluxes (Fσ,ǫ )ǫ∈Ẽ(DK,σ )\{σ} such that:
|DK,σ |
(̺K − ̺∗K ) + FK,σ +
δt
X
Fσ,ǫ = 0
ǫ∈Ẽ(DK,σ )\{σ}
If DK,σ is associated to σ ∈ EN , DK,σ = Dσ and the preceding relation is exactly (10), thanks
to the definition of ̺σ by Equation (20). If σ ∈ Eint , σ = K|L, summing the equation for DK,σ
and DL,σ , we get, since FK,σ = −FL,σ by their definition (17):
i
1h
|DK,σ |̺K + |DL,σ |̺L − |DK,σ |̺∗K + |DL,σ |̺∗L +
δt
X
Fσ,ǫ = 0
ǫ∈Ē(Dσ )
which is once again (10), thanks to (20).
3.2.2. The Crouzeix-Raviart element Since the shape functions for the Crouzeix-Raviart
element are linear over each cell K ∈ T , the field wK may be itself a Crouzeix-Raviart
function. A first possible choice is to derive it by direct interpolation of the fluxes at the
faces (FK,σ )σ∈E(K) :
X FK,σ
w K (x) =
nK,σ ϕσ (x)
|σ|
σ∈E(K)
However, if the fluxes (FK,σ )σ∈E(K) are computed by relation (17), i.e. take the form
K
FK,σ = |σ| ̺m
is:
σ uσ · nK,σ , another possible choice for w
wK (x) =
X
̺m
σ uσ ϕσ (x)
(22)
σ∈E(K)
This is this latter formula which is chosen for the numerical experiments presented hereafter.
AN L2 -STABLE APPROXIMATION OF THE NAVIER–STOKES CONVECTION OPERATOR11
3.2.3. The Rannacher-Turek element For the Rannacher-Turek element, the divergence of
discrete functions is not constant, so w K cannot belong to this discrete space. We thus
suppose that each cell K ∈ T is a rectangular parallellepiped, and we use the following
interpolation formula:
X FK,σ
wK (x) =
nK,σ ασ (x · nK,σ )
(23)
|σ|
σ∈E(K)
where the functions ασ are affine interpolation functions which are determined in such a
way that (21) holds. For instance, let us suppose that d = 2 and that the considered element
is (x1,W , x1,E ) × (x2,S , x2,N ). Using the notations introduced in Figure 3, we get:
 x − x −F
x1 − x1,W FE 
W
1,E
1
+
 δx1
δx2
δx1
δx2 

w K (x) = 

 x − x −F
x
−
x
F
2,N
2
S
2
2,S
N
+
δx2
δx1
δx2
δx1
Integrating the quantity w K · n over each dual face of the mesh leads to an expression of
the flux at each dual face as a linear combination of the fluxes at the primal ones; for the
present example, this relation takes the form:
(24)
Fσ,ǫ = αW FW + αE FE + αS FS + αN FN
where the coefficients αW , αE , αS and αN are given in the following table:
Fσ,ǫ
FW|S
FS|E
FE|N
FN|W
αW
− 3/8
− 1/8
1/8
3/8
αE
1/8
3/8
−3/8
−1/8
αS
3/8
−3/8
−1/8
1/8
αN
−1/8
1/8
3/8
−3/8
FN
N
FW
FE
FW
δx2
x2,N
F E|
FN
|W
δx1
FS
|E
|S
x1,W
FS
x2,S
x1,E
Figure 3. Rannacher-Turek element in Cartesian coordinates – Local notations for the definition
of the interpolation field w K .
3.2.4. The Rannacher-Turek element in axisymmetrical coordinates Let us now suppose
that we are using axisymmetrical coordinates, with d = 2, the first coordinate axis being
associated to the distance r to the symmetry axis and the second coordinate axis being
parallel to the symmetry axis (coordinate z ). We assume that the mesh is a (possibly nonuniform) rectangular grid. Then, with the notations defined on Figure 4, we get:


ψ(r)
1 − ψ(r)
(−F
)
+
F
W
E
2


r2 − rW
r
r

with
ψ(r)
=
(25)
w K (x) = 
2
2

 zN − z
−FS
FN
z − zS
rE − rW
+
2 − r2 )
2 − r2 )
δz
π (rE
δz π (rE
W
W
12
G. ANSANAY-ALEX ET AL.
Note that the divergence of this function is indeed constant, since, with this system of
coordinates, div w = ∂r (rw r )/r + ∂z wz .
In addition, it may be checked that the integral of each shape function ϕσ defines the
same dual mesh than in the Cartesian system of coordinates, i.e. is equal to the volume of
the torus having for section the triangle delimited by the diagonals of the element and σ ,
as sketched on Figure 4. As previously, integrating the quantity wK · n over each dual face
of the mesh leads to a relation of the form:
(26)
Fσ,ǫ = αW FW + αE FE + αS FS + αN FN
The coefficients αW , αE , αS and αN are given as a function of the geometrical features of
the element in the following table:
Fσ,ǫ
αW
αE
FW|S
− (1 − β)/2
β/2
FS|E
− γ/2
(1 − γ)/2
FE|N
γ/2
−(1 − γ)/2
FN|W
(1 − β)/2
−β/2
αS
rW
−
+ 2β
8 r̄
rE
− 2γ
8 r̄
rE
−γ
8 r̄
rW
+β
−
8 r̄
αN
rW
−β
8 r̄
rE
−
+γ
8 r̄
rE
−
+ 2γ
8 r̄
rW
− 2β
8 r̄
where r̄, β and γ are defined as:
r̄ =
rW + rE
,
2
β=
rW + δr/6
,
4 r̄
γ=
rE − δr/6
4 r̄
(27)
FE
FN
N
FW
FW
δz
zN
F E|
FN
|W
δr
FS
|E
|S
rW
FS
zS
z
rE
r
Figure 4. Rannacher-Turek element in axisymmetrical coordinates – Local notations for the
definition of the interpolation field w K .
3.3. The discrete momentum balance equation
The discretization of (1b) is obtained by applying the above-defined convection operator
to each component of the velocity and using the standard finite element technique to
approximate the pressure gradient and viscosity terms. We thus obtain the following discrete
AN L2 -STABLE APPROXIMATION OF THE NAVIER–STOKES CONVECTION OPERATOR13
momentum balance equation:
∀σ ∈ E \ ED , for 1 ≤ i ≤ d,
X
|Dσ |
Fσ,ǫ (u(i) )ǫ + ad (u, ϕ(i)
(̺σ (u(i) )σ − ̺∗σ (u(i) )∗σ ) +
σ )
δt
ǫ∈
Ē(D
)
σ
Z
Z
XZ
(i)
−
g · ϕ(i)
f
·
ϕ
dx
+
pn divϕ(i)
dx
=
σ dγ
σ
σ
K∈T
Ω
K
(28)
∂ΩN
where, for any discrete velocity field v and w:
XZ 1
ad (v, w) = µ
∇v : ∇w + divv divw dx
3
K∈T K
This relation is obtained by remarking that, since the viscosity µ is supposed to be
constant, the divergence of the stress tensor τ given by Equation (2) reads div(τ (u)) =
µ ∆u + µ/3 ∇div(u).
3.4. Concluding remarks
Extension to the MAC scheme. The content of this section extends in a
straightforward way to the MAC scheme [22, 14], and thus allows to build for this spatial
approximation a discrete convection operator satisfying the kinetic energy theorem. A
surprising consequence of the computation of the mass fluxes at the faces of the dual mesh
(for the MAC scheme as well as for the finite element schemes considered here) is that the
flux at a face ǫ included in the element K generally involves the value of the density in the
neighbours of K .
Comparison with the convection operator proposed in [34, 35]. In [34, 35], in
the two-dimensional incompressible case and for the Crouzeix-Raviart element, the authors
introduce a convection operator which is obtained by an upwind finite volume discretization
based on the dual cell. This work is adapted for the Rannacher-Turek element in [38]. This
operator is written in a non-conservative form, in the following sense. Let us assume that,
for any diamond cell of the mesh Dσ , the following discrete divergence-free condition holds:
X
Fσ,ǫ = 0
(29)
ǫ∈Ē(Dσ )
Then a convection operator which, applied to a discrete function u, reads:
X
Fσ,ǫ uǫ
(Cu)σ =
ǫ∈Ē(Dσ )
can equivalently be written:
(Cu)σ =
X
Fσ,ǫ (uǫ − uσ )
ǫ∈Ē(Dσ )
The first form may be considered as a discretization of the convection term written in
divergence form (i.e., for a given transport field q of fluxes Fσ,ǫ , div(u q)), and the second
one as a discretization of the same term in gradient form (i.e. q · ∇u). This is this latter
form which is used in [34, 35, 38].
In [34, 35, 38], the fluxes Fσ,ǫ are obtained from a direct interpolation of the transport
field (i.e., with the notations of the present paper, up to the multiplication by a constant
density, w = u). For the Crouzeix-Raviart discretization, the resulting convection operator
thus coincides (for the incompressible case) with an upwind version of the operator proposed
in this work. This is no more the case for the Rannacher-Turek element (the condition (29)
is not fulfilled by the choice made in [38]).
14
G. ANSANAY-ALEX ET AL.
4. A FRACTIONAL STEP SCHEME
In this section, we address the solution of the full system (1). To this purpose, we build an
incremental projection-like algorithm.
Let us consider a partition 0 = t0 < t1 < . . . < tN = T of the time interval (0, T ), which we
suppose uniform for the sake of simplicity. Let δt be the constant time step δt = tn+1 − tn
for n = 0, 1, . . . , N − 1. In a semi-discrete time setting, the proposed algorithm consists in
the following two step scheme:
The densities ̺n−1 , ̺n , ̺n+1 being given, un and pn being already computed, and
supposing that:
1 n
(̺ − ̺n−1 ) + div(̺n un ) = 0
(30)
δt
1 - Solve for ũn+1 :
1 n n+1
(̺ ũ
− ̺n−1 un ) + div(ũn+1 ⊗ ̺n un ) + ∇pn − div τ (ũn+1 ) = f n+1 (31)
δt
2 - Solve for pn+1 and un+1 :
1 n n+1
̺ (u
− ũn+1 ) + ∇(pn+1 − pn ) = 0
δt
1 n+1
(̺
− ̺n ) + div(̺n+1 un+1 ) = 0
δt
(32a)
(32b)
The equation (30) is the mass balance equation written at the previous time step. In the
fully discrete setting, it takes the form of (16):
X
|K| n
n−1
FK,σ (̺n , un ) = 0
(33)
̺K − ̺K
)+
∀K ∈ T ,
δt
σ∈E(K)
Where the mass fluxes are given by (17):
n
FK,σ (̺n , un ) = |σ| (̺n )m
σ uσ · nK,σ
the density at the face being given by the centered choice (so (18) with ̺ = ̺n ).
Step 1 consists in a semi-implicit solution of the momentum balance equation to obtain
a predicted velocity. Let us now explain how the process developed in Section 3 is applied
to obtain a L2 -stable convection operator. Since the mass balance at t = tn+1 is not solved
when the prediction step is performed, the starting point is the mass balance at the previous
time step (33), which in fine yields to a time shift of the densities, i.e. taking ̺n (resp. ̺n−1 )
when ̺n+1 (resp. ̺n ) would be the natural time level. We first build, for each cell of the
mesh, a lifting wK (̺n , un ) of the mass fluxes FK,σ (̺nK , un ) by either (22), (23) or (25),
depending on the type of the cell and of the problem (simplices, rectangles or rectangular
parallellepipeds in cartesian coordinates and rectangles with two sides aligned with the
symmetry axis in axisymmetrical coordinates, respectively). Then the mass flux through
the dual edge ǫ ∈ Ē , Fσ,ǫ (̺n , un ), is obtained by integrating wK (̺n , un ) · nσ,ǫ over ǫ. Note
that, for the Rannacher-Turek elements, this computation is avoided in practice, using
directly (24) (or its 3D counterpart) or (26). The density at the face at time k = n − 1 and
k = n is given by Equation (20):
∀σ ∈ Eint , σ = K|L,
|Dσ | ̺kσ = |DK,σ | ̺kK + |DL,σ | ̺kL
∀σ ∈ EN , σ ∈ E(K),
̺kσ = ̺kK
Finally, the convection operator is given by Equation (11), applied to each component of
the velocity:
∀σ ∈ E \ ED ,
1
1
− ̺σn−1 unσ +
Cu σ (̺n , un , ũn+1 ) = (̺nσ ũn+1
σ
δt
|Dσ |
X
ǫ∈Ē(Dσ )
Fσ,ǫ (̺n , un ) ũn+1
ǫ
AN L2 -STABLE APPROXIMATION OF THE NAVIER–STOKES CONVECTION OPERATOR15
where ũn+1
is given by the finite volume centered choice (i.e. writing Equation (12) for
ǫ
each component of the velocity). The other terms of (31) are discretized using finite element
techniques, as explained in Section 3.3, thus yielding System (28), with u = ũn+1 , u∗ = un ,
p = pn , ̺ = ̺n and ̺∗ = ̺n−1 .
Step 2 is a pressure correction step, which boils down to the usual projection step used in
incompressible flow solvers when the density is constant (e.g. [28]). The time-derivative term
in (32a) is lumped, as in the discretization of the prediction step, and the gradient operator
is also the same as in the prediction step. The mass balance (Equation (32b)) is discretized
according to (16), with the mass fluxes given by (17), the density at the face being given by
(18), with, for this equation, ̺ = ̺n+1 , ̺∗ = ̺n and u = un+1 . At the discrete level, taking
the divergence of (32a) and using (32b) to eliminate the unknown velocity un+1 yields a
linear elliptic problem for the pressure, the operator of which looks like a finite volume
discretization (however inconsistent) of the system:
−div
̺n+1
̺n
1
1
∇(pn+1 − pn ) = − div(̺n+1 ũn+1 ) − 2 (̺n+1 − ̺n )
δt
δt
The description of this computation at the discrete level may be found in [16, section 3.4]
Once the pressure is computed, the first relation yields the updated velocity.
By construction, the convection operator satisfies the stability stated in Theorem 3.1.
In the constant density incompressible case, combining this result with arguments of the
analysis of the pressure correction methods (see [36] and [20] for the seminal works, in
the time semi-discrete and discrete case respectively), the scheme may be shown to be
unconditionally stable, in the sense that the velocity satisfies discrete versions of the L∞ (L2 )
and L2 (H1 ) a priori estimates of the continuous problem. For the sake of completeness,
let us sketch the proof of such an estimate, supposing, for short, that the velocity obeys
an homogeneous Dirichlet boundary condition on the whole boundary, that the volume
forcing term f = 0, and denoting by ̺ the constant density. Let n ≥ 0; multiplying each
discrete momentum balance equation by the corresponding unknown and summing, we get,
by Theorem 3.1:
i
h
XZ
1 X
n+1
n+1
n+1 2
n 2
, ũ
) − δt
pn div(ũn+1 ) dx = 0 (34)
|Dσ | ̺ |ũ
| − |u | + δt ad (ũ
2
K
K∈T
σ∈Eint
The first equation of the projection step reads:
∀σ ∈ Eint , for 1 ≤ i ≤ d,
|Dσ | ̺
1/2
(u(i) )n+1
−
σ
δt
1/2
XZ
dx
pn+1 div ϕ(i)
σ
|Dσ | ̺
K∈T K
1/2 (i) n+1
= |Dσ | ̺
(ũ )σ − δt
|Dσ | ̺
1/2
XZ
K∈T
K
pn div ϕσ(i) dx
Squaring this relation and summing over σ ∈ Eint and i = 1, . . . , d yields:
i
h
XZ
1 X
|Dσ | ̺ |un+1 |2 − |ũn+1 |2 − δt
pn+1 div(un+1 ) dx
2
K
σ∈Eint
K∈T
XZ
δt2 n 2
δt2 n+1 2
|p
|T = −δt
|p |T
pn div(ũn+1 ) dx +
+
̺
̺
K∈T K
(35)
where | · |2T is a discrete semi-norm analog to the usual finite volume H1 seminorm. Remarking that the second term in the left hand side vanishes because of the
incompressibility constraint and summing (34) and (35) over the time-steps, we get, for
n ≥ 0:
n+1
X
δt2 n+1 2
1 X
n+1 2
|p
|T
|Dσ | ̺ |u
| +
δt ad (ũk , ũk ) +
2
̺
σ∈Eint
k=1
1 X
δt2 0 2
|p |T
≤
|Dσ | ̺ |u0 |2 +
2
̺
σ∈Eint
(36)
16
G. ANSANAY-ALEX ET AL.
which is the estimate we are searching for. A similar computation is performed (and detailed)
in [16] to prove the stability of a similar scheme, even if more complex (in particular,
involving an additional renormalization step for the pressure to cope with the fact that
̺ is varying with time), for the compressible barotropic case. Note that these results also
hold (with a simpler proof) for the semi-implicit coupled (i.e. without the prediction and
pressure correction technique) scheme.
If an additional balance equation must be solved for another variable on which the density
depends (as in Section 5.2), let say y , this is performed before the first step, with the following
time discretization:
1 n n+1
(̺ y
− ̺n−1 y n ) + div(̺n y n+1 un ) − div(∇y n+1 ) = 0
δt
(37)
and with a standard finite volume scheme. In this case, for a convenient discretization of
the convection term (upwind or MUSCL, for instance), the condition (30) allows to obtain
a monotone scheme [26, 4]. For K ∈ T , the density ̺n+1
is then evaluated as a function of
K
n+1
yK
at the end of this step, to be used afterwards in the projection step.
Finally, to initialize the algorithm, it may be suitable to calculate ̺−1 and u0 by
interpolation of the initial data and to compute ̺0 by the solution of the mass balance:
1 0
(̺ − ̺−1 ) + div(̺0 u0 ) = 0
δt
For this particular initialization step, the discretization of the convection term must then be
of upwind type, to preserve the positivity of the density. By construction, this initialization
allows to obtain energy estimates for the velocity (i.e. an estimate of the form of (36)).
Perhaps even more importantly, when an additional balance equation as (37) must be solved,
it also yields a discrete maximum principle preserving computation of y 1 .
5. NUMERICAL EXPERIMENTS
Theoretical proofs of convergence for schemes similar to the proposed one, i.e. combining
a Crouzeix-Raviart finite element discretization for the diffusion and a finite volume
discretization for the convection, are available in some cases: incompressible stationary flows
[34, 35], convection-diffusion equations [1, 10, 13]. However, the only analysis for NavierStokes equations [34, 35] is performed in the incompressible framework and for an upwind
approximation for the convection, the resulting scheme being only first order in space. Our
goal here is thus to perform numerical experimentations to check whether these convergence
results extend in essentially three directions: obtaining, with the centered approximation,
second order in space convergence for the velocity, in the unstationary case and for variable
density flows. In addition, we also address some classical benchmarks to assert the robustness
of the scheme. We begin with incompressible flows, then turn to variable density flows.
5.1. Incompressible flows
This section is devoted to the solution of unstationary incompressible Navier–Stokes
equations:
̺ ∂t u + div(u ⊗ u) + ∇p − µ∆u = f
(38)
divu = 0
We first assess the accuracy of the scheme against a well-known analytical solution, namely
the so-called Taylor-Green vortices, then compute various benchmarks of the literature:
backward-facing step flows and flows behind obstacles.
AN L2 -STABLE APPROXIMATION OF THE NAVIER–STOKES CONVECTION OPERATOR17
5.1.1. Green–Taylor vortices The so-called Green-Taylor vortex flows are a known solutions
to System (38) posed on R2 × [0, T ] with a zero forcing term f . The solution is given by:
8π 2
− cos(2πx1 ) sin(2πx2 )
u(x, t) =
exp −
t
sin(2πx1 ) cos(2πx2 )
Re
16π 2
cos(4πx1 ) + cos(4πx2 )
exp −
t
p(x, t) = −
4
Re
(39)
The Reynolds number chosen here is Re = 100 (̺ = 1 and µ = 0.01), the computational
domain is set to (0, 1/2)2 , and the final time is T = 1. The velocity is prescribed on the
whole boundary, and the initial and boundary conditions are chosen to match the analytical
solution.
For the Rannacher-Turek element, the domain is meshed by n × n regular grids, with
n = 16, 32, 64 and 128. For the Crouzeix-Raviart element, the meshes are obtained as
follows: first, we build a regular grid; then, this latter is perturbed by moving each inner
vertex a of the mesh to a random point of a circle centered on a and of radius equal to the
length of the smallest edge issued from a multiplied by 0.1; finally, each cell of this mesh
is cut along its diagonals in four triangles. The resulting mesh is quite general, in the sense
that it does not enjoy any particular symmetry property. Four meshes are built, referred
to by mesh 1, mesh 2, mesh 3 and mesh 4 and obtained from an initial 16 × 16, 32 × 32,
64 × 64 and 128 × 128 regular grid, respectively.
0.1
0.1
0.01
0.01
mesh 1
16 × 16
0.001
0.001
32 × 32
mesh 2
mesh 3
64 × 64
0.0001
0.0001
mesh 4
128 × 128
1e-05
0.001
0.01
0.1
1e-05
0.001
0.01
time step
0.1
time step
Figure 5. Green-Taylor vortices problem: time convergence in L2 norm for the velocity, for various
meshes. left: quadrangles, right: triangles.
0.01
0.01
16 × 16
mesh 1
32 × 32
mesh 2
0.001
0.001
64 × 64
mesh 3
128 × 128
mesh 4
0.0001
0.001
0.01
time step
0.1
0.0001
0.001
0.01
0.1
time step
Figure 6. Green-Taylor vortices problem: time convergence in L2 norm for the pressure, for various
meshes. left: quadrangles, right: triangles.
18
G. ANSANAY-ALEX ET AL.
The L2 errors for the velocity and the pressure at t = 1, as a function of the time step
and for the various considered meshes, are reported on Figure 5 and 6 respectively. From
large to small time steps, curves first decrease, according to a surprising second-order time
convergence rate (as the time discretization is as given is Section 4, i.e. a first order backward
Euler scheme). In fact, computations with the semi-implicit coupled scheme show that this
latter is much more accurate at large time step; reported errors are thus essentially splitting
errors, which indeed are known to behave as δt2 [21]. Then, at small time steps, a plateau
is obtained, which corresponds to a second order for the velocity and first order for the
pressure spatial error.
5.1.2. Backward-facing step We now address flows over a backward-facing step. The
computational domain (chosen to be the same as in [7]) starts at the channel expansion,
and the inlet velocity is prescribed, with a parabolic profile, at the upper part of height h
of the left wall. The velocity is set to zero at the top and bottom walls, and a ”do-nothing”
(or homogeneous Neumann) boundary condition is imposed at the right-hand side of the
domain. The expansion rate is H/h = 1.9423, and the domain length is set to L = 60 (Figure
7).
Figure 7. Computational domain for the backward-facing step flow.
The mesh is a regular uniform 600 × 291 grid, and the computation is performed with the
Rannacher-Turek element. The steady state is obtained by a fictitious transient, starting
from a zero velocity.
This test, experimentally studied in [3], is commonly used to assert the accuracy of
numerical schemes, thanks to the dependency of the reattachment length on Reynolds
number. This latter quantity is defined here as Re = ̺umax h/µ, where umax is the maximum
value of the velocity in the inlet section. On Figure 8, we compare the results of our
computations for various Reynolds number (from Re = 50 to Re = 1000) with the numerical
results published in [7]; a very good agreement is observed.
On Figure 9, we present the obtained transient development phase, for Re = 2500, at the
same times than [7, fig. 5]; once again, results compare satisfactorily.
5.1.3. Flow past a cylinder We address in this section some test cases which are part of a
benchmark proposed in [33]. The first considered case is two-dimensional, the second one
is three-dimensional. We give here a brief description of each case and refer to [33] for a
complete presentation.
2D flow, Re = 100 This test corresponds to the 2D-2 case in [33]. The geometry for this
test is sketched on Figure 10. The fluid enters the domain on the left boundary, with an
imposed velocity profile:
ux (0, y) = 4um y
H −y
,
H2
uy (0, y) = 0
where H = 0.41m is the height of the channel and um = 1.5 m/s; a zero velocity is prescribed
at the other boundaries except for the right-hand side, where we use an inlet/outlet
boundary condition which ensures the stability of the problem even in presence of inward
velocities [2].
AN L2 -STABLE APPROXIMATION OF THE NAVIER–STOKES CONVECTION OPERATOR19
14
12
10
8
6
4
2
Chiang et al.
RT - diamonds
0
0
200
400
600
800
1000
Figure 8. Computed recirculation length past a backward-facing step, as a function of the
Reynolds number. Ordinate: xR /(H − h), where xR is the reattachment length.
t=1
t=5
t=8
t = 14
t = 20
Figure 9. Streamlines past a backward-facing step for Re = 2500, at different times.
The density is ̺ = 1 and the viscosity is µ = 0.01, so the Reynolds number, defined as
Re = ̺ūD/µ, where D = 0.1 is the diameter of the cylinder and ū = 2 ux (0, H/2)/3, is equal
to 100.
20
G. ANSANAY-ALEX ET AL.
ux = uy = 0
0.16m
0.15m
ux = uy = 0
0.1m
0.15m
ux = uy = 0
2.2m
Figure 10. Geometry for the 2D flow around a cylinder.
A ”coarse version” of the meshes used for the presented computation is sketched on
Figure 11; real meshes are considerably refined with respect to this one, by diminishing
the discretization step along the characteristic lines (the boundaries and the concentric
circles around the cylinder).
Figure 11. A ”coarse version” of the 2D mesh.
The space discretization is performed with the Rannacher–Turek elements in their
parametric variant. The cells are not rectangular, so the definition of the discrete convection
operator given in the preceding section needs to be generalized: in fact, we simply keep for
the expression of the mass fluxes at the dual faces the same linear combination of the fluxes
on the primal faces as in the rectangular case (i.e. we suppose that Relation (24) applies).
Since the integral over a deformed element K of a shape function associated to a face of K is
no longer |K|/4, the geometrical interpretation of this term as a finite-volume discretization
over a dual cell built from the diagonal lines of the primal mesh does not hold. However, the
discrete trilinear form associated to the convection term is still antisymmetrical. Finally,
note also that the deviation from the rectangular case attenuates as the mesh is refined.
In all our computations, the time step is δt = 5.10−4 s.
Figure 12. 2D flow with Re = 100 - x-component of the velocity.
The flow is unsteady (see Figure 12 for a vizualization at a given time), and the main
characteristic flow quantities quoted in [33] are the maximum drag coefficient cD max , the
maximum lift coefficient cLmax , the Strouhal number St and an instantaneous pressure
difference ∆P between the front and end points of the cylinder, i.e. the points (0.15m, 0.20m)
and (0.25m, 0.20m) (see [33, section 2.2] for a precise definition of these quantities). The
AN L2 -STABLE APPROXIMATION OF THE NAVIER–STOKES CONVECTION OPERATOR21
obtained values for two space discretizations (computations C#1 and C#2) are gathered
in the following table, together with the results of some participants using the same finite
element as in the present work (computations RT#1 and RT#2), although in the nonparametric variant, and a plausible range for the results derived from the set of the
contributions to the benchmark. Values entering this reference interval are typeset in bold
in the table.
mesh
space unks
time steps
cD max
cLmax
St
∆P
C#1
294 882
660
3.2395
0.9334
0.3030
2.4533
C#2
352 722
664
3.2511
0.9983
0.3012
2.4961
RT#1
167 232
188
3.2498
1.0081
0.2927
2.4410
RT#2
667 264
612
3.2314
0.9999
0.2973
2.4707
3.22 – 3.24
0.99 – 1.01
0.295 – 0.305
2.46 – 2.50
Reference range
The obtained results either enter the reference range or are very close to it, without too
much refining the mesh (the finest one only leads to a little bit more than half the number
of space unknowns used in computation RT#2).
0.16m
1.95m
0.1m
0.41m
0.1m
0.15m
0.45m
0.41m
Figure 13. Geometry for the 3D flow around a cylinder.
3D flow, Re = 20 We now turn to the steady three-dimensional flow referred to as the
3D-1Z case in [33]. The geometry of the computational domain is sketched on Figure 13.
The flow is governed by the system of equations (38). The inlet velocity profile is parabolic:
ux (0, y, z) = 16um yz
(H − y)(H − z)
, uy (0, y, z) = uz (0, y, z) = 0
H4
with um = 0.45 m/s. The viscosity of the fluid is µ = 10−3 , and its density is ̺ = 1, so the
Reynolds number, defined as Re = ̺ūD/µ with ū = 4 ux (0, H/2, H/2)/9, is equal to 20.
The mesh is obtained by first adapting the mesh used for the previous case to triangulate
a cut of the domain along a x, z -plane, and then building on this basis tree-dimensional cells
by extrusion along the y -axis, with a uniform step. The resulting cells are thus hexahedra,
and the space discretization is performed with the parametric version of the RannacherTurek element. The convection operator is obtained by extending the definition of Section
3 as previously.
As required in [33], we compute the drag coefficient cD , the lift coefficient cL , and the
pressure difference ∆P between the front and end points of the cylinder, i.e. the points
22
G. ANSANAY-ALEX ET AL.
(0.45, 0.20, 0.205) and (0.55, 0.20, 0.205) (see [33, section 2.3] for a precise definition of these
quantities). The obtained values (computations C#3) are gathered in the following table,
together with the results of some participants using the same finite element as in the present
work (computations RT#3, RT#4 and RT#5), although in the non-parametric variant, and
a plausible range for the results derived from the set of the contributions to the benchmark.
We also give more accurate reference values more recently obtained [24, 5] using high degree
finite element discretizations and sophisticated post-processing techniques; these values are
probably the exact ones, up to the digits given below. The three computed characteristic
quantities lie in the reference intervals given by the benchmark outcomes, even if the low
degree of the approximation and the direct computation of the forces exerting on the cylinder
do not allow an accuracy comparable to [24, 5].
mesh
space unks
C#3
2 271 870
RT#3
98 128
RT#4
771 392
RT#5
6 116 608
Exact range
Values from [24, 5]
cD
cL
∆P
6.175
5.8431
5.9731
6.1043
6.05 – 6.25
6.18533
0.00814
0.0061
0.0059
0.0079
0.008 – 0.01
0.009401
0.1673
0.1482
0.1605
0.1672
0.165 – 0.175
0.171007
5.2. Variable density flows
In this section, we consider the system of equations governing a two-component flow:
∂t ̺ + div(̺u) = 0
∂t (̺u) + div(̺u ⊗ u) + ∇p − divτ = f
(40)
∂t (̺y) + div(̺yu) − λ ∆y = g
where the density ̺ is given as a function of the unknown y by:
̺ = ̺(y) =
1
1−y
y
+
̺1
̺2
(41)
The component densities ̺1 and ̺2 are supposed to be two constant positive real numbers.
Let us suppose that the velocity is prescribed to zero on the whole boundary of the
computational domain ∂Ω. In this condition, integrating the mass balance in (40) over Ω
yields the total mass balance:
Z
d
̺ dx = 0
(42)
dt Ω
On the other side, the density ̺ is given as a function of y , itself solution of the third
equation of (40). Integrating this relation over Ω and supposing that the diffusion flux of
the mass of component 1 vanishes at the boundary, i.e. λ∇y · n = 0 on ∂Ω, we obtain:
Z
d
̺y dx = 0
(43)
dt Ω
By an easy manipulation of the equation of state (41), we get:
̺2
̺ = ̺2 + (1 − ) ̺y
̺1
and thus, since ̺ appears as an affine function of ̺y , the relations (42) and (43) are
fortunately compatible. With the proposed time-stepping procedure, this property does
not hold anymore, because of the time shift of the density in the computation of y , which is
done by solving (37) at the first step of the algorithm. Thus a renormalisation of the density
is necessary to ensure the existence of a solution to the projection step:
Z
̺n dx
n+1
Ω
̺(y n+1 )
̺
=Z
n+1
̺(y
) dx
Ω
AN L2 -STABLE APPROXIMATION OF THE NAVIER–STOKES CONVECTION OPERATOR23
Note that this relation is reminiscent of the scaling of the density obtained through its
dependency versus the so-called thermodynamical pressure in the asymptotic model for low
Mach number flows [27].
5.2.1. Convergence to an analytical solution In this section, we choose the following solution
to system (40):
1
̺(x, t) = 1 + sin(πt) cos(πx1 ) + cos(πx2 )
4
1
sin(πx1 )
̺(x, t)u(x, t) = − cos(πt)
sin(πx2 )
4
p(x, t) = sin(πt) sin(πx1 ) + sin(πx2 )
We suppose that ̺1 = 0.25 and ̺2 = 2, so relation (41) yields y = (2 − ̺)/(7̺). The viscosity
is supposed to be µ = 0.01 and, in the third relation of (40), we choose λ = 0. The
computational domain is Ω = (0, 1) × (0, 1), so the normal velocity is always zero at the
boundary, and the integral of the density over Ω does not vary with time. With this choice
for ̺ and ̺u, the mass balance (first relation of (40)) is verified. The right-hand side in the
momentum balance and in the y transport equation, the initial conditions and the boundary
conditions (prescribed value for u) are chosen to match the analytical solution.
The domain is meshed by n × n regular grids, with n = 16, 32, 64 and 128. The
discretization of the convection term for y is performed by an upwind finite volume scheme.
Errors obtained at t = 0.5 are displayed on Figures 14 and 15, in L2 norm for the velocity
and the pressure, and, for y , in the following (usual in the finite volume context) discrete
L2 norm:
X
||y||L2h =
|K| y(xK )2
K∈T
where xK is the mass center of the cell K . Curves show a decrease at large time step which
correspond to an approximate first order in time convergence, and then a plateau, the value
on which corresponds to a second order space convergence for the velocity and a first order
space convergence for the pressure.
For simplicial meshes with a Crouzeix-Raviart element discretization, the same behaviour
is observed.
0.1
0.1
16 × 16
16 × 16
0.01
32 × 32
32 × 32
0.001
64 × 64
64 × 64
0.01
128 × 128
0.0001
0.001
128 × 128
0.01
time step
0.1
0.001
0.01
0.1
time step
Figure 14. Variable density analytical solution: time convergence in L2 norm for the velocity and
the pressure, for various meshes.
5.2.2. A Rayleigh-Taylor instability flow We now address a case studied in [37, 15],
consisting in a Rayleigh-Taylor instability flow. The data are exactly the same as in the
case referred to by ”Re = 1000” in [15]. The chosen mesh is the same as in [15] for the finite
volume computation, namely a 256 × 512 uniform grid. A MUSCL technique with a Van
Leer limiter is implemented for the solution of the balance equation for y .
24
G. ANSANAY-ALEX ET AL.
0.01
16 × 16
32 × 32
0.001
64 × 64
128 × 128
0.001
0.01
0.1
time step
Figure 15. Variable density analytical solution: time convergence in discrete L2 norm for the y
variable, for various meshes.
Results are plotted on Figure 16, and seem to be quite close to those presented in [15,
Figure 1, p.898].
6. CONCLUSION
We have presented in this paper a discrete operator for the approximation with low-order
non-conforming finite element spaces of the convection terms in Navier-Stokes equation
in variable density flows. This operator is built by a finite volume technique, based on a
dual mesh. We prove that this operator satisfies a discrete counterpart of the the kinetic
conservation identity. This stability property has been observed to greatly improve the
robustness of computations, specifically with relatively coarse meshes as often encountered
in real-life applications. In addition, the assembling cost of this operator is low, and it does
not extend the stencil of the scheme beyond the stencil of the diffusion terms.
This discretization is now routinely used in simulations performed with the ISIS [23] code,
implemented on the basis of the software component library PELICANS [31], both freewares
being developed at the French Institut de Radioprotection et de Sûreté Nucléaire (IRSN).
It is one of the ingredient of entropy preserving schemes for the simulation of monophasic
[16] or diphasic [19] compressible barotropic flows. The assessment of its efficiency in the
context of Large Eddy Simulations is foreseen in a next future.
REFERENCES
1. P. Angot, V. Dolejšı́, M. Feistauer, and J. Felcman. Analysis of a combined barycentric
finite volume–nonconforming finite element method for nonlinear convection-diffusion problems.
Application of Mathematics, 43:263–310, 1998.
2. G. Ansanay-Alex, F. Babik, L. Gastaldo, A. Larcher, C. Lapuerta, J.-C. Latché, and D. Vola. A
finite volume stability result for the convection operator in compressible flows . . . and some finite
element applications. In Finite Volumes for Complex Applications V - Problems and Perspectives
- Aussois, France, pages 185–192, 2008.
3. B.F. Armaly, F. Durst, J.C.F. Pereira, and B. Schönung. Experimental and theoretical investigation
of backward-facing step flow. Journal of Fluid Mechanics, 127:473–496, 1983.
4. F. Babik, T. Gallouët, J.-C. Latché, S. Suard, and D. Vola. On two fractional step finite volume
and finite element schemes for reactive low Mach number flows. In Finite Volumes for Complex
Applications IV - Problems and Perspectives - Marrakech, Morocco, pages 505–514, 2005.
5. M. Braack and T. Richter. Solutions of 3D Navier-Stokes benchmark problems with adaptative
finite elements. Computers & Fluids, 35:372–392, 2006.
6. E. Burman and P. Hansbo. A stabilized non-conforming finite element method for incompressible
flow. Computer Methods in Applied Mechanics and Engineering, 195:2881–2899, 2005.
7. T.P. Chiang, Tony W.H. Sheu, and C.C. Fang. Numerical investigation of vortical evolution in a
backward-facing step expansion flow. Applied Mathematical Modelling, 23:915–932, 1999.
8. P. G. Ciarlet. Handbook of numerical analysis volume II : Finite elements methods – Basic error
estimates for elliptic problems. In P. Ciarlet and J.L. Lions, editors, Handbook of Numerical
Analysis, Volume II, pages 17–351. North Holland, 1991.
AN L2 -STABLE APPROXIMATION OF THE NAVIER–STOKES CONVECTION OPERATOR25
t = 0.5
t = 1.0
t = 1.4
t = 1.5
t = 1.6
t = 1.7
t = 1.8
t = 1.9
t = 2.0
t = 2.1
t = 2.2
t = 2.3
t = 2.4
t = 2.5
Figure 16. Density in a Rayleigh-Taylor instability flow.
9. M. Crouzeix and P.A. Raviart. Conforming and nonconforming finite element methods for solving
the stationary Stokes equations. RAIRO Série Rouge, 7:33–75, 1973.
10. V. Dolejšı́, M. Feistauer, J. Felcman, and A. Kliková. Error estimates for barycentric finite volumes
combined with nonconforming finite elements applied to nonlinear convection-diffusion problems.
Application of Mathematics, 47:301–340, 2002.
11. A. Ern and J.-L. Guermond. Theory and practice of finite elements. Number 159 in Applied
Mathematical Sciences. Springer, New York, 2004.
12. R. Eymard, T. Gallouët, R. Herbin, and J.-C. Latché. A convergent finite element-finite volume
scheme for the compressible Stokes problem. Part II: the isentropic case. to appear in Mathematics
of Computation, 2009.
13. R. Eymard, D. Hilhorst, and M. Vohralı́k. A combined finite volume–nonconforming/mixed-hybrid
finite element scheme for degenerate parabolic problems. Numerische Mathematik, 105:73–131,
2006.
14. J.H. Ferziger and M. Perić. Computational methods for fluid dynamics. Springer-Verlag, 2002.
15. Y. Fraigneau, J.-L. Guermond, and L. Quartapelle.
Approximation of variable density
incompressible flows by means of finite elements and finite volumes. Communications in Numerical
Methods in Engineering, 17:893–902, 2001.
16. T. Gallouët, L. Gastaldo, R. Herbin, and J.-C. Latché. An unconditionally stable pressure
correction scheme for compressible barotropic Navier-Stokes equations. Mathematical Modelling
and Numerical Analysis, 42:303–331, 2008.
26
G. ANSANAY-ALEX ET AL.
17. T. Gallouët, R. Herbin, and J.-C. Latché. A convergent finite element-finite volume scheme for the
compressible Stokes problem. Part I: the isothermal case. Mathematics of Computation, 267:1333–
1352, 2009.
18. L. Gastaldo, R. Herbin, and J.-C. Latché. A discretization of phase mass balance in fractional step
algorithms for the drift-flux model. to appear in IMA Journal of Numerical Analysis, 2009.
19. L. Gastaldo, R. Herbin, and J.-C. Latché. An unconditionally stable finite element-finite volume
pressure correction scheme for the drift-flux model. to appear in Mathematical Modelling and
Numerical Analysis, 2010.
20. J.-L. Guermond. Some implementations of projection methods for Navier-Stokes equations.
Mathematical Modelling and Numerical Analysis, 30(5):637–667, 1996.
21. J.-L. Guermond. Un résultat de convergence d’ordre deux en temps pour l’approximation des
équations de Navier-Stokes par une technique de projection incrémentale. Mathematical Modelling
and Numerical Analysis, 33(1):169–189, 1999.
22. F.H. Harlow and J.E.. Welsh. Numerical calculation of time-dependent viscous incompressible flow
of fluid with free surface. Physics of Fluids, 8:2182–2189, 1965.
23. ISIS. A CFD computer code for the simulation of reactive turbulent flows
. https://gforge.irsn.fr/gf/project/isis.
24. V. John. Higher order finite element methods and multigrid solvers in a benchmark problem for the
3D Navier-Stokes equations. International Journal for Numerical Methods in Fluids, 40:775–798,
2002.
25. D. Kuzmin and S. Turek. Flux-corrected transport. Principles, Algorithms and Applications.,
chapter Algebraic Flux Correction III. Incompressible Flow Problems. Springer-Verlag, 2005.
26. B. Larrouturou. How to preserve the mass fractions positivity when computing compressible multicomponent flows. Journal of Computational Physics, 95:59–84, 1991.
27. A. Majda and J. Sethian. The derivation and numerical solution of the equations for zero Mach
number solution. Combustion Science and Techniques, 42:185–205, 1985.
28. M. Marion and R. Temam. Navier-Stokes equations: Theory and approximation. In P. Ciarlet and
J.L. Lions, editors, Handbook of Numerical Analysis, Volume VI. North Holland, 1998.
29. K. Ohmori and T. Ushijima. A technique of upstream type applied to a linear nonconforming finite
element approximation of convective diffusion equations. RAIRO analyse numérique, 18:309–322,
1984.
30. A. Ouazzi and S. Turek. Unified edge-oriented stabilization of nonconforming finite element methods
for incompressible flows problems. Journal of Numerical Mathematics, 15:299–322, 2007.
31. PELICANS. Collaborative development environment.
https://gforge.irsn.fr/gf/project/pelicans.
32. R. Rannacher and S. Turek. Simple nonconforming quadrilateral Stokes element. Numerical
Methods for Partial Differential Equations, 8:97–111, 1992.
33. M. Schäfer and S. Turek. Benchmark computations of laminar flow around a cylinder. In E.H.
Hirschel, editor, Flow Simulation with High–Performance Computers II, volume 52 of Notes on
Numerical Fluid Mechanics, pages 547–566, 1996.
34. F. Schieweck and L. Tobiska. A nonconforming finite element method of upstream type applied to
the stationary Navier-Stokes equation. Mathematical Modelling and Numerical Analysis, 23:627–
647, 1989.
35. F. Schieweck and L. Tobiska. An optimal order error estimate for an upwind discretization of the
Navier-Stokes equations. Numerical Methods for Partial Differential Equations, 12:407–421, 1996.
36. J. Shen. On error estimates of projection methods for Navier-Stokes equations: First-order schemes.
SIAM Journal on Numerical Analysis, 29:57–77, 1992.
37. G. Tryggvason. Numerical simulations of the Rayleigh-Taylor instability. Journal of Computational
Physics, 75:253–282, 1988.
38. S. Turek. Efficient solvers for incompressible flow problems: an algorithmic approach in view of
computational aspects. Springer-Verlag, 1999.