IMA Journal of Numerical Analysis (2014) Page 1 of 31
doi:10.1093/imanum/drnxxx
An exactly divergence-free finite element method for a generalized
Boussinesq problem
R ICARDO OYARZ ÚA †,
GIMNAP-Departamento de Matemática, Universidad del Bı́o-Bı́o, Casilla 5-C, Concepción,
Chile, CI2 MA, Universidad de Concepción, Casilla 160-C, Concepción, Chile,
T ONG Q IN‡
Mathematics Department, University of British Columbia, Vancouver, BC, Canada, V6T 1Z2
AND
D OMINIK S CH ÖTZAU§
Mathematics Department, University of British Columbia, Vancouver, BC, Canada, V6T 1Z2.
[Received on March 2013]
IMA J. Numer. Anal., Vol. 34, pp. 1104–1135, 2014
We propose and analyze a mixed finite element method with exactly divergence-free velocities for the
numerical simulation of a generalized Boussinesq problem, describing the motion of a non-isothermal
incompressible fluid subject to a heat source. The method is based on using divergence-conforming
elements of order k for the velocities, discontinuous elements of order k − 1 for the pressure, and standard
continuous elements of order k for the discretization of the temperature. The H 1 -conformity of the
velocities is enforced by a discontinuous Galerkin approach. The resulting numerical scheme yields
exactly divergence-free velocity approximations; thus, it is provably energy-stable without the need to
modify the underlying differential equations. We prove the existence and stability of discrete solutions,
and derive optimal error estimates in the mesh size for small and smooth solutions.
Keywords: Generalized Boussinesq equations, non-isothermal incompressible flow problems, divergenceconforming elements, discontinuous Galerkin methods
1. Introduction
The numerical simulation of incompressible non-isothermal fluid flow problems has become increasingly important for the design and analysis of devices in many branches of engineering. Relevant
industrial applications include heat pipes, heat exchangers, chemical reactors, or cooling processes.
Temperature-dependent flows have also become of great interest in geophysical or oceanographic flows
with applications to weather and climate predictions.
The last decade has seen a significant interest in the development and analysis of efficient finite
element methods for such problems. We mention here only (Bernardi et al., 1995; Boland & Layton,
† Corresponding
author. Email: royarzua@ubiobio.cl
tqin@math.ubc.ca
§ Email: schoetzau@math.ubc.ca
‡ Email:
c The author 2014. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.
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R. OYARZÚA, T. QIN, AND D. SCHÖTZAU
1990a,b; Cox et al., 2007; Farhloul & Zine, 2011; Pérez et al., 2008a,b; Tabata & Tagami, 2005) and
the references therein. In particular, in (Pérez et al., 2008b) a conforming method is presented and
analyzed for approximating non-isothermal incompressible fluid flow problems. However, the analysis
there hinges on technical assumptions which may be difficult to verify in practice. The work Tabata &
Tagami (2005) studies a finite element method for time-dependent non-isothermal incompressible fluid
flow problems. Here, the governing equations are discretized by the backward Euler method in time and
conforming finite elements in space.
In this paper, we propose an alternative approach for the numerical approximation (in space) of
a non-isothermal flow problem. As a model problem, we consider the generalized Boussinesq model
analyzed theoretically in Lorca & Boldrini (1996): it couples the stationary incompressible NavierStokes equations for the fluid variables (velocity and pressure) with a convection-diffusion equation for
the temperature variable. The coupling is non-linear through a temperature-dependent viscosity, and
through a buoyancy term typically acting in direction opposite to gravity.
Following Cockburn et al. (2007), we employ divergence-conforming Brezzi-Douglas-Marini (BDM)
elements of order k for the approximation of the velocity, discontinuous elements of order k − 1 for the
pressure, and continuous elements of order k for the temperature. To enforce H 1 -continuity of the velocities, we use an interior penalty discontinuous Galerkin (DG) technique. The resulting mixed finite
element method has the distinct property that it yields exactly divergence-free velocity approximations.
Thus, it exactly preserves an essential constraint of the governing equations and is provably energystable without the need for symmetrization of the convective discretization; see Cockburn et al. (2005,
2007). We also refer to Linke (2009) for a discussion on the importance of exact mass conservation of
colliding flows in a cross-shaped domain.
We show the existence and stability of discrete solutions by mimicking the fixed point arguments
presented in Lorca & Boldrini (1996) for the continuous problem. A crucial aspect of this argument is
the construction of a suitable lifting of the temperature boundary data into the computational domain.
On the discrete level, this is a delicate manner, as the numerical construction of discrete liftings may
be computationally expensive. One option is to choose the discrete harmonic extension of the discrete
boundary datum, which requires one elliptic solve. However, in our theoretical analysis, this comes at
the cost of a relatively strict small data assumption. We also discuss the most practical choice of straightforward nodal interpolation, which seems to work fine in our (non-exhaustive) numerical experiments.
We then derive optimal error estimates for problems with small and sufficiently smooth solutions. In
particular, we show that the velocity errors in the DG energy norm, the pressure errors in the L2 -norm,
and the temperature errors in the H 1 -norm converge of order O(hk ) in the mesh size h. This convergence
rates are numerically confirmed for a test problem with a smooth solution.
The rest of the paper is structured as follows. In Section 2, we introduce a generalized Boussinesq
model problem, and review the results from Lorca & Boldrini (1996) regarding existence and uniqueness
of solutions. In Section 3, we present our finite element discretization, and review the stability properties of the discrete formulation. In Section 4, we establish the existence and stability of approximate
solutions under a small data assumption. In Section 5, we state and prove our a-priori error estimates.
In Section 6, we present numerical results for a test problem with a smooth solution. We end the paper
with concluding remarks in Section 7.
We end this section by fixing some notation. To that end, let O be a domain in Rd , d = 2, 3, with
Lipschitz boundary ∂ O. We write C(O), C(∂ O) for the standard spaces of continuous functions, and
k · kC(O) , k · kC(∂ Ω ) for the associated maximum norms. For r > 0 and p ∈ [1, ∞], we denote by L p (O)
and W r,p (O) the usual Lebesgue and Sobolev spaces endowed with the norms k · kL p (O) and k · kW r,p (O) ,
EXACTLY DIVERGENCE-FREE FEM FOR A GENERALIZED BOUSSINESQ PROBLEM
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respectively. Note that W 0,p (O) = L p (O). If p = 2, we write H r (O) in place of W r,2 (O), and denote
the corresponding Lebesgue and Sobolev norms by k · k0,O and k · kr,O , respectively. For r > 0, we
write | · |r,O for the H r -seminorm. The space H01 (O) is the space of functions in H 1 (O) with vanishing
trace on Γ , and L02 (O) is the space of L2 -functions with vanishing mean value over O. Spaces of vectorvalued functions are denoted in bold face. For example, Hr (O) = [H r (O)]d for r > 0. For simplicity, we
also write k · kr,O and | · |r,O for the corresponding norms and seminorms on these spaces. Furthermore,
we will use the vector-valued Hilbert spaces
H(div ; O) = w ∈ L2 (O) : div w ∈ L2 (O) ,
H0 (div ; O) = w ∈ H(div ; O) : w · n∂ O = 0 on ∂ O ,
(1.1)
H0 (div 0 ; O) = w ∈ H0 (div ; O) : div w ≡ 0 in Ω ,
with nO denoting the unit outward normal on ∂ O. These spaces are endowed with the norm
kwk2div,O = kwk20,O + kdiv wk20,O .
In the subsequent analysis, we denote by C∞ > 0 the embedding constant such that
kuk1,O 6 C∞ kukW1,∞ (O) ,
kθ k1,O 6 C∞ kθ kW 1,∞ (O) ,
(1.2)
for all u ∈ W1,∞ (O) and θ ∈ W 1,∞ (O). Finally, we shall frequently use the notation C and c, with
or without subscripts, bars, tildes or hats, to denote generic positive constants independent of the discretization parameters.
2. Weak formulation of a generalized Boussinesq problem
In this section, we introduce a model problem, cast it into weak form, discuss the stability properties
of the forms involved, and review some theoretical properties regarding existence and uniqueness of
solutions.
2.1
Model problem
We consider the stationary generalized Boussinesq problem analyzed theoretically in Lorca & Boldrini
(1996). The governing partial differential equations then are given by
−div(ν(θ )∇u) + (u · ∇)u + ∇ p − g θ = 0
in Ω ,
(2.1)
div u = 0
in Ω ,
(2.2)
−div (κ(θ ) ∇θ ) + u · ∇θ = 0
in Ω ,
(2.3)
u=0
on Γ ,
(2.4)
θ = θD
on Γ .
(2.5)
Here, Ω is a polygon or polyhedron in Rd , d = 2, 3 with Lipschitz boundary Γ = ∂ Ω . The unknowns
are the fluid velocity u, the pressure p, and the temperature θ . The given data are the non-vanishing
boundary temperature θD ∈ H 1/2 (Γ ), and the external force per unit mass g ∈ L2 (Ω ), usually acting in
direction opposite to gravity. We assume that
θD ∈ C(Γ ),
(2.6)
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R. OYARZÚA, T. QIN, AND D. SCHÖTZAU
so that nodal interpolation of θD is well defined.
The functions ν(·) and κ(·) are the fluid viscosity and the thermal conductivity, respectively. We
assume that ν and κ are Lipschitz continuous and satisfy
|ν(θ1 ) − ν(θ2 )| 6 νlip |θ1 − θ2 |,
|κ(θ1 ) − κ(θ2 )| 6 κlip |θ1 − θ2 |,
(2.7)
for all values of θ1 , θ2 , with Lipschitz constants νlip , κlip > 0. Moreover, we suppose that ν and κ are
bounded from above and from below, that is, there are positive constants such that
0 < ν1 6 ν(θ ) 6 ν2 ,
0 < κ1 6 κ(θ ) 6 κ2 ,
(2.8)
for all values of θ .
The variational formulation of problem (2.1)–(2.5) amounts to finding (u, p, θ ) ∈ H10 (Ω ) × L02 (Ω ) ×
1
H (Ω ) such that θ |Γ = θD and
AS (θ ; u, v) + OS (u; u, v) − B(v, p) − D(θ , v) = 0,
B(u, q) = 0,
(2.9)
AT (θ ; θ , ψ) + OT (u; θ , ψ) = 0,
for all (v, q, ψ) ∈ H10 (Ω ) × L02 (Ω ) × H01 (Ω ). Here, the forms are given by
Z
AS (ψ; u, v) =
Z
ν(ψ) ∇u : ∇v,
OS (w; u, v) =
Ω
Z
AT (ϕ; θ , ψ) =
κ(ϕ)∇θ · ∇ψ,
(v · ∇θ )ψ,
(2.11)
Z
Ω
Z
Z
q div v,
Ω
2.2
(2.10)
OT (v; θ , ψ) =
Ω
B(v, q) =
((w · ∇)u) · v,
Ω
θ g · v.
D(θ , v) =
(2.12)
Ω
Stability
Next, let us discuss the stability properties of the forms appearing in (2.9).
We start by discussing boundedness of the forms. Due to the bounds (2.8), the following continuity
properties hold:
|AS (·; u, v)| 6 ν2 kuk1,Ω kvk1,Ω ,
|AT (·; θ , ψ)| 6 κ2 kθ k1,Ω kψk1,Ω ,
|B(v, q)| 6 CB kvk1,Ω kqk0,Ω ,
u, v ∈ H1 (Ω ),
(2.13)
1
θ , ψ ∈ H (Ω ),
(2.14)
v ∈ H1 (Ω ), q ∈ L2 (Ω ).
(2.15)
Moreover, from the Lipschitz continuity of ν and κ in (2.7) and Hölder’s inequality it readily follows
that, for θ1 , θ2 ∈ H 1 (Ω ), u ∈ W1,∞ (Ω ), θ ∈ W 1,∞ (Ω ),
|AS (θ1 ; u, v) − AS (θ2 ; u, v)| 6 νlip kukW1,∞ (Ω ) kθ1 − θ2 k1,Ω kvk1,Ω ,
|AT (θ1 ; θ , ψ) − AT (θ2 ; θ , ψ)| 6 κlip kθ kW 1,∞ (Ω ) kθ1 − θ2 k1,Ω kψk1,Ω ,
v ∈ H1 (Ω ),
1
ψ ∈ H (Ω ).
(2.16)
(2.17)
EXACTLY DIVERGENCE-FREE FEM FOR A GENERALIZED BOUSSINESQ PROBLEM
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The forms OS and OT are linear in each argument. Hölder’s inequality and standard Sobolev embeddings then give the following bounds:
|OS (w; u, v)| 6 CS kwk1,Ω kuk1,Ω kvk1,Ω ,
|OT (w; θ , ψ)| 6 CT kwk1,Ω kθ k1,Ω kψk1,Ω ,
w, u, v ∈ H1 (Ω ),
(2.18)
w ∈ H1 (Ω ), θ , ψ ∈ H 1 (Ω ).
(2.19)
Similarly, we have
|D(θ , v)| 6 CD kgk0,Ω kθ k1,Ω kvk1,Ω ,
θ ∈ H 1 (Ω ), v ∈ H1 (Ω ).
(2.20)
Next, we review the positivity properties of the forms in (2.10) and (2.11). By the Poincaré inequality
and the bounds (2.8), the elliptic forms AS and AT are coercive:
|AS (·; v, v)| > αS kvk21,Ω ,
v ∈ H10 (Ω ),
(2.21)
|AT (·; ψ, ψ)| > αT kψk21,Ω ,
ψ ∈ H01 (Ω ).
(2.22)
To discuss the convective form OS and OT , we introduce the kernel
X = v ∈ H10 (Ω ) : B(v, q) = 0 ∀ q ∈ L02 (Ω ) = v ∈ H10 (Ω ) : div v ≡ 0 in Ω .
(2.23)
Clearly, X ⊂ H0 (div 0 ; Ω ). Then, integration by parts shows that,
OS (w; v, v) = 0,
w ∈ X, v ∈ H1 (Ω ),
(2.24)
OT (w; ψ, ψ) = 0,
w ∈ X, ψ ∈ H 1 (Ω ).
(2.25)
Finally, the bilinear form B satisfies the continuous inf-sup condition
B(v, q)
> β kqk0,Ω ,
v∈H1 (Ω )\{0} kvk1,Ω
sup
∀ q ∈ L02 (Ω ),
(2.26)
0
with an inf-sup constant β > 0 only depending on Ω ; see Girault & Raviart (1986), for instance.
2.3
Results concerning existence and uniqueness
In this section, we review some results regarding the existence and uniqueness of solutions of (2.9). To
that end, it is enough to study the reduced problem of (2.9) on the kernel X. in (2.23). It consists in
finding (u, θ ) ∈ X × H 1 (Ω ) such that θ |Γ = θD and
AS (θ ; u, v) + OS (u; u, v) − D(θ , v) = 0,
AT (θ ; θ , ψ) + OT (u; θ , ψ) = 0,
(2.27)
for all (v, ψ) ∈ X × H01 (Ω ).
The following equivalence property is standard; see Girault & Raviart (1986).
L EMMA 2.1 If (u, p, θ ) ∈ H10 (Ω ) × L02 (Ω ) × H 1 (Ω ) is a solution of (2.9), then u ∈ X and (u, θ ) is also
a solution of (2.27). Conversely, if (u, θ ) ∈ X × H 1 (Ω ) is a solution of (2.27), then there exists a unique
pressure p ∈ L02 (Ω ) such that (u, p, θ ) is a solution of (2.9).
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R. OYARZÚA, T. QIN, AND D. SCHÖTZAU
The following existence result for the reduced problem (2.27) is proved in (Lorca & Boldrini, 1996,
Theorem 2.1). To state it, we write the temperature θ as
θ = θ0 + θ1 ,
(2.28)
where θ0 ∈ H01 (Ω ) and θ1 is such that
θ1 ∈ H 1 (Ω ),
θ1 |Γ = θD .
(2.29)
T HEOREM 2.1 Assume (2.7) and (2.8). Then, for any g ∈ L2 (Ω ), there is a lifting θ1 ∈ H 1 (Ω ) of
θD ∈ H 1/2 (Γ ) satisfying (2.29) such that the reduced problem (2.27) has a solution (u, θ = θ0 + θ1 ) ∈
H10 (Ω ) × H 1 (Ω ). Furthermore, there exist constants Cu and Cθ only depending on kgk0,Ω , and the
stability constants in Section 2.2, such that
kuk1,Ω 6 Cu kθ1 k1,Ω ,
kθ k1,Ω 6 Cθ kθ1 k1,Ω .
(2.30)
The work (Lorca & Boldrini, 1996, Section 7) also establishes the uniqueness of small solutions
to problem (2.27), albeit under additional smoothness assumptions on the domain. Here, we restrict
ourselves to proving the following (more straightforward) uniqueness result, whose proof is motivated
by a similar argument in Cox et al. (2007) for Stokes-Oldroyd problems.
T HEOREM 2.2 Let (u, θ ) ∈ X ∩ W1,∞ (Ω ) × W 1,∞ (Ω ) be a solution to problem (2.27), and assume
that there exists a sufficiently small constant M > 0 such that
max{kgk0,Ω , kukW1,∞ (Ω ) , kθ kW 1,∞ (Ω ) } 6 M.
(2.31)
Then, the solution is unique. (A precise condition on M can be found in (2.43).)
Proof. Let (u, θ ) and (u? , θ ? ) be two solutions of problem (2.27), both satisfying assumption (2.31).
By subtracting the two corresponding variational formulations from each other, it follows that
[AS (θ ; u, v) − AS (θ ? ; u? , v)] + [OS (u; u, v) − OS (u? ; u? , v)] − D(θ − θ ? , v) = 0,
(2.32)
[AT (θ ; θ , ψ) − AT (θ ? ; θ ? , ψ)] + [OT (u; θ , ψ) − OT (u? ; θ ? , ψ)] = 0,
(2.33)
and
∈ H01 (Ω ).
for all v ∈ X and ψ
In (2.32), we write
[AS (θ ; u, v) − AS (θ ? ; u? , v)] = AS (θ ; u − u? , v) + [AS (θ ; u? , v) − AS (θ ? ; u? , v)],
[OS (u; u, v) − OS (u? ; u? , v)] = OS (u; u − u? , v) + OS (u − u? ; u? , v).
(2.34)
Similarly, in (2.33),
[AT (θ ; θ , ψ) − AT (θ ? ; θ ? , ψ)] = AT (θ ; θ − θ ? , ψ) + [AT (θ ; θ ? , ψ) − AT (θ ? ; θ ? , ψ)],
[OT (u; θ , ψ) − OT (u? ; θ ? , ψ)] = OT (u; θ − θ ? , ψ) + OT (u − u? ; θ ? , ψ).
(2.35)
Then, by choosing the test function v = u − u? ∈ X in (2.32), and using (2.34), the coercivity property (2.21), and the fact that OS (u; u − u? , u − u? ) = 0, see (2.24), we obtain
αS ku − u? k21,Ω 6 |AS (θ ; u? , u − u? ) − AS (θ ? ; u? , u − u? )|
+ |OS (u − u? ; u? , u − u? )| + |D(θ − θ ? , u − u? )|.
(2.36)
EXACTLY DIVERGENCE-FREE FEM FOR A GENERALIZED BOUSSINESQ PROBLEM
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Analogously, by taking ψ = θ − θ ? ∈ H01 (Ω ) in (2.33), and using (2.35), the coercivity (2.22) for AT ,
and the fact that OT (u; θ − θ ? , θ − θ ? ) = 0, cf. (2.25), we find that
αT kθ − θ ? k21,Ω 6 |AT (θ ; θ ? , θ − θ ? ) − AT (θ ? ; θ ? , θ − θ ? )| + |OT (u − u? ; θ ? , θ − θ ? )|.
(2.37)
From (2.16) and (2.17) and since ku? kW1,∞ (Ω ) 6 M and kθ ? kW 1,∞ (Ω ) 6 M by assumption (2.31), the
right-hand sides in (2.36) and (2.37) can be bounded by
|AS (θ ; u? , u − u? ) − AS (θ ? ; u? , u − u? )| 6 νlip Mkθ − θ ? k1,Ω ku − u? k1,Ω ,
(2.38)
and
|AT (θ ; θ ? , θ − θ ? ) − AT (θ ? ; θ ? , θ − θ ? )| 6 κlip Mkθ − θ ? k21,Ω ,
respectively. Hence, by using these inequalities in (2.36) and (2.37), respectively, and the continuity of
OS , OT , D, we find that
αS ku − u? k21,Ω 6νlip Mkθ − θ ? k1,Ω ku − u? k1,Ω + CS ku? k1,Ω ku − u? k21,Ω
+ CD kgk0,Ω kθ − θ ? k1,Ω ku − u? k1,Ω ,
(2.39)
as well as
αT kθ − θ ? k21,Ω 6 κlip Mkθ − θ ? k21,Ω +CT ku − u? k1,Ω kθ ? k1,Ω kθ − θ ? k1,Ω .
(2.40)
We continue bounding the right-hand sides of (2.39) and (2.40) by applying the embedding esti2
2
mate (1.2), assumption (2.31), and the inequality |ab| 6 a2 + b2 . This results in
αS ku − u? k21,Ω 6M(CD + νlip )kθ − θ ? k1,Ω ku − u? k1,Ω +CSC∞ Mku − u? k21,Ω
6M(CSC∞ +
CD νlip
M
+
)ku − u? k21,Ω + (CD + νlip )kθ − θ ? k21,Ω ,
2
2
2
(2.41)
respectively,
αT kθ − θ ? k21,Ω 6κlip Mkθ − θ ? k21,Ω +CTC∞ Mku − u? k1,Ω kθ − θ ? k1,Ω
M
CTC∞ 6M κlip +
kθ − θ ? k21,Ω + CTC∞ ku − u? k21,Ω .
2
2
(2.42)
Finally, adding up (2.41) and (2.42), and bringing all the terms to the left-hand side of the resulting
inequality, we conclude that
αS − M(CSC∞ + K) ku − u? k21,Ω + αT − M(κlip + K) kθ − θ ? k21,Ω 6 0,
with K := (CTC∞ +CD + νlip )/2. Thus, if M satisfies
M < min
n
αT o
αS
,
,
CSC∞ + K κlip + K
then θ = θ ? and u = u? . This completes the proof.
(2.43)
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3. Finite element discretization
In this section, we introduce our finite element method for approximating problem (2.1)–(2.5), review
the discrete stability properties of the forms involved, and discuss the reduced version of the discrete
variational problem.
3.1
Preliminaries
We consider a family of regular and shape-regular triangulations Th of mesh size h that partition the
domain Ω into simplices {K} (i.e., triangles for d = 2 and tetrahedra for d = 3). For each K we denote
by nK the unit outward normal vector on the boundary ∂ K, and by hK the elemental diameter. As usual,
we define the mesh size by h = maxK∈Th hK . We denote by EI (Th ) the set of all interior edges (faces)
of Th , by EB (Th ) the set of all boundary edges (faces), and define Eh (Th ) = EI (Th ) ∪ EB (Th ). The
(d − 1)-dimensional diameter of an edge (face) e is denoted by he .
We will use standard average and jump operators. To define them, let K + and K − be two adjacent
elements of Th , and e = ∂ K + ∩ ∂ K − ∈ EI (Th ). Let u and τ be a piecewise smooth vector-valued,
respectively matrix-valued function, and let us denote by u± , τ ± the traces of u, τ on e, taken from
within the interior of K ± . Then, we define the jump of u, respectively the mean value of τ at x ∈ e by
1
{{τ}} = (τ + + τ − ),
2
JuK = u+ ⊗ nK + + u− ⊗ nK − ,
(3.1)
where for u = (u1 , ..., ud ) and n = (n1 , ..., nd ), we denote by u ⊗ n the tensor product matrix [u ⊗ n]i, j =
ui n j , 1 6 i, j 6 d. For a boundary edge (face) e = ∂ K + ∩ Γ , we set JuK = u+ ⊗ n, with n denoting the
unit outward normal vector on Γ , and {{τ}} = τ + .
3.2
Exactly divergence-free finite element approximation
For an approximation order k > 1 and a mesh Th on Ω , we consider the discrete spaces
n
o
Vh = v ∈ H0 (div ; Ω ) : v|K ∈ [Pk (K)]d , K ∈ Th ,
Qh = q ∈ L02 (Ω ) : q|K ∈ Pk−1 (K), K ∈ Th ,
Ψh = ψ ∈ C (Ω ) : ψ|K ∈ Pk (K), K ∈ Th ,
(3.2)
Ψh,0 = Ψh ∩ H01 (Ω ).
Here, the space Pk (K) denotes the usual space of polynomials of total degree less or equal than k on
element K. The space Vh is non-conforming in H10 (Ω ), while Qh and Ψh are conforming in L02 (Ω )
and H 1 (Ω ), respectively. In fact, the space Vh is the space of divergence-conforming Brezzi-DouglasMarini (BDM) elements; see Brezzi & Fortin (1991).
Consistent with our choice (3.2) for the discrete spaces, we need to introduce discontinuous versions
of AS and OS , respectively. For the discrete vector Laplacian, we take the interior penalty form Arnold
(1982); Arnold et al. (2002) given by
AhS (ψ; u, v) =
Z
ν(ψ) ∇h u : ∇h v −
Ω
−
Z
∑
e∈Eh (Th ) e
Z
∑
e∈Eh (Th ) e
{{ν(ψ)∇h v}} : JuK +
{{ν(ψ)∇h u}} : JvK
∑
e∈Eh (Th
a0
h
) e
(3.3)
Z
e
ν(ψ)JuK : JvK.
EXACTLY DIVERGENCE-FREE FEM FOR A GENERALIZED BOUSSINESQ PROBLEM
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Here, a0 > 0 is the interior penalty parameter, and we denote by ∇h the broken gradient operator. As
discussed in Cockburn et al. (2007), other choices for AhS are equally feasible (such as LDG or BR
methods), provided that the stability properties in Section 3.3 below hold.
For the convection term, we take the standard upwind form LeSaint & Raviart (1974) defined by
OhS (w; u, v) =
Z
(w · ∇h )u · v +
Ω
Z
∑
K∈Th ∂ K\Γ
1
(w · nK − |w · nK |)(ue − u) · v,
2
(3.4)
where ue is the trace of u taken from within the exterior of K. We note that convective forms with
no upwinding can also be chosen in our setting, such as the trilinear form in (Di Pietro & Ern, 2012,
Section 6).
The remaining forms are the same as in the continuous case.
Next, we introduce an approximation θD,h to the boundary datum θD , which we take in the trace
space
θD,h ∈ Λh = { ξ ∈ C(Γ ) : ξ |e ∈ Pk (e), e ∈ EB (Th ) }.
(3.5)
Then the discrete formulation for problem (2.1)–(2.5) is to find (uh , ph , θh ) ∈ Vh × Qh ×Ψh such
that θh |Γ = θD,h and
AhS (θh ; uh , v) + OhS (uh ; uh , v) − B(v, ph ) − D(θh , v) = 0,
B(uh , q) = 0,
(3.6)
AT (θh ; θh , ψ) + OT (uh ; θh , ψ) = 0,
for all (v, q, ψ) ∈ Vh × Qh ×Ψh,0 .
A key feature of the method (3.6) is that the discrete velocity uh is exactly divergence-free. To
discuss this property, we introduce the discrete kernel of B
Xh = { v ∈ Vh : B(v, q) = 0 ∀ q ∈ Qh } .
(3.7)
Since Vh ⊂ H0 (div ; Ω ) and div Vh ⊆ Qh , it can be readily seen that
Xh = { v ∈ Vh : div v ≡ 0 in Ω } ;
we refer to Cockburn et al. (2007) for details. Hence, Xh ⊂ H0 (div 0 ; Ω ). In particular, the following
result holds.
L EMMA 3.1 An approximate velocity uh ∈ Vh obtained by (3.6) is exactly divergence-free, i.e., it
satisfies div uh ≡ 0 in Ω .
An important consequence of Lemma 3.1 is the provable energy-stability of the numerical scheme
in (3.6), without the need for symmetrization or other modifications of the convective terms; see also
the discusssion in Cockburn et al. (2005, 2007). These stability properties are established in the next
subsection.
3.3
3.3.1
Discrete stability properties
Broken spaces and norms. We introduce the broken space
Hr (Th ) = { v ∈ L2 (Ω ) : v|K ∈ Hr (K), K ∈ Th },
r > 0.
(3.8)
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We shall mostly work with r = 1 and r = 2; in these cases we use the broken norms
kvk21,Th =
∑
k∇h vk20,K +
K∈Th
2
∑ a0 h−1
e kJvKk0,e ,
v ∈ H1 (Th ),
(3.9)
v ∈ H2 (Th ).
(3.10)
e∈Eh
kvk22,Th = kvk21,Th +
∑
h2K |v|22,K ,
K∈Th
By the inverse estimate |p|2,K 6 Ch−1
K |p|1,K for all K ∈ Th , p ∈ Pk (K), we see that
kvk2,Th 6 Ckvk1,Th ,
v ∈ Vh .
(3.11)
We recall the following broken version of the usual Sobolev embeddings: for d = 2, 3, and any
p ∈ I(d) ⊂ R there exists a constant C > 0 such that
kvkL p (Ω ) 6 Ckvk1,Th ,
v ∈ H1 (Th ),
(3.12)
where I(2) = [1, ∞) and I ? (3) = [1, 6]. For d = 2, this has been proved in (Girault et al., 2005,
Lemma 6.2). In the case d = 3, the proof follows along the lines of to (Waluga, 2012, Lemma 5.15,
Theorem 5.16). In the following, we shall explicitly write Cemb for the embedding constant in the case
p = 3.
Moreover, we introduce the broken C1 -space given by
C1 (Th ) = u ∈ H1 (Th ) : u|K ∈ C1 (K), K ∈ Th ,
(3.13)
equipped with the broken W 1,∞ -norm
kukW1,∞ (Th ) = max kukW1,∞ (K) .
(3.14)
K∈Th
We shall also make use of the augmented H 1 -norm
kψk21,Eh = kψk21,Ω +
∑
2
h−1
e kψk0,e ,
ψ ∈ H 1 (Ω ).
(3.15)
e∈Eh (Th )
3.3.2 Continuity. First, we establish continuity properties of the elliptic forms AhS and AT , respectively. To that end, we recall that by (2.14), the form AT is a bounded bilinear form over H 1 (Ω ) ×
H 1 (Ω ). To bound the DG form AhS , we proceed in a standard way; see Arnold et al. (2002), for instance.
Indeed, by using the standard trace inequalities
−1/2
1/2
v ∈ H 1 (K),
(3.16)
kvk0,∂ K 6 C hK kvk0,K + hK |v|1,K ,
−1/2
kpk0,∂ K 6 ChK
kpk0,K ,
p ∈ Pk (K),
(3.17)
and the inverse inequality in (3.11), we obtain the following result.
L EMMA 3.2 There holds
|AhS (·; u, v)| 6 Ckuk2,Th kvk1,Th ,
u ∈ H2 (Th ), v ∈ Vh ,
(3.18)
|AhS (·; u, v)| 6 C̃A kuk1,Th kvk1,Th ,
u, v ∈ Vh .
(3.19)
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EXACTLY DIVERGENCE-FREE FEM FOR A GENERALIZED BOUSSINESQ PROBLEM
Moreover, the elliptic forms are Lipschitz continuons with respect to the first argument. For the
conforming form AT , this follows from (2.17). The following result holds for the DG form AhS .
L EMMA 3.3 Let ψ1 , ψ2 ∈ H 1 (Ω ), u ∈ C1 (Th ), and v ∈ Vh . Then there holds
h
AS (ψ1 ; u, v) − AhS (ψ2 ; u, v) 6 C̃lip νlip kψ1 − ψ2 k1,Eh kukW1,∞ (Th ) kvk1,Th .
(3.20)
In addition, if u ∈ H10 (Ω ), then
h
AS (ψ1 ; u, v) − AhS (ψ2 ; u, v) 6 C̃lip νlip kψ1 − ψ2 k1,Ω kukW1,∞ (Th ) kvk1,Th .
(3.21)
The constant C̃lip > 0 is independent of the mesh size.
Proof. As before, we note that
h
AS (ψ1 ; u, v) − AhS (ψ2 ; u, v) 6 |T1 | + |T2 | + |T3 | + |T4 |,
with
Z
Z
(ν(ψ1 ) − ν(ψ2 ))∇h u : ∇h v,
T1 =
T2 =
Ω
Z
T3 =
∑
e∈Eh (Th ) e
(ν(ψ1 ) − ν(ψ2 )){{∇v}} : JuK,
T4 =
∑
e∈Eh (Th ) e
∑
e∈Eh (Th
(ν(ψ1 ) − ν(ψ2 )){{∇u}} : JvK,
a0
h
) e
Z
e
(ν(ψ1 ) − ν(ψ2 ))JuK : JvK.
For T1 , the Lipschitz continuity of ν in (2.7) readily yields the bound
|T1 | 6 νlip kψ1 − ψ2 k0,Ω kukW1,∞ (Th ) k∇h vk0,Ω .
To estimate T2 , we notice that, since u ∈ C1 (Th ), we have k{{∇h u}}e kL∞ (e) 6 kukW1,∞ (Th ) for all
e ∈ Eh (Th ). Hence, from the Lipschitz continuity of ν it follows that
|T2 | 6 νlip kukW1,∞ (Th )
∑
kψ1 − ψ2 k0,e kJvKk0,e .
e∈Eh (Th )
By applying the discrete Cauchy-Schwarz inequality, the shape-regularity of the meshes, and the trace
inequality (3.16), the sum over the edges (faces) can be bounded by
∑
e∈Eh (Th )
kψ1 − ψ2 k0,e kJvKk0,e 6
he kψ1 − ψ2 k20,e
∑
1/2
e∈Eh (Th )
6C
∑
hK kψ1 − ψ2 k20,∂ K
∑
e∈Eh (Th )
1/2
kvk1,Th
K∈Th
6 Ckψ1 − ψ2 k1,Ω kvk1,Th .
This yields
|T2 | 6 Cνlip kukW1,∞ (Th ) kψ1 − ψ2 k1,Ω kvk1,Th .
2
h−1
e kJvKk0,e
1/2
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For the term T3 , we have kJuKkL∞ (e) 6 2kukL∞ (Ω ) 6 2kukW1,∞ (Th ) for any e ∈ Eh (Th ). Hence, the
Lipschitz continuity of ν, the Cauchy-Schwarz inequality the shape-regularity of the meshes, and the
polynomial trace inequality (3.17),
|T3 | 6 Cνlip kukW1,∞ (Th )
∑
kψ1 − ψ2 k0,e k{{∇v}}k0,e
e∈Eh (Th )
6 Cνlip kukW1,∞ (Th )
∑
2
h−1
e kψ1 − ψ2 k0,e
1/2
∑
hK k∇vk20,∂ K
1/2
K∈Th
e∈Eh (Th )
6 Cνlip kukW1,∞ (Th ) kψ1 − ψ2 k1,Eh k∇h vk0,Ω .
Similarly, T4 can be bounded by:
|T4 | 6 Cνlip kukW1,∞ (Th ) kψ1 − ψ2 k1,Eh kvk1,Th .
Gathering the above bounds for T1 through T4 implies the estimate (3.20).
If u ∈ H10 (Ω ), then T3 = T4 = 0, and the second bound (3.21) follows from the estimates for T1
and T2 .
Second, we notice that the forms B and D are bounded by
|B(v, q)| 6 C̃B kvk1,Th kqk0,Ω ,
|D(ψ, v)| 6 C̃D kgk0,Ω kψk1,Ω kvk1,Th ,
v ∈ H1 (Th ), q ∈ L02 (Ω ),
1
1
v ∈ H (Th ), ψ ∈ H (Ω ).
(3.22)
(3.23)
The estimate for B is straightforward, and the one for D follows from the embedding (3.12) with p = 4
and Hölder’s inequality.
Third, we discuss the convective forms OhS and OT , respectively. In contrast to OS and due to the
upwind terms, the discrete form OhS is not linear in the first argument. However, as established in the
following lemma, it is Lipschitz continuous.
L EMMA 3.4 There exists a constant C̃S > 0, independent of the mesh size, such that
|OhS (w1 ; u, v) − OhS (w2 ; u, v)| 6 C̃S kw1 − w2 k1,Th kuk1,Th kvk1,Th ,
(3.24)
for any w1 , w2 , u ∈ H2 (Th ) and v ∈ Vh .
Proof. The proof of this property in the case d = 2 can be found in Cockburn et al. (2005), and
makes use of the embedding (3.12) with p = 4. In the case d = 3, we proceed similarly: we use the
shape-regularity of the meshes, Hölder’s inequality, the embedding (3.12) with p = 4, and the trace
1/4
estimate hK kzkL4 (∂ K) 6 C kzkL4 (K) + k∇zkL2 (K) , z ∈ W 1,4 (K), from (Karakashian & Jureidini, 1998,
Section 7). We omit further details.
The conforming temperature form OT is still trilinear, and there holds
|OT (w; ϕ, ψ)| 6 C̃T kwk1,Th kϕk1,Ω kψk1,Ω ,
w ∈ H1 (Th ), ψ, ϕ ∈ H 1 (Ω ).
(3.25)
This follows similarly from Hölder’s inequality and the embedding (3.12). We use the following variant
of (3.25).
L EMMA 3.5 There is a constant C̃T,2 > 0 such that
|OT (w; θ , ψ)| 6 C̃T,2 kθ kL3 (Ω ) kwk1,Th kψk1,Ω ,
w ∈ H0 (div 0 ; Ω ), θ , ψ ∈ H 1 (Ω ).
(3.26)
EXACTLY DIVERGENCE-FREE FEM FOR A GENERALIZED BOUSSINESQ PROBLEM
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Proof. Integration by parts yields and using that div w ≡ 0 in Ω , w · n = 0 on Γ yield
Z
(w · ∇θ )ψ = −
OT (w; θ , ψ) =
Ω
Z
θ (w · ∇ψ).
Ω
From Hölder’s inequality we obtain
|OT (w; θ , ψ)| 6 Ckθ kL3 (Ω ) k∇ψk0,Ω kwkL6 (Ω ) .
Hence, the embeddings in (3.12) with p = 3, p = 6 yield the assertion.
3.3.3 Coercivity and inf-sup condition. First, we point out that coercivity of AT over the discrete
spaces is implied by (2.22). Due to the bounds of ν in (2.8) the DG form AhS is also elliptic, and we have
AhS (·, v, v) > α̃S kvk21,Th ,
v ∈ Vh ,
(3.27)
provided that a0 > 0 is sufficiently large independently of the mesh size; cf. Arnold et al. (2002).
To state the positivity of OhS and OT , let w ∈ H0 (div 0 ; Ω ). Then we have
OhS (w; u, u) =
1
2 e∈E∑(T
I
Z
e
h)
|w · n||Ju ⊗ nK|2 ds > 0,
u ∈ Vh .
(3.28)
Here, in the integrals over edges (faces) e, the vector n denotes any unit vector normal to e. This is a
standard property of the upwind form OS , see, e.g., LeSaint & Raviart (1974); Cockburn et al. (2005).
Moreover, integration by parts readily implies that
OT (w; θ , θ ) = 0,
θ ∈ H 1 (Ω ).
(3.29)
Finally, we recall the discrete inf-sup condition for B:
B(vh , qh )
> β̃ kqh k0,Ω
vh ∈Vh \{0} kvh k1,Th
sup
∀ qh ∈ Qh ,
(3.30)
with β̃ > 0 independent of the mesh size. The proof of (3.30) follows along the lines of Hansbo &
Larson (2002) from the surjectivity of div : H10 (Ω ) → L02 (Ω ) and the properties of the BDM projection.
We omit further details.
3.4 The reduced problem
The reduced version of (3.6) consists in finding (uh , θh ) ∈ Xh ×Ψh such that θh |Γ = θD,h and
AhS (θh ; uh , v) + OS (uh ; uh , v) − D(θh , v) = 0,
AT (θh ; θh , ψ) + OT (uh ; θh , ψ) = 0,
(3.31)
for all (v, ψ) ∈ Xh ×Ψh,0 .
Due to the discrete stability properties of Section 3.3, the discrete analog of Lemma 2.1 hold.
L EMMA 3.6 If (uh , ph , θh ) ∈ Vh × Qh ×Ψh is a solution of (3.6), then uh ∈ Xh and (uh , θh ) is also a
solution of (3.31). Conversely, if (uh , θh ) ∈ Xh ×Ψh is a solution of (3.31), then there exists a unique
pressure ph ∈ Qh such that (uh , ph , θh ) is a solution of (3.6).
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In what follows, we shall discuss the existence for the reduced problem (3.31). We notice that the
uniqueness of discrete solutions is an open issue. Indeed, adapting Theorem 2.2 to the discrete setting
requires controlling the augmented norm (3.15) appearing in the discrete counterpart of (2.38). This is
in contrast to conforming Galerkin (CG) methods, where a discrete version of Theorem 2.2 can easily
be established. On the other hand, in our (non-exhaustive) numerical tests presented in Section 6, we
did not observe any difficulties to that extent.
4. Existence of discrete solutions
In this section, we establish the existence of discrete solutions of (3.31) following the continuous arguments proposed in Lorca & Boldrini (1996) and based on Brouwer’s fixed point theorem. We propose a
general approach of constructing discrete liftings based on computing harmonic extensions, and discuss
the most practical choice of straightforward nodal interpolation.
4.1
Stability and existence
We start by proving the following stability property of the discrete solutions under a small data assumption. As in the continuous case, we write the discrete temperature θh as θh = θh,0 + θh,1 , with θh,0 ∈ Ψh,0
and
θh,1 ∈ Ψh ,
θh,1 |Γ = θD,h .
(4.1)
L EMMA 4.1 Let (uh , θh ) be a solution of (3.31) with θh = θh,0 + θh,1 as in (4.1). Assume that
1
C̃dep kgk0,Ω kθh,1 kL3 (Ω ) 6 ,
2
(4.2)
with
C̃dep =
C̃DC̃T,2
,
α̃S αT
(4.3)
then there exist constants C̃u and C̃θ only depending on kgk0,Ω and the stability constants in Section 3.3,
such that
kuh k1,Th 6 C̃u kθh,1 k1,Ω ,
kθh k1,Ω 6 C̃θ kθh,1 k1,Ω .
(4.4)
(Explicit expressions for C̃u and C̃θ can be found in (4.8) and (4.9), respectively.)
Proof. We choose the test function (v, ψ) = (uh , θh,0 ) in (3.31), and use (3.29) to obtain the two
equations
AhS (θh ; uh , uh ) + OhS (uh ; uh , uh ) = D(θh,0 , uh ) + D(θh,1 , uh ),
AT (θh ; θh,0 , θh,0 ) = − AT (θh ; θh,1 , θh,0 ) − OT (uh ; θh,1 , θh,0 ).
(4.5)
In the first identity of (4.5), the coercivity of AhS in (3.27), the positivity of OhS in (3.28), and the boundedness of D in (3.23) imply
kuk1,Th 6 α̃S−1C̃D kgk0,Ω kθh,0 k1,Ω + α̃S−1C̃D kgk0,Ω kθh,1 k1,Ω .
(4.6)
In the second equation of (4.5), we employ the coercivity and boundedness of AT in (2.22) and (2.14),
respectively, along with the bound for OT in Lemma 3.5. We conclude that
kθh,0 k1,Ω 6 αT−1 κ2 kθh,1 k1,Ω + αT−1C̃T,2 kθh,1 kL3 (Ω ) kuh k1,Th .
(4.7)
EXACTLY DIVERGENCE-FREE FEM FOR A GENERALIZED BOUSSINESQ PROBLEM
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Then, using the bound (4.7) in (4.6) yields
kuh k1,Th 6 α̃S−1 αT−1C̃DC̃T,2 kgk0,Ω kθh,1 kL3 (Ω ) kuh k1,Th + α̃S−1 αT−1C̃D kgk0,Ω αT + κ2 kθh,1 k1,Ω .
Hence, referring to assumption (4.2), we obtain
kuh k1,Th 6 C̃u kθh,1 k1,Ω
with
C̃u = 2α̃S−1 αT−1C̃D kgk0,Ω αT + κ2 .
(4.8)
Moreover, by using the triangle inequality, estimate (4.8), the definition of C̃dep and assumption (4.2)
we find that
kθh k1,Ω 6 kθh,0 k1,Ω + kθh,1 k1,Ω
6 (αT−1 κ2 + 1)kθh,1 k1,Ω + αT−1C̃T,2 kθh,1 kL3 (Ω ) kuh k1,Th
6 (αT−1 κ2 + 1)kθh,1 k1,Ω + 2αT−1C̃dep kgk0,Ω kθh,1 kL3 (Ω ) (αT + κ2 )kθh,1 k1,Ω
6 (αT−1 κ2 + 1)kθh,1 k1,Ω + αT−1 (αT + κ2 )kθh,1 k1,Ω .
Hence,
kθh k1,Ω 6 C̃θ kθh,1 k1,Ω
with
C̃θ = 2(1 + αT−1 κ2 ).
This completes the proof.
We are now ready to state our main existence result.
(4.9)
T HEOREM 4.1 Let θh,1 be a discrete lifting satisfying (4.2). Then there exists a discrete solution
(uh , θh ) ∈ Xh ×Ψh to the reduced problem (3.31) satisfying the stability bound (4.4).
The proof of Theorem 4.1 is carried out in detail in Section 4.3.
It is useful to derive from Theorem 4.1 an existence result for any discrete boundary datum θD,h . We
do this at the cost of more restrictive smallness assumptions and stability bounds as compared to those
in (4.2), (4.4). To that end, we establish the following lemma.
L EMMA 4.2 For any θD,h ∈ Λh , there is a discrete lifting θh,1 ∈ Ψh , θh,1 |Γ = θD,h , which satisfies
kθh,1 k1,Ω 6 Clift kθD,h k1/2,Γ ,
with a constant Clift > 0 independent of the mesh size and θD,h .
Proof. There is a continuous lifting of θD,h , i.e., a function θ ∈ H 1 (Ω ) such that
θ |Γ = θD,h ,
kθ k1,Ω 6 CkθD,h k1/2,Γ .
Denoting by θh,1 ∈ Ψh the Scott-Zhang quasi-interpolant of θ ; see Scott & S.Zhang (1990). As it is
stable in H 1 (Ω ) and reproduces polynomial boundary conditions, we have
kθh,1 k1,Ω 6 Ckθ k1,Ω 6 CkθD,h k1/2,Γ .
This implies the assertion.
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C OROLLARY 4.1 Let θD,h ∈ Λh be a discrete boundary datum. Assume that
C̃depCembClift ||g||0,Ω ||θD,h ||1/2,Γ 6 1/2,
(4.10)
with C̃dep defined in (4.3), Cemb > 0 the embedding constant in (3.12) for p = 3, and Clift the constant in
Lemma 4.2, Then the lifting θh,1 of Lemma 4.2 gives rises to a solution (uh , θh ) to (3.31) which satisfies
the stability bounds
kuh k1,Th 6 C̃uClift kθD,h k1/2,Γ ,
kθh k1,Ω 6 C̃θ Clift kθD,h k1/2,Γ ,
(4.11)
where C̃u and C̃θ are the constants in (4.4).
Proof. We apply Theorem 4.1 for the discrete lifting θh,1 constructed in Lemma 4.2. Hence, using the
embedding (3.12) with p = 3 yields
C̃dep kgk0,Ω kθh,1 kL3 (Ω ) 6 C̃depCemb kgk0,Ω kkθh,1 k1,Ω
1
6 C̃depCembClift kgk0,Ω kkθD,h k1/2,Γ 6 .
2
Hence, the assertion follows from Theorem 4.1 and the particular choice of θh,1 .
R EMARK 4.1 We point out that the discrete lifting θh,1 constructed in Lemma 4.2 cannot be easily
computed numerically. On the other hand, it is well known that the discrete lifting θh,1 ∈ Ψh with
minimum H 1 -norm is given by the discrete generalized harmonic extension of θD,h . It can be computed
by solving the elliptic problem: find θh,1 ∈ Ψh such that θh,1 |Γ = θD,h and
(θh,1 , v)1,Ω = 0,
for all v ∈ Ψh,0 ,
with (·, ·)1,Ω denoting the inner product on H 1 (Ω ). Indeed, if ψh,1 ∈ Ψh is another lifting with ψh,1 |Γ =
θD,h , then ψh,1 − θh,1 ∈ Ψh,0 . Then, by Galerkin orthogonality (θh,1 , ψh,1 − θh,1 )1,Ω = 0, and
kθh,1 k21,Ω = (θh,1 , θh,1 )1,Ω
6 (θh,1 , θh,1 )1,Ω + (ψh,1 − θh,1 , ψh,1 − θh,1 )1,Ω
6 (θh,1 , θh,1 )1,Ω + (ψh,1 , ψh,1 )1,Ω − (ψh,1 , θh,1 )1,Ω
6 (θh,1 , θh,1 )1,Ω + (ψh,1 , ψh,1 )1,Ω − (ψh,1 − θh,1 , θh,1 )1,Ω − (θh,1 , θh,1 )1,Ω
6 (ψh,1 , ψh,1 )1,Ω = kψh,1 k21,Ω .
Hence, using the discrete harmonic extension θh,1 in Corollary 4.1 gives rise to the same existence result
and stability bounds, and leads to a systematic (albeit expensive) approach to compute a suitable discrete
lifting for any discrete boundary datum.
R EMARK 4.2 We note that the stability bounds in (4.11), (4.4) will only be useful in an error analysis if the H 1/2 -norms of the discrete boundary data θD,h , respectively the H 1 -norms of the associated
liftings θh,1 can be bounded independently of the mesh size.
4.2
Nodal boundary data and liftings
The choice of the discrete boundary datum and the associated discrete liftings is crucial in the application
of Theorem 4.1 or Corollary 4.1. In addition, the construction of the liftings may be computationally
EXACTLY DIVERGENCE-FREE FEM FOR A GENERALIZED BOUSSINESQ PROBLEM
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expensive. We shall focus mainly on nodal interpolation of the boundary data, and in Remark 4.4, we
mention another possibility, which might be applicable in particular cases.
Let N (Th ) a set of unisolvent nodes associated with the conforming space Ψh , see, e.g., (Girault &
Raviart, 1986, Appendix A). We disjointly split N (Th ) = NI (Th ) ∪ NB (Th ) into interior and boundary
nodes. With each node N ∈ N (Th ), we associate the (global) Lagrange basis function lN (xx) ∈ Ψh . We
then denote by I : C(Ω ) → Ψh , v 7→ I v, the classical nodal interpolation operator given by
I v(xx) =
v(N)lN (xx).
∑
(4.12)
N∈N (Th )
The restriction of I to the boundary (nodes) is denoted by IΓ : C(Γ ) → Λh . Evidently, we have the
property that (I v)|Γ = IΓ (v|Γ ). In view of assumption (2.6), we now take the discrete boundary
datum θD,h ∈ Λh as the nodal interpolant of θD :
θD,h = IΓ θD .
(4.13)
We first show that kθD,h k1/2,Γ can be bounded independently of the mesh size (under additional smoothness assumption on the exact temperature); cf. Remark 4.2.
L EMMA 4.3 If the exact temperature θ of (2.1)– (2.5) belongs to H 2 (Ω ), then kθD,h kH 1/2 (Γ ) is bounded
as h → 0.
Proof. We first note that θ ∈ H 2 (Ω ) implies θ ∈ C(Ω ), and hence the nodal interpolant I of v is
well-defined. Hence,
kθD − θD,h k1/2,Γ 6 kθ − I θ k1,Ω 6 hkθ k2,Ω .
Then, by the triangle inequality,
kθD,h kH 1/2 (Γ ) 6 kθD − θD,h kH 1/2 (Γ ) + kθD kH 1/2 (Γ ) 6 Chkθ k2,Ω + kθD kH 1/2 (Γ ) ,
which implies the assertion.
R EMARK 4.3 The argument in Lemma 4.3 is somewhat adhoc, but sufficient for our purposes. We
also mention that the stability result kθD,h kC(Γ ) 6 CkθD kC(Γ ) can be readily shown. Moreover, stability
bounds for nodal interpolands in fractional-order Sobolev spaces can be found in (Belgacem & Brenner,
2001, Theorem 2.6).
The associated lifting of θD,h in (4.13) can now be taken as the discfrete harmonic extension as
discussed in Remark 4.1, thereby ensuring that Corollary 4.1 holds. However, the computationally most
practical discrete lifting is given by
θh,1 (xx) =
∑
θD (N)lN (xx) ∈ Ψh .
(4.14)
N∈NB (Th )
This lifting corresponds to a standard way of imposing non-homogenous boundary conditions in a finite
element implementation, where, in the resulting matrix system, the unknown coefficients at boundary
nodes N are simply set to θD (N). Obviously, the choice (4.14) allows one to satisfy condition (4.2) for
all functions g , and to prove existence of discrete solutions, provided the mesh size is sufficiently small.
Indeed, in this case θh,1 is zero outside a layer of elements adjacent to ∂ Ω , and hence the L3 -norm
of θh,1 can be made as small as possible for sufficiently small mesh sizes. Our numerical results will be
based on this choice.
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R EMARK 4.4 A theoretical construction of a discrete and stable lifting has been given in Scott &
S.Zhang (1990): For θD ∈ H 1/2 (Γ ), there is a lifting θh,1 ∈ Ψh , satisfying θD,h = θh,1 |Γ , and
kθD,h k1/2,Γ 6 kθh,1 k1,Ω 6 CkθD k1/2,Γ .
Although in principle it is possible to compute θh,1 if a stable lifting of θD in H 1 (Ω ) is explicitly known,
the numerical evaluation of this extension is costly and not feasible in practice.
A particular situation arises when the lifting θ1 in Theorem 2.1 is explicitly known or can be explicitly constructed, say from a known lifting of the boundary conditions. If, in addition, θ1 is sufficiently
smooth, we may simply take θh,1 as the nodal interpolant of θ1 . This would allow one again to satisfy
condition 4.2, and to obtain existence of discrete solutions for sufficiently small mesh sizes. Moreover,
kθh,1 k1,Ω can be bounded independently of h if θ1 is sufficiently smooth.
4.3
Proof of Theorem 4.1
To prove Theorem 4.1, we shall now make use of Brouwer’s fixed point theorem in the following
form Brezis (2011): Let K be a non-empty compact convex subset of a finite dimensional normed
space, and let L be a continuous mapping of K into itself. Then L has a fixed point in K . We
proceed in several steps.
Step 1: We introduce the finite dimensional set
(uh , θh ) ∈ Xh ×Ψh : kuh k1,Th 6 C̃u kθh,1 k1,Ω ,
K =
and θh = θh,0 + θh,1
kθh k1,Ω 6 C̃θ kθh,1 k1,Ω
,
(4.15)
with C̃u and C̃θ the constants defined in (4.8) and (4.9), respectively. It is convex and compact. We then
define the mapping
L : (zh , ϕh ) ∈ Xh ×Ψh 7→ (uh , θh := θh,0 + θh,1 ) ∈ Xh ×Ψh
as the solution to the following linearized version of problem (3.31): find (uh , θh ) ∈ Xh ×Ψh such that
AhS (ϕh ; uh , v) + OhS (zh ; uh , v) − D(ϕh , v) = 0,
AT (ϕh ; θh,0 , ψ) + OT (zh ; θh,0 , ψ) = −AT (ϕh ; θh,1 , ψ) − OT (zh ; θh,1 , ψ)
(4.16)
for all v ∈ Xh and ψ ∈ Ψh,0 . With the stability properties in Section 3.3, it is not difficult to see that
problem (4.16) is uniquely solvable, and hence the operator L is well defined.
Step 2: Let us prove that L maps from K into K . To that end, let (zh , ϕh ) ∈ K be given, and
denote by (uh , θh ) ∈ Vh ×Ψh the solution to the problem (4.16). Then, as in the proof of Lemma 4.1, we
take the test function (v, ψ) = (uh , θh,0 ). In the first of the two resulting equations, we use the coercivity
of AhS in (3.27), the positivity of OhT in (3.28), and the boundedness of D in (3.23). This results in
kuh k21,Th 6 α̃S−1 |D(ϕh , uh )| 6 α̃S−1C̃D kgk0,Ω kϕh k1,Ω kuh k1,Th .
Division by kuk1,Th and the bound kϕh k1,Ω 6 C̃θ kθh,1 k1,Ω then give
kuh k1,Th 6 α̃S−1C̃DC̃θ kgk0,Ω kθh,1 k1,Ω = C̃u kθh,1 k1,Ω ,
EXACTLY DIVERGENCE-FREE FEM FOR A GENERALIZED BOUSSINESQ PROBLEM
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where we have also used the identity
C̃u = α̃S−1C̃D kgk0,Ω C̃θ .
(4.17)
In the second of the two resulting equations, we use the coercivity of AT in (2.22), property (3.29), the
boundedness of AT and OT in (2.14) and Lemma 3.5, respectively, the bound kzh k1,Th 6 C̃u kθh,1 k1,Ω ,
and division by kθh,0 k1,Ω , to find that
kθh,0 k1,Ω 6 αT−1 κ2 kθh,1 k1,Ω + αT−1C̃T,2C̃u kθh,1 k1,Ω kθh,1 kL3 (Ω ) .
Then, from the identity (4.17) and assumption (4.2),
kθh,0 k1,Ω 6 αT−1 κ2 kθh,1 k1,Ω + α̃S−1 αT−1C̃DC̃T,2 kgk0,Ω C̃θ kθh,1 k1,Ω kθh,1 kL3 (Ω )
6 αT−1 κ2 kθh,1 k1,Ω +
C̃θ
kθh,1 k1,Ω .
2
Then, the triangle inequality and the definition C̃θ = 2(1 + αT−1 κ2 ) in (4.9) imply
kθh k1,Ω 6 kθh,0 k1,Ω + kθh,1 k1,Ω
6 (1 + αT−1 κ2 )kθh,1 k1,Ω +
C̃θ
kθh,1 k1,Ω 6 C̃θ kθh,1 k1,Ω .
2
Hence, we have (uh , θh ) ∈ K . It is now clear that the existence of a fixed point of L : K → K is
equivalent to the solvability of (3.31) as stated in the assertion.
Step 3: To apply Brouwer’s fixed point theorem, it remains to show that L is a continuous operator.
To do so, assume we are given (z, ϕ) ∈ K and a sequence {(zm , ϕm )}m∈N ⊂ K , such that
m→∞
kzm − zk1,Th −→ 0
and
m→∞
kϕm − ϕk1,Ω −→ 0.
We note that by the trace inequality (3.16) and for a fixed mesh size, there also holds lim kϕm −ϕk1,Eh =
m→∞
0. Thus, setting (u, θ ) = L (z, ϕ) and (um , θm ) = L (zm , ϕm ), m ∈ N, we need to prove that
m→∞
kum − uk1,Th −→ 0
and
m→∞
kθm − θ k1,Ω −→ 0.
(4.18)
From the definition of L in (4.16) we see that there hold
AhS (ϕm ; um , v) + OhS (zm ; um , v) − D(ϕm , v) = 0,
AT (ϕm ; θm , ψ) + OT (zm ; θm , ψ) = 0,
and
AhS (ϕ; u, v) + OhS (z; u, v) − D(ϕ, v) = 0,
AT (ϕ; θ , ψ) + OT (z; θ , ψ) = 0,
for all v ∈ Xh , ψ ∈ Ψh,0 and m ∈ N. Subtracting the two systems from each other yields the equations
AhS (ϕm ; um , v) − AhS (ϕ; u, v) + OhS (zm ; um , v) − OhS (z; u, v) − D(ϕm − ϕ, v) = 0,
(4.19)
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for all v ∈ Xh , and
AT (ϕm ; θm , ψ) − AT (ϕ; θ , ψ) + OT (zm ; θm , ψ) − OT (z; θ , ψ) = 0,
(4.20)
for all ψ ∈ Ψh,0 .
We first consider (4.19). Elementary manipulations then yield
AhS (ϕm ; u − um , v) + OhS (zm ; u − um , v) = − [AhS (ϕ; u, v) − AhS (ϕm ; u, v)]
− [OhS (z; u, v) − OhS (zm ; u, v)] + D(ϕm − ϕ, v).
We take v = u − um , use the ellipticity property of AhS and OhS in (3.27) and (3.28), respectively, as well
as the continuity of OhS and D, to get
α̃S ku − um k21,Th 6 AhS (ϕ; u, u − um ) − AhS (ϕm ; u, u − um )
+ C̃S kz − zm k1,Th kuk1,Th ku − um k1,Th + C̃D kgk0,Ω kϕ − ϕm k1,Ω ku − um k1,Th .
With the continuity property (3.20) for AhS and division by ku − um k1,Th , it follows that
ku − um k1,Th 6 C kϕ − ϕm k1,Eh kukW1,∞ (Th ) + kz − zm k1,Th kuk1,Th + kϕ − ϕm k1,Ω .
Hence, we find that
lim ku − um k1,Th = 0.
m→∞
(4.21)
Next, we consider equation (4.20). By proceeding as before, we rewrite it as
AT (ϕm ; θ − θm , ψ) + OT (zm ; θ − θm , ψ) = − [AT (ϕ; θ , ψ) − AT (ϕm ; θ , ψ)]
− [OT (z; θ , ψ) − OT (zm ; θ , ψ)].
Then, we take ψ = θ − θm ∈ Ψh,0 , note that OT (zm ; θ − θm , θ − θm ) = 0, by (3.29), and apply the
continuity property (2.17), the ellipticity (2.22), and the bound (3.25) for OT . Dividing the resulting
inequality by kθ − θm k1,Ω results in
kθ − θm k1,Ω 6 C kϕ − ϕm k1,Ω kθ kW 1,∞ (Ω ) + kz − zm k1,Th kθ k1,Ω .
Therefore,
lim kθ − θm k1,Ω = 0.
m→∞
(4.22)
Referring to (4.21) and (4.22) shows the claim in (4.18), which completes the proof.
5. Error analysis
In this section, we carry out the error analysis of the finite element approximation in (3.6). We start by
stating our error bounds. Then, we present the details of the proofs in several steps.
EXACTLY DIVERGENCE-FREE FEM FOR A GENERALIZED BOUSSINESQ PROBLEM
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5.1 Error estimates
We shall prove the following error estimates.
T HEOREM 5.1 Let θD,h be the nodal interpoland of θD in (4.13), and assume that (2.6) and the small data
assumption (4.10) hold true. Let (u, p, θ ) be a solution of (2.9), and let (uh , ph , θh ) be an approximate
solution obtained by (3.6) with the discrete lifting θh,1 of Lemma 4.2 or the harmonic extension in
Remark 4.1 and satisfying the stability bounds (4.11) in Corollary 4.1. Assume further that
n
o
max kgk0,Ω , kukW1,∞ (Ω ) , kθ kW 1,∞ (Ω ) 6 min{M, M̃},
(5.1)
with M and M̃ sufficiently small, as specified in (2.43) and (5.18) below. We further suppose that, for
k = 1,
p ∈ H 1 (Ω ),
θ ∈ W 1,∞ (Ω ) ∩ H 2 (Ω ),
(5.2)
u ∈ C1 (Ω ) ∩ H2 (Ω ) ∩ X,
and, for k > 2,
u ∈ Hk+1 (Ω ) ∩ X,
p ∈ H k (Ω ),
θ ∈ H k+1 (Ω ).
(5.3)
Then there exist two constants C > 0 independent of the mesh size such that
ku − uh k2,Th + kθ − θh k1,Ω 6 Chk ( kukk+1,Ω + kθ kk+1,Ω ),
(5.4)
kp − ph k0,Ω 6 Chk ( kpkk,Ω + kukk+1,Ω + kθ kk+1,Ω ).
(5.5)
and
The proof of Theorem 5.1 is presented in Section 5.2.
R EMARK 5.1 In our analysis, we shall need the base regularity (u, θ ) ∈ C1 (Ω ) ×W 1,∞ (Ω ) as assumed
in the lowest-order case k = 1 in (5.2); cf. Lemma 3.3 and (2.17). Notice that for k > 2, the regularity
assumption (u, θ ) ∈ Hk+1 (Ω ) × H k+1 (Ω ) in (5.3) implies (u, θ ) ∈ C1 (Ω ) ×C1 (Ω ).
R EMARK 5.2 Observe that under the small solution assumption (5.1), the exact solution to (2.9) is
unique, in agreement to Theorem 2.1. On the other hand and as mentioned above, an analogous uniqueness result for the discrete solution remains an open question.
5.2 Proof of Theorem 5.1
We present the proof of Theorem 5.1 in several steps.
5.2.1 Preliminaries. Let (u, p, θ ) be a solution of problem (2.9), and (uh , ph θh ) a finite element
approximation obtained by its discrete counterpart (3.6). To simplify the subsequent analysis, we write
eu = u − uh , eθ = θ − θh and e p = p − ph . As usual, we shall then decompose these errors into
eu = ξ u + χ u = (u − ṽh ) + (ṽh − uh ),
eθ = ξθ + χθ = (θ − ψ̃h ) + (ψ̃h − θh ),
(5.6)
e p = ξ p + χ p = (p − q̃h ) + (q̃h − ph ),
where we take ṽh as the BDM projection of u, ψ̃h = I θ ∈ Ψh is the nodal projection of θ , as introduced
in Section 4.2, and q̃h is the L2 -projection of p into Qh .
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We recall that for u ∈ X, we have ṽh ∈ Xh ; see, e.g., Brezzi & Fortin (1991). Then, we also have
χ u ∈ Xh . The following approximation properties are standard:
kξξ u k2,Th 6 Chk kukk+1,Ω ,
kξθ k1,Ω 6 Chk kθ kk+1,Ω ,
kξ p k0,Ω 6 Chk kpkk,Ω .
(5.7)
Then, according to the triangle inequality and the inverse inequality (3.11), we see that
χ u k2,Th 6 C hk kukk+1,Ω +Ckχ
χ u k1,Th ,
keu k2,Th 6kξξ u k2,Th + kχ
keθ k1,Ω 6kξθ k1, Ω + kχθ k1,Ω 6 C hk kθ kk+1,Ω + kχθ k1,Ω ,
(5.8)
ke p k0,Ω 6kξ p k0,Ω + kχ p k0,Ω 6 C hk kpkk,Ω + kχ p k0,Ω .
χ u k1,Th , kχθ k1,Ω ,
Hence, to prove the error estimate (5.1), we need to show the optimal convergence of kχ
and kχ p k0,Ω .
To do so, we shall employ the following Galerkin orthogonality property.
L EMMA 5.1 Assume that u ∈ H2 (Ω ) ∩ X. Then we have
h
AS (θ ; u, v) − AhS (θh ; uh , v) + OhS (u; u, v) − OhS (uh ; uh , v) − B(v, e p ) − D(eθ , v) = 0,
B(eu , q) = 0,
AT (θ ; θ , ψ) − AT (θh ; θh , ψ) + OT (u; θ , ψ) − OT (uh ; θh , ψ) = 0,
for all (v, q, ψ) ∈ Vh × Qh ×Ψh,0 .
Proof. As we assume H2 (Ω )-regularity for the velocity field u, it can be readily seen by integration by
parts that the exact solution (u, p, θ ) satisfies
AhS (θ ; u, v) + OhS (u; u, v) − B(v, p) − D(θ , v) = 0,
for all v ∈ Vh ; see also Arnold et al. (2002). This implies the first equation. The second and third
equations are readily verified.
5.2.2 Error estimates in the velocity and temperature.
χ u k1,Th and kχθ k1,Ω .
kχ
We now start by analyzing the convergence of
L EMMA 5.2 There exists a constant C1 > 0 independent of the mesh size such that
χ u k21,Th 6C1 kξξ u k2,Th + kξθ k1,Ω kχ
χ u k1,Th
(α̃S − C̃SC∞ M̃)kχ
χ u k1,Th kχθ k1,Ω .
+ M̃(C̃lip νlip + C̃D )kχ
Proof. First, note that χ u ∈ Xh . From the ellipticity of AhS in (3.27) and elementary calculations, it is
not difficult to see that
χ u k21,Th 6 AhS (θh ; χ u , χ u ) = A1S + A2S + A3S + A4S ,
α̃S kχ
(5.9)
EXACTLY DIVERGENCE-FREE FEM FOR A GENERALIZED BOUSSINESQ PROBLEM
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with the terms A1S through A4S given by
A1S = AhS (θh ; u, χ u ) − AhS (ψ̃h ; u, χ u ),
A2S = AhS (ψ̃h ; u, χ u ) − AhS (θ ; u, χ u ),
A3S = AhS (θ ; u, χ u ) − AhS (θh ; uh , χ u ),
A4S = −AhS (θh ; ξ u , χ u ).
Similarly, thanks to the positivity of OhS in (3.28), we obtain
0 6 OhS (uh ; χ u , χ u ) = O1S + O2S + O3S + O4S ,
(5.10)
with O1S through O4S given by
O1S = OhS (uh ; u, χ u ) − OhS (ṽh ; u, χ u ),
O2S = OhS (ṽh ; u, χ u ) − OhS (u; u, χ u ),
O3S = OhS (u; u, χ u ) − OhS (uh ; uh , χ u ),
O4S = −OhS (uh ; ξ u , χ u ).
From the first error equation in Lemma 5.1, it further follows that
A3S + O3S = D(eθ , χ u ) = D(ξθ , χ u ) + D(χθ , χ u ),
(5.11)
χ u , e p ) = 0 since χ u ∈ Xh is exactly divergence-free.
where we have used the fact that B(χ
Next, we bound each of the terms on the right hand sides of (5.9), (5.10) and (5.11), respectively.
We start by estimating those in (5.9). To that end, we use bound (3.21), the continuity of AhS in (3.18),
and the fact that kukW1,∞ (Th ) = kukW1,∞ (Ω ) 6 M̃ (since u ∈ C1 (Ω )). We find that
χ u k1,Th 6 M̃C̃lip νlip kχθ k1,Ω kχ
χ u k1,Th ,
|A1S | 6 C̃lip νlip kθh − ψ̃h k1,Ω kukW1,∞ (Th ) kχ
χ u k1,Th 6 M̃C̃lip νlip kξθ k1,Ω kχ
χ u k1,Th ,
|A2S | 6 C̃lip νlip kθ − ψ̃h k1,Ω kukW1,∞ (Th ) kχ
(5.12)
χ u k1,Th .
|A4S | 6 Ckξξ u k2,Th kχ
We proceed similarly for the terms in (5.10). We use the continuity of OhS , cf. (3.24), the continuous
dependence of uh in (4.11), and note that kuk1,Ω 6 C∞ kukW1,∞ (Ω ) 6 C∞ M̃ by (1.2). This results in
χ u k21,Th 6 C̃SC∞ M̃kχ
χ u k21,Th ,
|O1S | 6 C̃S kuk1,Ω kχ
χ u k1,Th 6 C̃SC∞ M̃kξξ u k2,Th kχ
χ u k1,Th ,
|O2S | 6 C̃S kξξ u k1,Th kuk1,Ω kχ
χ u k1,Th
|O4S | 6 C̃S kuh k1,Th kξξ u k1,Th kχ
(5.13)
χ u k1,Th .
6 C̃SC̃uClift kθD,h kH 1/2 (Γ ) kξξ u k2,Th kχ
In the bound for |O4S |, we emphasize that kθD,h kH 1/2 (Γ ) is bounded independently of the mesh size, in
agreement with Lemma 4.3.
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Finally, to estimate the terms in (5.11) we employ the continuity of D and the hypothesis that
kgk0,Ω 6 M̃. We conclude that
χ u k1,Th ,
|D(ξθ , χ u )| 6 M̃C̃D kξθ k1,Ω kχ
χ u k1,Th .
|D(χθ , χ u )| 6 M̃C̃D kχθ k1,Ω kχ
(5.14)
Hence, from (5.9), (5.10) and (5.11), and the upper bounds (5.12), (5.13) and (5.14) the assertion
follows.
A corresponding upper bound for kχθ k1,Ω is established in a similar fashion.
L EMMA 5.3 There exists a constant C2 > 0 independent of the mesh size such that
χ u k1,Th kχθ k1,Ω .
(αT − κlip M̃)kχθ k21,Ω 6 C2 kξξ u k2,Th + kξθ k1,Ω kχθ k1,Ω + C̃TC∞ M̃kχ
Proof. We proceed similarly to the proof of Lemma 5.2. Indeed, by adding and subtracting suitable
terms and noting that χθ ∈ H01 (Ω ), the ellipticity (2.22) of AT and property (3.29) for OT imply that
αT kχθ k21,Ω 6 AT (θh ; χθ , χθ ) + OT (uh ; χθ , χθ )
= A1T + A2T + A3T + A4T + O1T + O2T + O3T + O4T ,
(5.15)
with
A1T = AT (θh ; θ , χθ ) − AT (ψ̃h ; θ , χθ ),
A2T = AT (ψ̃h ; θ , χθ ) − AT (θ ; θ , χθ ),
A3T = AT (θ ; θ , χθ ) − AT (θh ; θh , χθ ),
A4T = −AT (θh ; ξθ , χθ ),
and
χ u ; θ , χθ ),
O1T = −OT (χ
O2T = −OT (ξξ u ; θ , χθ ),
O3T = OT (u; θ , χθ ) − OT (uh ; θh , χθ ),
O4T = −OT (uh ; ξθ , χθ ).
As before, the third error equation in Lemma 5.1 yields
A3T + O3T = 0.
Now, by the continuity properties of AT in (2.14), (2.17), and since kθ kW 1,∞ (Ω ) 6 M̃, we see that
|A1T | 6 κlip kθ kW 1,∞ (Ω ) kθh − ψ̃h k1,Ω kχθ k1,Ω 6 κlip M̃kχθ k21,Ω ,
|A2T | 6 κlip kθ − ψ̃h k1,Ω kθ kW 1,∞ (Ω ) kχθ k1,Ω 6 κlip M̃kξθ k1,Ω kχθ k1,Ω ,
(5.16)
|A4T | 6 κ2 kξθ k1,Ω kχθ k1,Ω .
On the other hand, by employing the bound kθ k1,Ω 6 C∞ kθ kW 1,∞ (Ω ) 6 C∞ M̃, the inequality (2.14),
and the continuous dependence in (4.11), we obtain
χ u k1,Th kθ k1,Ω kχθ k1,Ω 6 C̃TC∞ M̃kχ
χ u k1,Th kχθ k1,Ω ,
|O1T | 6 C̃T kχ
|O2T | 6 C̃T kξξ u k1,Th kθ k1,Ω kχθ k1,Ω 6 C̃TC∞CT M̃kξξ u k2,Th kχθ k1,Ω ,
|O4T | 6 C̃T kuh k1,Th kξθ k1,Ω kχθ k1,Ω 6 C̃TC̃uClift kθD,h kH 1/2 (Γ ) kξθ k1,Ω k1,Ω kχθ k1,Ω .
(5.17)
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The desired result follows from (5.15) and the estimates in (5.16) and (5.17), noting again that
kθD,h kH 1/2 (Γ ) is bounded independently of the mesh size; cf. Lemma 4.3.
We are now ready to prove the error bound (5.4) of Theorem 5.1.
L EMMA 5.4 There is a constant C > 0 independent of the mesh size such that
ku − uh k2,Th + kθ − θh k1,Ω 6 Chk (kukk+1,Ω + kθ kk+1,Ω ).
χ u k1,Th and kχθ k1,Ω . To this end, we set
Proof. Starting from (5.8), it is enough to bound kχ
L(u, θ ) = kξξ u k2,Th + kξθ k1,Ω .
Adding the two bounds in Lemma 5.2 and Lemma 5.3 results in
χ u k21,Th + (αT − κlip M̃)kχθ k21,Ω 6 CL(u, θ ) kχ
χ u k1,Th + kχθ k1,Ω
(α̃S − C̃SC∞ M̃)kχ
χ u k1,Th kχθ k1,Ω
+ M̃(C̃lip νlip + C̃D )kχ
χ u k1,Th kχθ k1,Ω .
+ C̃T C∞ M̃kχ
2
2
An application of the inequality |ab| 6 a2 + b2 allows
us to bring the last two terms above to the righthand side. By setting K̃ = (C̃lip νlip ) + C̃D + C̃T C∞ /2, we obtain
χ u k21,Th +(αT − (κlip + K̃)M̃)kχθ k21,Ω
(α̃S − (C̃SC∞ + K̃)M̃)kχ
χ u k1,Th + kχθ k1,Ω .
6 CL(u, θ ) kχ
Hence, if we choose M̃ such that
αT o
α̃S
,
,
C̃SC∞ + K̃ κlip + K̃
(5.18)
χ u k1,Th + kχθ k1,Ω 6 C L(u, θ ).
kχ
(5.19)
M̃ < inf
n
we readily obtain
From the approximation properties in (5.7), we conclude that
L(u, θ ) 6 C hk kukk+1,Ω + kθ kk+1,Ω ,
which implies the desired estimate (5.4).
5.2.3 Error in the pressure. Next, we bound the error in the pressure.
L EMMA 5.5 There is a constant C > 0 independent of the mesh size such that
ke p k0,Ω 6 Chk kukk+1,Ω + kθ kk+1,Ω + kpkk,Ω .
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R. OYARZÚA, T. QIN, AND D. SCHÖTZAU
Proof. From (5.8), it remains to bound kχ p k0,Ω . To that end, we invoke the discrete inf-sup condition (3.30) and the boundedness of B in (3.22) to find that
kχ p k0,Ω 6β̃ −1
B(v, χ p )
v∈Vh \{0} kvk1,Th
sup
6β̃ −1
B(v, −ξ p )
B(v, e p )
+ β̃ −1 sup
kvk
1,Th
v∈Vh \{0} kvk1,Th
v∈Vh \{0}
6β̃ −1
B(v, e p )
+ β̃ −1C̃B kξ p k0,Ω .
v∈Vh \{0} kvk1,Th
sup
(5.20)
sup
Then, from the first error equation in Lemma 5.1, we find that, for any v ∈ Vh ,
B(v, e p ) 6 |D(eθ , v)| + |T1 | + |T2 | + |T3 | + |T4 |,
(5.21)
with
T1 = [AhS (θ ; u, v) − AhS (θh ; u, v)],
T2 = AhS (θh ; eu , v),
T3 = [OhS (u; u, v) − OhS (uh ; u, v)],
T4 = OhS (uh ; eu , v).
Next, we bound the terms T1 through T4 appearing on the right hand side of (5.21). For T1 , we use the
triangle inequality, the continuity bound in Lemma 3.3, and the assumption kukW1,∞ (Th ) = kukW1,∞ (Ω ) 6
M̃. We obtain
|T1 | 6 C̃lip νlip keθ k1,Ω kukW1,∞ (Th ) kvk1,Th 6 C̃lip νlip M̃keθ k1,Ω kvk1,Th .
Furthermore, from the bound (3.18),
|T2 | 6 Ckeu k2,Th kvk1,Th .
From the Lipschitz continuity of OhS in (3.24), the stability bound (4.11), and the inequality kuk1,Ω 6
C∞ M̃, we have the estimates
|T3 | 6C̃S keu k1,Th kuk1,Th kvk1,Th 6 C̃SC∞ M̃keu k2,Th kvk1,Th ,
|T4 | 6C̃S kuh k1,Th keu k1,Th kvk1,Th 6 C̃SC̃uClift kθD,h kH 1/2 (Γ ) keu k2,Th kvk1,Th .
Finally, note that, by (3.23) and assumption (5.1),
|D(eθ , v)| 6 C̃D kgk0,Ω keθ k1,Ω kvk1,Th 6 C̃D M̃keθ k1,Ω kvk1,Th .
The above estimates imply
|B(v, e p )| 6 C3 keθ k1,Ω + keu k2,Th kvk1,Th .
Hence, the desired estimate (5.5) follows from the inequalities in (5.20), (5.21), and (5.22).
This completes the proof of Theorem 5.1.
(5.22)
EXACTLY DIVERGENCE-FREE FEM FOR A GENERALIZED BOUSSINESQ PROBLEM
27 of 31
6. A numerical test
In this section, we present computed errors and orders of convergence for a two-dimensional Boussinesq
problem (2.1)–(2.3) with a smooth solution. Our goal is to confirm the convergence rates in Theorem 5.1.
Our implementation is based on the deal.II finite element library1 , in conjunction with the direct linear
solver UMFPACK, see Davis (2004). We employ a variant of our method (3.6) adapted to quadrilateral meshes. Specifically, for an order k > 1, we employ divergence-conforming Raviart-Thomas (RT)
elements R T k of order k for the velocities, discontinuous tensor product polynomials Qk for the pressures, and conforming Qk polynomials for the temperatures. While the velocity-pressure pair is not
optimally matched in terms of approximation properties, the resulting mixed method also yields exactly
divergence-free velocity approximation (since div R T k = Qk ); we also refer to Cockburn et al. (2005,
2007) for details.
In our test, the computational domain is taken as Ω = (−1, 1)2 , and we consider a sequence of
uniformly refined square meshes {Thl }l of mesh size hl = 2−l . We take g = (0, 1)> , ν = 1, and choose
the temperature-dependent viscosity and thermal conductivity of the exponential form
ν(θ ) = exp(−θ ),
κ(θ ) = exp(θ ).
(6.1)
We then prescribe boundary data and additional right-hand sides so that the test solution is given by the
smooth functions
u1 (x, y) = sin(y),
p(x, y) = 1 + sin(xy),
u2 (x, y) = sin(x)
θ (x, y) = 1 + cos(xy)
The temperature boundary conditions are enforced as in (4.14). For the velocity field, the inhomogeneous boundary condition u = uD is essentially enforced in normal direction at the RT degrees of
freedom, while standard DG terms are used to incorporate it in tangential direction. The additional
right-hand sides are discretized in a straightforward fashion. Finally, we select the penalty parameter as
a0 = (k + 1)2 , where k is the approximation order.
We use a simple iteration scheme to deal with the non-linearities. Given the velocity uhn−1 at iteration
level n > 1, we obtain the temperature θhn by solving the discrete version of the convection-diffusion
problem (2.3), where we take the flow field explicitly as un−1
h . Then, we get the updated fluid variables
(unh , pnh ) by solving a discrete Oseen problem of the form (2.1)–(2.2), where the temperature is now taken
explicitly as θhn , and the convective term is linearized with un−1
h . Proceeding in this way, we obtain a
n
h
n
sequence of iterates (uh , pn , θh ). The iteration is terminated once the difference of the entire coefficient
vectors between two consecutive iterates is sufficiently small, i.e.,
kcoeffn+1 − coeffn kl 2 6 tol,
where k · kl 2 is the standard l 2 -norm in Rdof , with dof denoting the total number of degrees of freedom,
and tol is a fixed tolerance chosen as tol = 10−8 . As initial guess, we simply take u0h = 0.
The computed errors and convergence rates for the velocities are listed in Tables 1 and 2. From
Table 1, it can be seen that H 1 -norm errors for the temperature, and the DG-norm errors in the velocity
converge of order O(hk ), in agreement with Theorem 5.1. In addition, the property that the approximate
velocities are exactly divergence-free is verified by evaluating k∇ · uh kL∞ (Ω ) over a set of quadrature
points.
1 See
www.dealii.com.
28 of 31
R. OYARZÚA, T. QIN, AND D. SCHÖTZAU
In Table 2, we display the computed L2 -norm errors and convergence rates for the pressure, the
temperature, and the velocity. For the latter two unknowns, optimal rates of order O(hk+1 ) are observed
(although this is not corroborated by our theoretical results). For the L2 -norm errors in the pressure, the
convergence rate is between O(hk ) and O(hk+1 ). In Cockburn et al. (2007), the same phenomenon has
been observed for the Navier-Stokes equations in isolation; this is a reflection of the fact that R T k − Qk
are not optimally matched in terms of approximation properties, in contrast to the simplicial elements
studied in our theoretical analysis.
k
1
2
3
4
l
1
2
3
4
5
1
2
3
4
1
2
3
4
1
2
3
keθ k1,Ω
2.201e-01
9.332e-02
4.267e-02
2.079e-02
1.030e-02
4.108e-02
9.169e-03
2.223e-03
5.513e-04
1.451e-03
1.805e-04
2.240e-05
2.791e-06
1.703e-04
9.917e-06
5.981e-07
1.24
1.13
1.04
1.01
2.16
2.04
2.01
3.01
3.01
3.00
4.10
4.05
keu k1,h
5.982e-01
2.831e-01
1.315e-01
6.195e-02
2.927e-02
1.591e-01
4.546e-02
1.199e-02
3.060e-03
7.545e-03
9.239e-04
1.077e-04
1.206e-05
9.295e-04
6.241e-05
4.010e-06
1.08
1.11
1.09
1.08
1.81
1.92
1.97
3.03
3.10
3.16
3.90
3.96
k∇ · uh kL∞ (Ω )
1.443e-15
5.329e-15
1.199e-14
2.931e-14
6.584e-14
4.732e-15
1.521e-14
2.429e-14
5.556e-14
1.484e-14
3.040e-14
6.105e-14
1.131e-13
4.174e-14
7.412e-14
1.645e-13
Table 1. H 1 -norm errors and convergence rates for θh and uh , L∞ -norm for ∇ · uh .
In Figure 1, we plot the residual kcoeffn+1 − coeffn kl 2 against the number of iteration n for the case
k = 1. We see that the residuals decrease in a linear fashion. While this is not surprising for the type
of fixed-point iteration applied, our theoretical analysis does not provide any indication of a contraction
property. For k > 1, in our experiment, almost the same phenomena is observed; for the sake of brevity,
these plots have been omitted.
EXACTLY DIVERGENCE-FREE FEM FOR A GENERALIZED BOUSSINESQ PROBLEM
1
2
3
4
ke p k0,Ω
9.86e-02
3.50e-02
1.32e-02
4.62e-03
1.53e-03
1.044e-02
1.611e-03
2.389e-04
3.735e-05
6.910e-04
7.407e-05
7.033e-06
6.318e-07
6.456e-05
2.375e-06
9.623e-08
l
1
2
3
4
5
1
2
3
4
1
2
3
4
1
2
3
keθ k0,Ω
7.16e-02
1.92e-02
4.96e-03
1.25e-03
3.14e-04
6.328e-03
6.998e-04
8.498e-05
1.055e-05
1.520e-04
1.009e-05
6.455e-07
4.083e-08
1.201e-05
3.402e-07
1.057e-08
1.49
1.41
1.52
1.60
2.70
2.75
2.68
3.22
3.40
3.48
4.76
4.63
keu k0,Ω
6.75e-02
2.12e-02
5.02e-03
1.21e-03
3.09e-04
1.101e-02
1.306e-03
1.600e-04
1.975e-05
4.713e-04
3.295e-05
2.049e-06
1.242e-07
4.212e-05
1.121e-06
3.270e-08
1.90
1.95
1.98
2.00
3.18
3.04
3.01
3.91
3.97
3.98
5.14
5.01
Table 2. L2 -norm errors and convergence rates for ph , θh and uh .
2
10
l=1
l=2
l=3
l=4
0
10
−2
10
residual
k
−4
10
−6
10
−8
10
−10
10
1
2
3
4
5
6
iteration number n
7
8
9
FIG. 1. Residuals vs. number of iterations for k = 1, l = 1, ..., 4
10
29 of 31
1.67
2.08
2.06
1.97
3.08
3.03
3.02
3.84
4.01
4.04
5.23
5.10
30 of 31
R. OYARZÚA, T. QIN, AND D. SCHÖTZAU
7. Conclusions
We have introduced a new mixed finite element method for the numerical simulation of a generalized
Boussinesq problem with exactly divergence-free BDM elements of order k for the velocities, discontinuous elements of order k − 1 for the pressure, and standard continuous elements of order k for the
discretization of the temperature. The resulting method yields exactly divergence-free velocity approximations, and thus it is energy-stable without additional modifications of the convection terms. Under
suitable hypotheses on the data, we have shown the existence and stability of discrete solutions. Moreover, we have shown optimal a-priori error estimates with respect to the mesh size h for problems with
smooth and sufficiently small solutions. More precisely, the broken H 1 -norm errors in the velocity, the
H 1 -norm errors in the temperature, and the L2 -norm errors in the pressure are proved to converge with
order O(hk ). These rates were confirmed in a numerical test for a problem with a smooth solution.
The uniqueness of (small) discrete solutions remains an open theoretical problem: one of the difficulties in adapting Theorem 2.2 to the discrete level is the appearance of the augmented norm (3.15)
in the continuity estimate (3.20). In addition, our stability theory is based the availability of discrete
liftings whose actual computation may be expensive. Ongoing research is concerned with finding ways
to overcome these issues.
The numerical results shown in this paper are non-exhaustive. Additional testing is necessary to fully
assess the performance of the proposed scheme. This includes tests for physically relevant problems
with realistic parameters and three-dimensional geometries, the development of efficient linearization
strategies (such as Newton’s methods), and the design of iterative solvers or preconditioners. Some of
these computational aspects will be addressed in a forthcoming paper.
Finally, we emphasize that using conforming elements for the temperature unknown makes the
analysis simpler, but may not yield robust approximations in highly convection-dominated problems.
In this regime, discontinuous discretizations may be more appropriate for the temperature equation as
well. This is also the subject of ongoing work.
Funding
R.O. was supported in part by the Natural Sciences and Engineering Research Council of Canada
(NSERC), BECAS CHILE para postdoctorado en el extranjero (convocatoria 2011) and FONDECYT
project 11121347. T.Q. and D.S. were supported in part by the Natural Sciences and Engineering Research Council of Canada (NSERC).
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