An exactly divergence-free ﬁnite element method for a...

IMA Journal of Numerical Analysis (2014) of 31doi:10.1093/imanum/drnxxx

An exactly divergence-free finite element method for a generalizedBoussinesq problem

RICARDO OYARZUA †,GIMNAP-Departamento de Matematica, Universidad del Bıo-Bıo, Casilla 5-C, Concepcion,

Chile, CI2MA, Universidad de Concepcion, Casilla 160-C, Concepcion, Chile,

TONG QIN‡Mathematics Department, University of British Columbia, Vancouver, BC, Canada, V6T 1Z2

AND

DOMINIK SCHOTZAU§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 thenumerical simulation of a generalized Boussinesq problem, describing the motion of a non-isothermalincompressible fluid subject to a heat source. The method is based on using divergence-conformingelements of order k for the velocities, discontinuous elements of order k−1 for the pressure, and standardcontinuous elements of order k for the discretization of the temperature. The H1-conformity of thevelocities is enforced by a discontinuous Galerkin approach. The resulting numerical scheme yieldsexactly divergence-free velocity approximations; thus, it is provably energy-stable without the need tomodify 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, divergence-conforming elements, discontinuous Galerkin methods

1. Introduction

The numerical simulation of incompressible non-isothermal fluid flow problems has become increas-ingly important for the design and analysis of devices in many branches of engineering. Relevantindustrial applications include heat pipes, heat exchangers, chemical reactors, or cooling processes.Temperature-dependent flows have also become of great interest in geophysical or oceanographic flowswith applications to weather and climate predictions.

The last decade has seen a significant interest in the development and analysis of efficient finiteelement methods for such problems. We mention here only (Bernardi et al., 1995; Boland & Layton,

†Corresponding author. Email: [email protected]‡Email: [email protected]§Email: [email protected]

c© The author 2014. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

2 of 31 R. OYARZUA, T. QIN, AND D. SCHOTZAU

1990a,b; Cox et al., 2007; Farhloul & Zine, 2011; Perez et al., 2008a,b; Tabata & Tagami, 2005) andthe references therein. In particular, in (Perez et al., 2008b) a conforming method is presented andanalyzed for approximating non-isothermal incompressible fluid flow problems. However, the analysisthere 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 fluidflow problems. Here, the governing equations are discretized by the backward Euler method in time andconforming finite elements in space.

In this paper, we propose an alternative approach for the numerical approximation (in space) ofa non-isothermal flow problem. As a model problem, we consider the generalized Boussinesq modelanalyzed theoretically in Lorca & Boldrini (1996): it couples the stationary incompressible Navier-Stokes equations for the fluid variables (velocity and pressure) with a convection-diffusion equation forthe temperature variable. The coupling is non-linear through a temperature-dependent viscosity, andthrough 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 thepressure, and continuous elements of order k for the temperature. To enforce H1-continuity of the ve-locities, we use an interior penalty discontinuous Galerkin (DG) technique. The resulting mixed finiteelement 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 energy-stable 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 ofcolliding flows in a cross-shaped domain.

We show the existence and stability of discrete solutions by mimicking the fixed point argumentspresented in Lorca & Boldrini (1996) for the continuous problem. A crucial aspect of this argument isthe 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 maybe computationally expensive. One option is to choose the discrete harmonic extension of the discreteboundary datum, which requires one elliptic solve. However, in our theoretical analysis, this comes atthe cost of a relatively strict small data assumption. We also discuss the most practical choice of straight-forward 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. Inparticular, we show that the velocity errors in the DG energy norm, the pressure errors in the L2-norm,and the temperature errors in the H1-norm converge of order O(hk) in the mesh size h. This convergencerates 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 Boussinesqmodel problem, and review the results from Lorca & Boldrini (1996) regarding existence and uniquenessof solutions. In Section 3, we present our finite element discretization, and review the stability prop-erties of the discrete formulation. In Section 4, we establish the existence and stability of approximatesolutions 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 paperwith 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, withLipschitz boundary ∂O . We write C(O), C(∂O) for the standard spaces of continuous functions, and‖ · ‖C(O), ‖ · ‖C(∂Ω) for the associated maximum norms. For r > 0 and p ∈ [1,∞], we denote by Lp(O)

and W r,p(O) the usual Lebesgue and Sobolev spaces endowed with the norms ‖ ·‖Lp(O) and ‖ ·‖W r,p(O),

EXACTLY DIVERGENCE-FREE FEM FOR A GENERALIZED BOUSSINESQ PROBLEM 3 of 31

respectively. Note that W 0,p(O) = Lp(O). If p = 2, we write Hr(O) in place of W r,2(O), and denotethe corresponding Lebesgue and Sobolev norms by ‖ · ‖0,O and ‖ · ‖r,O , respectively. For r > 0, wewrite | · |r,O for the Hr-seminorm. The space H1

0 (O) is the space of functions in H1(O) with vanishingtrace on Γ , and L2

0(O) is the space of L2-functions with vanishing mean value over O . Spaces of vector-valued functions are denoted in bold face. For example, Hr(O) = [Hr(O)]d for r> 0. For simplicity, wealso write ‖ · ‖r,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,

H0(div 0;O) =

w ∈H0(div ;O) : div w≡ 0 in Ω,

(1.1)

with nO denoting the unit outward normal on ∂O . These spaces are endowed with the norm

‖w‖2div,O = ‖w‖2

0,O +‖div w‖20,O .

In the subsequent analysis, we denote by C∞ > 0 the embedding constant such that

‖u‖1,O 6C∞‖u‖W1,∞(O), ‖θ‖1,O 6C∞‖θ‖W 1,∞(O), (1.2)

for all u ∈W1,∞(O) and θ ∈W 1,∞(O). Finally, we shall frequently use the notation C and c, withor without subscripts, bars, tildes or hats, to denote generic positive constants independent of the dis-cretization 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 propertiesof the forms involved, and review some theoretical properties regarding existence and uniqueness ofsolutions.

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 − gθ = 0 in Ω , (2.1)

divu = 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 unknownsare the fluid velocity u, the pressure p, and the temperature θ . The given data are the non-vanishingboundary temperature θD ∈ H1/2(Γ ), and the external force per unit mass g ∈ L2(Ω), usually acting indirection opposite to gravity. We assume that

θD ∈C(Γ ), (2.6)


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 arebounded 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)∈H1

0(Ω)×L20(Ω)×

H1(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,

AT(θ ;θ ,ψ) + OT(u;θ ,ψ) = 0,

(2.9)

for all (v,q,ψ) ∈H10(Ω)×L2

0(Ω)×H10 (Ω). Here, the forms are given by

AS(ψ;u,v) =∫

Ω

ν(ψ)∇u : ∇v, OS(w;u,v) =∫

Ω

((w ·∇)u) ·v, (2.10)

AT(ϕ;θ ,ψ) =∫

Ω

κ(ϕ)∇θ ·∇ψ, OT(v;θ ,ψ) =∫

Ω

(v ·∇θ)ψ, (2.11)

B(v,q) =∫

Ω

qdivv, D(θ ,v) =∫

Ω

θg ·v. (2.12)

2.2 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‖u‖1,Ω‖v‖1,Ω , u,v ∈H1(Ω), (2.13)

|AT(·;θ ,ψ)|6 κ2‖θ‖1,Ω‖ψ‖1,Ω , θ ,ψ ∈ H1(Ω), (2.14)

|B(v,q)|6CB‖v‖1,Ω‖q‖0,Ω , v ∈H1(Ω), q ∈ L2(Ω). (2.15)

Moreover, from the Lipschitz continuity of ν and κ in (2.7) and Holder’s inequality it readily followsthat, for θ1,θ2 ∈ H1(Ω), u ∈W1,∞(Ω), θ ∈W 1,∞(Ω),

|AS(θ1;u,v)−AS(θ2;u,v)|6 νlip‖u‖W1,∞(Ω)‖θ1−θ2‖1,Ω‖v‖1,Ω , v ∈H1(Ω), (2.16)

|AT(θ1;θ ,ψ)−AT(θ2;θ ,ψ)|6 κlip‖θ‖W 1,∞(Ω)‖θ1−θ2‖1,Ω‖ψ‖1,Ω , ψ ∈ H1(Ω). (2.17)


The forms OS and OT are linear in each argument. Holder’s inequality and standard Sobolev em-beddings then give the following bounds:

|OS(w;u,v)|6CS‖w‖1,Ω‖u‖1,Ω‖v‖1,Ω , w,u,v ∈H1(Ω), (2.18)

|OT(w;θ ,ψ)|6CT‖w‖1,Ω‖θ‖1,Ω‖ψ‖1,Ω , w ∈H1(Ω), θ ,ψ ∈ H1(Ω). (2.19)

Similarly, we have

|D(θ ,v)|6CD‖g‖0,Ω‖θ‖1,Ω‖v‖1,Ω , θ ∈ H1(Ω), v ∈H1(Ω). (2.20)

Next, we review the positivity properties of the forms in (2.10) and (2.11). By the Poincare inequalityand the bounds (2.8), the elliptic forms AS and AT are coercive:

|AS(·;v,v)|> αS‖v‖21,Ω , v ∈H1

0(Ω), (2.21)

|AT(·;ψ,ψ)|> αT‖ψ‖21,Ω , ψ ∈ H1

0 (Ω). (2.22)

To discuss the convective form OS and OT, we introduce the kernel

X =

v ∈H10(Ω) : B(v,q) = 0 ∀q ∈ L2

0(Ω)=

v ∈H10(Ω) : divv≡ 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 ∈ H1(Ω). (2.25)

Finally, the bilinear form B satisfies the continuous inf-sup condition

supv∈H1

0(Ω)\0

B(v,q)‖v‖1,Ω

> β‖q‖0,Ω , ∀q ∈ L20(Ω), (2.26)

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). Tothat end, it is enough to study the reduced problem of (2.9) on the kernel X. in (2.23). It consists infinding (u,θ) ∈ X×H1(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×H10 (Ω).

The following equivalence property is standard; see Girault & Raviart (1986).

LEMMA 2.1 If (u, p,θ) ∈H10(Ω)×L2

0(Ω)×H1(Ω) is a solution of (2.9), then u ∈X and (u,θ) is alsoa solution of (2.27). Conversely, if (u,θ) ∈X×H1(Ω) is a solution of (2.27), then there exists a uniquepressure p ∈ L2

0(Ω) such that (u, p,θ) is a solution of (2.9).


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 ∈ H10 (Ω) and θ1 is such that

θ1 ∈ H1(Ω), θ1|Γ = θD. (2.29)

THEOREM 2.1 Assume (2.7) and (2.8). Then, for any g ∈ L2(Ω), there is a lifting θ1 ∈ H1(Ω) ofθD ∈ H1/2(Γ ) satisfying (2.29) such that the reduced problem (2.27) has a solution (u,θ = θ0 +θ1) ∈H1

0(Ω)×H1(Ω). Furthermore, there exist constants Cu and Cθ only depending on ‖g‖0,Ω , and thestability constants in Section 2.2, such that

‖u‖1,Ω 6Cu‖θ1‖1,Ω , ‖θ‖1,Ω 6Cθ‖θ1‖1,Ω . (2.30)

The work (Lorca & Boldrini, 1996, Section 7) also establishes the uniqueness of small solutionsto problem (2.27), albeit under additional smoothness assumptions on the domain. Here, we restrictourselves to proving the following (more straightforward) uniqueness result, whose proof is motivatedby a similar argument in Cox et al. (2007) for Stokes-Oldroyd problems.

THEOREM 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‖g‖0,Ω , ‖u‖W1,∞(Ω), ‖θ‖W 1,∞(Ω)6M. (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)

and[AT(θ ;θ ,ψ)−AT(θ

?;θ?,ψ)] + [OT(u;θ ,ψ)−OT(u?;θ

?,ψ)] = 0, (2.33)

for all v ∈ X and ψ ∈ H10 (Ω).

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 prop-erty (2.21), and the fact that OS(u;u−u?,u−u?) = 0, see (2.24), we obtain

αS‖u−u?‖21,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)


Analogously, by taking ψ = θ −θ ? ∈ H10 (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‖θ −θ?‖2

1,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 ‖u?‖W1,∞(Ω) 6M and ‖θ ?‖W 1,∞(Ω) 6M by assumption (2.31), theright-hand sides in (2.36) and (2.37) can be bounded by

|AS(θ ;u?,u−u?)−AS(θ?;u?,u−u?)|6 νlipM‖θ −θ

?‖1,Ω‖u−u?‖1,Ω , (2.38)

and|AT(θ ;θ

?,θ −θ?)−AT(θ

?;θ?,θ −θ

?)|6 κlipM‖θ −θ?‖2

1,Ω ,

respectively. Hence, by using these inequalities in (2.36) and (2.37), respectively, and the continuity ofOS, OT, D, we find that

αS‖u−u?‖21,Ω 6νlipM‖θ −θ

?‖1,Ω‖u−u?‖1,Ω + CS‖u?‖1,Ω‖u−u?‖21,Ω

+ CD‖g‖0,Ω‖θ −θ?‖1,Ω‖u−u?‖1,Ω ,

(2.39)

as well as

αT‖θ −θ?‖2

1,Ω 6 κlipM‖θ −θ?‖2

1,Ω +CT‖u−u?‖1,Ω‖θ ?‖1,Ω‖θ −θ?‖1,Ω . (2.40)

We continue bounding the right-hand sides of (2.39) and (2.40) by applying the embedding esti-mate (1.2), assumption (2.31), and the inequality |ab|6 a2

2 + b2

2 . This results in

αS‖u−u?‖21,Ω 6M(CD +νlip)‖θ −θ

?‖1,Ω‖u−u?‖1,Ω +CSC∞M‖u−u?‖21,Ω

6M(CSC∞ +CD

2+

νlip

2)‖u−u?‖2

1,Ω +M2(CD +νlip)‖θ −θ

?‖21,Ω ,

(2.41)

respectively,

αT‖θ −θ?‖2

1,Ω 6κlipM‖θ −θ?‖2

1,Ω +CTC∞M‖u−u?‖1,Ω‖θ −θ?‖1,Ω

6M(

κlip +CTC∞

2

)‖θ −θ

?‖21,Ω +

M2

CTC∞‖u−u?‖21,Ω .

(2.42)

Finally, adding up (2.41) and (2.42), and bringing all the terms to the left-hand side of the resultinginequality, we conclude that(

αS−M(CSC∞ +K))‖u−u?‖2

1,Ω +(

αT−M(κlip +K))‖θ −θ

?‖21,Ω 6 0,

with K := (CTC∞ +CD +νlip)/2. Thus, if M satisfies

M < min

αS

CSC∞ +K,

αT

κlip +K

, (2.43)

then θ = θ ? and u = u?. This completes the proof.


3. Finite element discretization

In this section, we introduce our finite element method for approximating problem (2.1)–(2.5), reviewthe discrete stability properties of the forms involved, and discuss the reduced version of the discretevariational problem.

3.1 Preliminaries

We consider a family of regular and shape-regular triangulations Th of mesh size h that partition thedomain Ω into simplices K (i.e., triangles for d = 2 and tetrahedra for d = 3). For each K we denoteby 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 adjacentelements 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 fromwithin the interior of K±. Then, we define the jump of u, respectively the mean value of τ at x ∈ e by

JuK = u+⊗nK+ +u−⊗nK− , τ= 12(τ++ τ

−), (3.1)

where for u = (u1, ...,ud) and n = (n1, ...,nd), we denote by u⊗n the tensor product matrix [u⊗n]i, j =uin j, 1 6 i, j 6 d. For a boundary edge (face) e = ∂K+∩Γ , we set JuK = u+⊗n, with n denoting theunit 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

Vh =

v ∈ H0(div ;Ω) : v|K ∈ [Pk(K)]d , K ∈Th

,

Qh =

q ∈ L20(Ω) : q|K ∈ Pk−1(K), K ∈Th

,

Ψh =

ψ ∈ C (Ω) : ψ|K ∈ Pk(K), K ∈Th,

Ψh,0 =Ψh∩H10 (Ω).

(3.2)

Here, the space Pk(K) denotes the usual space of polynomials of total degree less or equal than k onelement K. The space Vh is non-conforming in H1

0(Ω), while Qh and Ψh are conforming in L20(Ω)

and H1(Ω), respectively. In fact, the space Vh is the space of divergence-conforming Brezzi-Douglas-Marini (BDM) elements; see Brezzi & Fortin (1991).

Consistent with our choice (3.2) for the discrete spaces, we need to introduce discontinuous versionsof 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) =

∫Ω

ν(ψ)∇hu : ∇hv− ∑e∈Eh(Th)

∫eν(ψ)∇hu : JvK

− ∑e∈Eh(Th)

∫eν(ψ)∇hv : JuK+ ∑

e∈Eh(Th)

a0

he

∫e

ν(ψ)JuK : JvK.(3.3)


Here, a0 > 0 is the interior penalty parameter, and we denote by ∇h the broken gradient operator. Asdiscussed in Cockburn et al. (2007), other choices for Ah

S are equally feasible (such as LDG or BRmethods), 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) =

∫Ω

(w ·∇h)u ·v+ ∑K∈Th

∫∂K\Γ

12(w ·nK−|w ·nK |)(ue−u) ·v, (3.4)

where ue is the trace of u taken from within the exterior of K. We note that convective forms withno 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

spaceθD,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 suchthat θh|Γ = θD,h and

AhS(θh;uh,v)+Oh

S(uh;uh,v)−B(v, ph)−D(θh,v) = 0,

B(uh,q) = 0,

AT(θh;θh,ψ)+OT(uh;θh,ψ) = 0,

(3.6)

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 divVh ⊆ Qh, it can be readily seen that

Xh = v ∈ Vh : divv≡ 0 in Ω ;

we refer to Cockburn et al. (2007) for details. Hence, Xh ⊂ H0(div 0;Ω). In particular, the followingresult holds.

LEMMA 3.1 An approximate velocity uh ∈ Vh obtained by (3.6) is exactly divergence-free, i.e., itsatisfies divuh ≡ 0 in Ω .

An important consequence of Lemma 3.1 is the provable energy-stability of the numerical schemein (3.6), without the need for symmetrization or other modifications of the convective terms; see alsothe discusssion in Cockburn et al. (2005, 2007). These stability properties are established in the nextsubsection.

3.3 Discrete stability properties

3.3.1 Broken spaces and norms. We introduce the broken space

Hr(Th) = v ∈ L2(Ω) : v|K ∈Hr(K), K ∈Th , r > 0. (3.8)


We shall mostly work with r = 1 and r = 2; in these cases we use the broken norms

‖v‖21,Th

= ∑K∈Th

‖∇hv‖20,K + ∑

e∈Eh

a0h−1e ‖JvK‖2

0,e, v ∈H1(Th), (3.9)

‖v‖22,Th

= ‖v‖21,Th

+ ∑K∈Th

h2K |v|22,K , v ∈H2(Th). (3.10)

By the inverse estimate |p|2,K 6Ch−1K |p|1,K for all K ∈Th, p ∈ Pk(K), we see that

‖v‖2,Th 6C‖v‖1,Th , v ∈ Vh. (3.11)

We recall the following broken version of the usual Sobolev embeddings: for d = 2,3, and anyp ∈ I(d)⊂ R there exists a constant C > 0 such that

‖v‖Lp(Ω) 6C‖v‖1,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 casep = 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

‖u‖W1,∞(Th)= max

K∈Th‖u‖W1,∞(K). (3.14)

We shall also make use of the augmented H1-norm

‖ψ‖21,Eh

= ‖ψ‖21,Ω + ∑

e∈Eh(Th)

h−1e ‖ψ‖2

0,e, ψ ∈ H1(Ω). (3.15)

3.3.2 Continuity. First, we establish continuity properties of the elliptic forms AhS and AT, respec-

tively. To that end, we recall that by (2.14), the form AT is a bounded bilinear form over H1(Ω)×H1(Ω). To bound the DG form Ah

S, we proceed in a standard way; see Arnold et al. (2002), for instance.Indeed, by using the standard trace inequalities

‖v‖0,∂K 6C(

h−1/2K ‖v‖0,K +h1/2

K |v|1,K), v ∈ H1(K), (3.16)

‖p‖0,∂K 6Ch−1/2K ‖p‖0,K , p ∈ Pk(K), (3.17)

and the inverse inequality in (3.11), we obtain the following result.

LEMMA 3.2 There holds

|AhS(·;u,v)|6C‖u‖2,Th‖v‖1,Th , u ∈H2(Th), v ∈ Vh, (3.18)

|AhS(·;u,v)|6 CA‖u‖1,Th‖v‖1,Th , u,v ∈ Vh. (3.19)


Moreover, the elliptic forms are Lipschitz continuons with respect to the first argument. For theconforming form AT, this follows from (2.17). The following result holds for the DG form Ah

S.

LEMMA 3.3 Let ψ1, ψ2 ∈ H1(Ω), u ∈ C1(Th), and v ∈ Vh. Then there holds∣∣∣AhS(ψ1;u,v)−Ah

S(ψ2;u,v)∣∣∣6 Clipνlip ‖ψ1−ψ2‖1,Eh ‖u‖W1,∞(Th)

‖v‖1,Th . (3.20)

In addition, if u ∈H10(Ω), then∣∣∣Ah

S(ψ1;u,v)−AhS(ψ2;u,v)

∣∣∣6 Clipνlip ‖ψ1−ψ2‖1,Ω ‖u‖W1,∞(Th)‖v‖1,Th . (3.21)

The constant Clip > 0 is independent of the mesh size.

Proof. As before, we note that∣∣∣AhS(ψ1;u,v)−Ah

S(ψ2;u,v)∣∣∣6 |T1|+ |T2|+ |T3|+ |T4|,

with

T1 =∫

Ω

(ν(ψ1)−ν(ψ2))∇hu : ∇hv, T2 = ∑e∈Eh(Th)

∫e(ν(ψ1)−ν(ψ2))∇u : JvK,

T3 = ∑e∈Eh(Th)

∫e(ν(ψ1)−ν(ψ2))∇v : JuK, T4 = ∑

e∈Eh(Th)

a0

he

∫e(ν(ψ1)−ν(ψ2))JuK : JvK.

For T1, the Lipschitz continuity of ν in (2.7) readily yields the bound

|T1|6 νlip‖ψ1−ψ2‖0,Ω‖u‖W1,∞(Th)‖∇hv‖0,Ω .

To estimate T2, we notice that, since u ∈ C1(Th), we have ‖∇hue‖L∞(e) 6 ‖u‖W1,∞(Th)for all

e ∈ Eh(Th). Hence, from the Lipschitz continuity of ν it follows that

|T2|6 νlip‖u‖W1,∞(Th) ∑e∈Eh(Th)

‖ψ1−ψ2‖0,e ‖JvK‖0,e.

By applying the discrete Cauchy-Schwarz inequality, the shape-regularity of the meshes, and the traceinequality (3.16), the sum over the edges (faces) can be bounded by

∑e∈Eh(Th)

‖ψ1−ψ2‖0,e ‖JvK‖0,e 6(

∑e∈Eh(Th)

he‖ψ1−ψ2‖20,e)1/2 (

∑e∈Eh(Th)

h−1e ‖JvK‖2

0,e)1/2

6C(

∑K∈Th

hK‖ψ1−ψ2‖20,∂K

)1/2‖v‖1,Th

6C‖ψ1−ψ2‖1,Ω‖v‖1,Th .

This yields|T2|6Cνlip‖u‖W1,∞(Th)

‖ψ1−ψ2‖1,Ω‖v‖1,Th .


For the term T3, we have ‖JuK‖L∞(e) 6 2‖u‖L∞(Ω) 6 2‖u‖W1,∞(Th)for any e ∈ Eh(Th). Hence, the

Lipschitz continuity of ν , the Cauchy-Schwarz inequality the shape-regularity of the meshes, and thepolynomial trace inequality (3.17),

|T3|6Cνlip‖u‖W1,∞(Th) ∑e∈Eh(Th)

‖ψ1−ψ2‖0,e‖∇v‖0,e

6Cνlip‖u‖W1,∞(Th)

(∑

e∈Eh(Th)

h−1e ‖ψ1−ψ2‖2

0,e)1/2(

∑K∈Th

hK‖∇v‖20,∂K

)1/2

6Cνlip‖u‖W1,∞(Th)‖ψ1−ψ2‖1,Eh‖∇hv‖0,Ω .

Similarly, T4 can be bounded by:

|T4|6Cνlip‖u‖W1,∞(Th)‖ψ1−ψ2‖1,Eh‖v‖1,Th .

Gathering the above bounds for T1 through T4 implies the estimate (3.20).If u ∈ H1

0(Ω), then T3 = T4 = 0, and the second bound (3.21) follows from the estimates for T1and T2.

Second, we notice that the forms B and D are bounded by

|B(v,q)|6 CB‖v‖1,Th‖q‖0,Ω , v ∈H1(Th), q ∈ L20(Ω), (3.22)

|D(ψ,v)|6 CD‖g‖0,Ω‖ψ‖1,Ω‖v‖1,Th , v ∈H1(Th), ψ ∈ H1(Ω). (3.23)

The estimate for B is straightforward, and the one for D follows from the embedding (3.12) with p = 4and Holder’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.

LEMMA 3.4 There exists a constant CS > 0, independent of the mesh size, such that

|OhS(w1;u,v)−Oh

S(w2;u,v)|6 CS‖w1−w2‖1,Th‖u‖1,Th‖v‖1,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), andmakes use of the embedding (3.12) with p = 4. In the case d = 3, we proceed similarly: we use theshape-regularity of the meshes, Holder’s inequality, the embedding (3.12) with p = 4, and the traceestimate h1/4

K ‖z‖L4(∂K) 6C(‖z‖L4(K)+‖∇z‖L2(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 CT‖w‖1,Th‖ϕ‖1,Ω‖ψ‖1,Ω , w ∈H1(Th), ψ,ϕ ∈ H1(Ω). (3.25)

This follows similarly from Holder’s inequality and the embedding (3.12). We use the following variantof (3.25).

LEMMA 3.5 There is a constant CT,2 > 0 such that

|OT(w;θ ,ψ)|6 CT,2‖θ‖L3(Ω)‖w‖1,Th‖ψ‖1,Ω , w ∈H0(div 0;Ω), θ ,ψ ∈ H1(Ω). (3.26)


Proof. Integration by parts yields and using that divw≡ 0 in Ω , w ·n = 0 on Γ yield

OT(w;θ ,ψ) =∫

Ω

(w ·∇θ)ψ =−∫

Ω

θ(w ·∇ψ).

From Holder’s inequality we obtain

|OT(w;θ ,ψ)|6C‖θ‖L3(Ω)‖∇ψ‖0,Ω‖w‖L6(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 discretespaces is implied by (2.22). Due to the bounds of ν in (2.8) the DG form Ah

S is also elliptic, and we have

AhS(·,v,v)> αS‖v‖2

1,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 Oh

S and OT , let w ∈H0(div 0;Ω). Then we have

OhS(w;u,u) =

12 ∑

e∈EI(Th)

∫e|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 astandard 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 ∈ H1(Ω). (3.29)

Finally, we recall the discrete inf-sup condition for B:

supvh∈Vh\0

B(vh,qh)

‖vh‖1,Th

> β‖qh‖0,Ω ∀qh ∈ Qh, (3.30)

with β > 0 independent of the mesh size. The proof of (3.30) follows along the lines of Hansbo &Larson (2002) from the surjectivity of div : H1

0(Ω)→ L20(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.

LEMMA 3.6 If (uh, ph,θh) ∈ Vh×Qh×Ψh is a solution of (3.6), then uh ∈ Xh and (uh,θh) is also asolution of (3.31). Conversely, if (uh,θh) ∈ Xh×Ψh is a solution of (3.31), then there exists a uniquepressure ph ∈ Qh such that (uh, ph,θh) is a solution of (3.6).


In what follows, we shall discuss the existence for the reduced problem (3.31). We notice that theuniqueness of discrete solutions is an open issue. Indeed, adapting Theorem 2.2 to the discrete settingrequires controlling the augmented norm (3.15) appearing in the discrete counterpart of (2.38). This isin contrast to conforming Galerkin (CG) methods, where a discrete version of Theorem 2.2 can easilybe established. On the other hand, in our (non-exhaustive) numerical tests presented in Section 6, wedid 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 argu-ments proposed in Lorca & Boldrini (1996) and based on Brouwer’s fixed point theorem. We propose ageneral approach of constructing discrete liftings based on computing harmonic extensions, and discussthe 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 assump-tion. As in the continuous case, we write the discrete temperature θh as θh = θh,0+θh,1, with θh,0 ∈Ψh,0and

θh,1 ∈Ψh, θh,1|Γ = θD,h. (4.1)

LEMMA 4.1 Let (uh,θh) be a solution of (3.31) with θh = θh,0 +θh,1 as in (4.1). Assume that

Cdep‖g‖0,Ω‖θh,1‖L3(Ω) 612, (4.2)

with

Cdep =CDCT,2

αSαT, (4.3)

then there exist constants Cu and Cθ only depending on ‖g‖0,Ω and the stability constants in Section 3.3,such that

‖uh‖1,Th 6 Cu‖θh,1‖1,Ω , ‖θh‖1,Ω 6 Cθ‖θh,1‖1,Ω . (4.4)

(Explicit expressions for Cu and Cθ 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 twoequations

AhS(θh;uh,uh)+Oh

S(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 Oh

S in (3.28), and the bound-edness of D in (3.23) imply

‖u‖1,Th 6 α−1S CD‖g‖0,Ω‖θh,0‖1,Ω + α

−1S CD‖g‖0,Ω‖θh,1‖1,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

‖θh,0‖1,Ω 6 α−1T κ2‖θh,1‖1,Ω +α

−1T CT,2‖θh,1‖L3(Ω)‖uh‖1,Th . (4.7)


Then, using the bound (4.7) in (4.6) yields

‖uh‖1,Th 6 α−1S α

−1T CDCT,2‖g‖0,Ω‖θh,1‖L3(Ω)‖uh‖1,Th + α

−1S α

−1T CD‖g‖0,Ω

(αT +κ2

)‖θh,1‖1,Ω .

Hence, referring to assumption (4.2), we obtain

‖uh‖1,Th 6 Cu‖θh,1‖1,Ω with Cu = 2α−1S α

−1T CD‖g‖0,Ω

(αT +κ2

). (4.8)

Moreover, by using the triangle inequality, estimate (4.8), the definition of Cdep and assumption (4.2)we find that

‖θh‖1,Ω 6 ‖θh,0‖1,Ω +‖θh,1‖1,Ω

6 (α−1T κ2 +1)‖θh,1‖1,Ω +α

−1T CT,2‖θh,1‖L3(Ω)‖uh‖1,Th

6 (α−1T κ2 +1)‖θh,1‖1,Ω +2α

−1T Cdep‖g‖0,Ω‖θh,1‖L3(Ω)(αT +κ2)‖θh,1‖1,Ω

6 (α−1T κ2 +1)‖θh,1‖1,Ω +α

−1T (αT +κ2)‖θh,1‖1,Ω .

Hence,

‖θh‖1,Ω 6 Cθ‖θh,1‖1,Ω with Cθ = 2(1+α−1T κ2). (4.9)

This completes the proof. We are now ready to state our main existence result.

THEOREM 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 thosein (4.2), (4.4). To that end, we establish the following lemma.

LEMMA 4.2 For any θD,h ∈Λh, there is a discrete lifting θh,1 ∈Ψh, θh,1|Γ = θD,h, which satisfies

‖θh,1‖1,Ω 6Clift‖θD,h‖1/2,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 ∈ H1(Ω) such that

θ |Γ = θD,h, ‖θ‖1,Ω 6C‖θD,h‖1/2,Γ .

Denoting by θh,1 ∈Ψh the Scott-Zhang quasi-interpolant of θ ; see Scott & S.Zhang (1990). As it isstable in H1(Ω) and reproduces polynomial boundary conditions, we have

‖θh,1‖1,Ω 6C‖θ‖1,Ω 6C‖θD,h‖1/2,Γ .

This implies the assertion.


COROLLARY 4.1 Let θD,h ∈Λh be a discrete boundary datum. Assume that

CdepCembClift||g||0,Ω ||θD,h||1/2,Γ 6 1/2, (4.10)

with Cdep defined in (4.3), Cemb > 0 the embedding constant in (3.12) for p = 3, and Clift the constant inLemma 4.2, Then the lifting θh,1 of Lemma 4.2 gives rises to a solution (uh,θh) to (3.31) which satisfiesthe stability bounds

‖uh‖1,Th 6 CuClift‖θD,h‖1/2,Γ , ‖θh‖1,Ω 6 CθClift‖θD,h‖1/2,Γ , (4.11)

where Cu and Cθ 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 theembedding (3.12) with p = 3 yields

Cdep‖g‖0,Ω‖θh,1‖L3(Ω) 6 CdepCemb‖g‖0,Ω‖‖θh,1‖1,Ω

6 CdepCembClift‖g‖0,Ω‖‖θD,h‖1/2,Γ 612.

Hence, the assertion follows from Theorem 4.1 and the particular choice of θh,1.

REMARK 4.1 We point out that the discrete lifting θh,1 constructed in Lemma 4.2 cannot be easilycomputed numerically. On the other hand, it is well known that the discrete lifting θh,1 ∈Ψh withminimum H1-norm is given by the discrete generalized harmonic extension of θD,h. It can be computedby 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 H1(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

‖θh,1‖21,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 = ‖ψh,1‖21,Ω .

Hence, using the discrete harmonic extension θh,1 in Corollary 4.1 gives rise to the same existence resultand stability bounds, and leads to a systematic (albeit expensive) approach to compute a suitable discretelifting for any discrete boundary datum.

REMARK 4.2 We note that the stability bounds in (4.11), (4.4) will only be useful in an error analy-sis if the H1/2-norms of the discrete boundary data θD,h, respectively the H1-norms of the associatedliftings 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 applicationof Theorem 4.1 or Corollary 4.1. In addition, the construction of the liftings may be computationally


expensive. We shall focus mainly on nodal interpolation of the boundary data, and in Remark 4.4, wemention 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 boundarynodes. With each node N ∈N (Th), we associate the (global) Lagrange basis function lN(xxx) ∈Ψh. Wethen denote by I : C(Ω)→Ψh, v 7→I v, the classical nodal interpolation operator given by

I v(xxx) = ∑N∈N (Th)

v(N)lN(xxx). (4.12)

The restriction of I to the boundary (nodes) is denoted by IΓ : C(Γ )→ Λh. Evidently, we have theproperty that (I v)|Γ = IΓ (v|Γ ). In view of assumption (2.6), we now take the discrete boundarydatum θD,h ∈Λh as the nodal interpolant of θD:

θD,h = IΓ θD. (4.13)

We first show that ‖θD,h‖1/2,Γ can be bounded independently of the mesh size (under additional smooth-ness assumption on the exact temperature); cf. Remark 4.2.

LEMMA 4.3 If the exact temperature θ of (2.1)– (2.5) belongs to H2(Ω), then ‖θD,h‖H1/2(Γ ) is boundedas h→ 0.

Proof. We first note that θ ∈ H2(Ω) implies θ ∈ C(Ω), and hence the nodal interpolant I of v iswell-defined. Hence,

‖θD−θD,h‖1/2,Γ 6 ‖θ −I θ‖1,Ω 6 h‖θ‖2,Ω .

Then, by the triangle inequality,

‖θD,h‖H1/2(Γ ) 6 ‖θD−θD,h‖H1/2(Γ )+‖θD‖H1/2(Γ ) 6Ch‖θ‖2,Ω +‖θD‖H1/2(Γ ),

which implies the assertion.

REMARK 4.3 The argument in Lemma 4.3 is somewhat adhoc, but sufficient for our purposes. Wealso mention that the stability result ‖θD,h‖C(Γ ) 6C‖θD‖C(Γ ) can be readily shown. Moreover, stabilitybounds 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 asdiscussed in Remark 4.1, thereby ensuring that Corollary 4.1 holds. However, the computationally mostpractical discrete lifting is given by

θh,1(xxx) = ∑N∈NB(Th)

θD(N)lN(xxx) ∈Ψh. (4.14)

This lifting corresponds to a standard way of imposing non-homogenous boundary conditions in a finiteelement implementation, where, in the resulting matrix system, the unknown coefficients at boundarynodes N are simply set to θD(N). Obviously, the choice (4.14) allows one to satisfy condition (4.2) forall functions ggg, 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-normof θh,1 can be made as small as possible for sufficiently small mesh sizes. Our numerical results will bebased on this choice.


REMARK 4.4 A theoretical construction of a discrete and stable lifting has been given in Scott &S.Zhang (1990): For θD ∈ H1/2(Γ ), there is a lifting θh,1 ∈Ψh, satisfying θD,h = θh,1|Γ , and

‖θD,h‖1/2,Γ 6 ‖θh,1‖1,Ω 6C‖θD‖1/2,Γ .

Although in principle it is possible to compute θh,1 if a stable lifting of θD in H1(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 explic-itly constructed, say from a known lifting of the boundary conditions. If, in addition, θ1 is sufficientlysmooth, we may simply take θh,1 as the nodal interpolant of θ1. This would allow one again to satisfycondition 4.2, and to obtain existence of discrete solutions for sufficiently small mesh sizes. Moreover,‖θh,1‖1,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 followingform Brezis (2011): Let K be a non-empty compact convex subset of a finite dimensional normedspace, and let L be a continuous mapping of K into itself. Then L has a fixed point in K . Weproceed in several steps.

Step 1: We introduce the finite dimensional set

K =

(uh,θh) ∈ Xh×Ψh : ‖uh‖1,Th 6 Cu‖θh,1‖1,Ω , ‖θh‖1,Ω 6 Cθ‖θh,1‖1,Ω

and θh = θh,0 +θh,1

, (4.15)

with Cu and Cθ the constants defined in (4.8) and (4.9), respectively. It is convex and compact. We thendefine 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)+Oh

S(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 thatproblem (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, anddenote by (uh,θh) ∈Vh×Ψh the solution to the problem (4.16). Then, as in the proof of Lemma 4.1, wetake the test function (v,ψ) = (uh,θh,0). In the first of the two resulting equations, we use the coercivityof Ah

S in (3.27), the positivity of OhT in (3.28), and the boundedness of D in (3.23). This results in

‖uh‖21,Th6 α

−1S |D(ϕh,uh)|6 α

−1S CD‖g‖0,Ω‖ϕh‖1,Ω‖uh‖1,Th .

Division by ‖u‖1,Th and the bound ‖ϕh‖1,Ω 6 Cθ‖θh,1‖1,Ω then give

‖uh‖1,Th 6 α−1S CDCθ‖g‖0,Ω‖θh,1‖1,Ω = Cu‖θh,1‖1,Ω ,


where we have also used the identity

Cu = α−1S CD‖g‖0,ΩCθ . (4.17)

In the second of the two resulting equations, we use the coercivity of AT in (2.22), property (3.29), theboundedness of AT and OT in (2.14) and Lemma 3.5, respectively, the bound ‖zh‖1,Th 6 Cu‖θh,1‖1,Ω ,and division by ‖θh,0‖1,Ω , to find that

‖θh,0‖1,Ω 6 α−1T κ2‖θh,1‖1,Ω +α

−1T CT,2Cu‖θh,1‖1,Ω‖θh,1‖L3(Ω).

Then, from the identity (4.17) and assumption (4.2),

‖θh,0‖1,Ω 6 α−1T κ2‖θh,1‖1,Ω + α

−1S α

−1T CDCT,2‖g‖0,ΩCθ‖θh,1‖1,Ω‖θh,1‖L3(Ω)

6 α−1T κ2‖θh,1‖1,Ω +

Cθ

2‖θh,1‖1,Ω .

Then, the triangle inequality and the definition Cθ = 2(1+α−1T κ2) in (4.9) imply

‖θh‖1,Ω 6 ‖θh,0‖1,Ω +‖θh,1‖1,Ω

6 (1+α−1T κ2)‖θh,1‖1,Ω +

Cθ

2‖θh,1‖1,Ω 6 Cθ‖θh,1‖1,Ω .

Hence, we have (uh,θh) ∈K . It is now clear that the existence of a fixed point of L : K →K isequivalent 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

‖zm− z‖1,Th

m→∞−→ 0 and ‖ϕm−ϕ‖1,Ωm→∞−→ 0.

We note that by the trace inequality (3.16) and for a fixed mesh size, there also holds limm→∞‖ϕm−ϕ‖1,Eh =

0. Thus, setting (u,θ) = L (z,ϕ) and (um,θm) = L (zm,ϕm), m ∈ N, we need to prove that

‖um−u‖1,Th

m→∞−→ 0 and ‖θm−θ‖1,Ωm→∞−→ 0. (4.18)

From the definition of L in (4.16) we see that there hold

AhS(ϕm;um,v)+Oh

S(zm;um,v)−D(ϕm,v) = 0,

AT(ϕm;θm,ψ)+OT(zm;θm,ψ) = 0,

and

AhS(ϕ;u,v)+Oh

S(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) − Ah

S(ϕ;u,v)+OhS(zm;um,v)−Oh

S(z;u,v)−D(ϕm−ϕ,v) = 0, (4.19)


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)+ Oh

S(zm;u−um,v) = − [AhS(ϕ;u,v)−Ah

S(ϕm;u,v)]

− [OhS(z;u,v)−Oh

S(zm;u,v)]+D(ϕm−ϕ,v).

We take v = u−um, use the ellipticity property of AhS and Oh

S in (3.27) and (3.28), respectively, as wellas the continuity of Oh

S and D, to get

αS‖u−um‖21,Th6∣∣Ah

S(ϕ;u,u−um)−AhS(ϕm;u,u−um)

∣∣+ CS ‖z− zm‖1,Th‖u‖1,Th‖u−um‖1,Th + CD‖g‖0,Ω‖ϕ−ϕm‖1,Ω‖u−um‖1,Th .

With the continuity property (3.20) for AhS and division by ‖u−um‖1,Th , it follows that

‖u−um‖1,Th 6C(‖ϕ−ϕm‖1,Eh‖u‖W1,∞(Th)

+‖z− zm‖1,Th‖u‖1,Th +‖ϕ−ϕm‖1,Ω

).

Hence, we find thatlim

m→∞‖u−um‖1,Th = 0. (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 thecontinuity property (2.17), the ellipticity (2.22), and the bound (3.25) for OT . Dividing the resultinginequality by ‖θ −θm‖1,Ω results in

‖θ −θm‖1,Ω 6C(‖ϕ−ϕm‖1,Ω‖θ‖W 1,∞(Ω)+ ‖z− zm‖1,Th‖θ‖1,Ω

).

Therefore,lim

m→∞‖θ −θm‖1,Ω = 0. (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 bystating our error bounds. Then, we present the details of the proofs in several steps.


5.1 Error estimates

We shall prove the following error estimates.

THEOREM 5.1 Let θD,h be the nodal interpoland of θD in (4.13), and assume that (2.6) and the small dataassumption (4.10) hold true. Let (u, p,θ) be a solution of (2.9), and let (uh, ph,θh) be an approximatesolution obtained by (3.6) with the discrete lifting θh,1 of Lemma 4.2 or the harmonic extension inRemark 4.1 and satisfying the stability bounds (4.11) in Corollary 4.1. Assume further that

max‖g‖0,Ω ,‖u‖W1,∞(Ω),‖θ‖W 1,∞(Ω)

6minM,M, (5.1)

with M and M sufficiently small, as specified in (2.43) and (5.18) below. We further suppose that, fork = 1,

u ∈ C1(Ω)∩H2(Ω)∩X, p ∈ H1(Ω), θ ∈W 1,∞(Ω)∩H2(Ω), (5.2)

and, for k > 2,u ∈Hk+1(Ω)∩X, p ∈ Hk(Ω), θ ∈ Hk+1(Ω). (5.3)

Then there exist two constants C > 0 independent of the mesh size such that

‖u−uh‖2,Th +‖θ −θh‖1,Ω 6Chk(‖u‖k+1,Ω +‖θ‖k+1,Ω ), (5.4)

and‖p− ph‖0,Ω 6Chk(‖p‖k,Ω +‖u‖k+1,Ω +‖θ‖k+1,Ω ). (5.5)

The proof of Theorem 5.1 is presented in Section 5.2.

REMARK 5.1 In our analysis, we shall need the base regularity (u,θ) ∈C1(Ω)×W 1,∞(Ω) as assumedin the lowest-order case k = 1 in (5.2); cf. Lemma 3.3 and (2.17). Notice that for k > 2, the regularityassumption (u,θ) ∈Hk+1(Ω)×Hk+1(Ω) in (5.3) implies (u,θ) ∈ C1(Ω)×C1(Ω).

REMARK 5.2 Observe that under the small solution assumption (5.1), the exact solution to (2.9) isunique, in agreement to Theorem 2.1. On the other hand and as mentioned above, an analogous unique-ness 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, phθh) a finite elementapproximation obtained by its discrete counterpart (3.6). To simplify the subsequent analysis, we writeeu = u−uh, eθ = θ −θh and ep = p− ph. As usual, we shall then decompose these errors into

eu = ξξξ u +χχχu = (u− vh)+(vh−uh),

eθ = ξθ +χθ = (θ − ψh)+(ψh−θh),

ep = ξp +χp = (p− qh)+(qh− ph),

(5.6)

where we take vh as the BDM projection of u, ψh =I θ ∈Ψh is the nodal projection of θ , as introducedin Section 4.2, and qh is the L2-projection of p into Qh.


We recall that for u ∈ X, we have vh ∈ Xh; see, e.g., Brezzi & Fortin (1991). Then, we also haveχχχu ∈ Xh. The following approximation properties are standard:

‖ξξξ u‖2,Th 6Chk‖u‖k+1,Ω , ‖ξθ‖1,Ω 6Chk‖θ‖k+1,Ω , ‖ξp‖0,Ω 6Chk‖p‖k,Ω . (5.7)

Then, according to the triangle inequality and the inverse inequality (3.11), we see that

‖eu‖2,Th 6‖ξξξ u‖2,Th +‖χχχu‖2,Th 6C hk‖u‖k+1,Ω +C‖χχχu‖1,Th ,

‖eθ‖1,Ω 6‖ξθ‖1,Ω +‖χθ‖1,Ω 6C hk‖θ‖k+1,Ω +‖χθ‖1,Ω ,

‖ep‖0,Ω 6‖ξp‖0,Ω +‖χp‖0,Ω 6C hk‖p‖k,Ω +‖χp‖0,Ω .

(5.8)

Hence, to prove the error estimate (5.1), we need to show the optimal convergence of ‖χχχu‖1,Th , ‖χθ‖1,Ω ,and ‖χp‖0,Ω .

To do so, we shall employ the following Galerkin orthogonality property.

LEMMA 5.1 Assume that u ∈H2(Ω)∩X. Then we have[Ah

S(θ ;u,v)−AhS(θh;uh,v)

]+[Oh

S(u;u,v)−OhS(uh;uh,v)

]−B(v,ep)−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 byparts that the exact solution (u, p,θ) satisfies

AhS(θ ;u,v)+Oh

S(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 thirdequations are readily verified.

5.2.2 Error estimates in the velocity and temperature. We now start by analyzing the convergence of‖χχχu‖1,Th and ‖χθ‖1,Ω .

LEMMA 5.2 There exists a constant C1 > 0 independent of the mesh size such that

(αS−CSC∞M)‖χχχu‖21,Th6C1

(‖ξξξ u‖2,Th +‖ξθ‖1,Ω

)‖χχχu‖1,Th

+ M(Clipνlip +CD)‖χχχu‖1,Th‖χθ‖1,Ω .

Proof. First, note that χχχu ∈ Xh. From the ellipticity of AhS in (3.27) and elementary calculations, it is

not difficult to see that

αS‖χχχu‖21,Th6 Ah

S(θh; χχχu,χχχu) = A1S +A2

S +A3S +A4

S, (5.9)


with the terms A1S through A4

S given by

A1S = Ah

S(θh;u,χχχu)−AhS(ψh;u,χχχu),

A2S = Ah

S(ψh;u,χχχu)−AhS(θ ;u,χχχu),

A3S = Ah

S(θ ;u,χχχu)−AhS(θh;uh,χχχu),

A4S = −Ah

S(θh;ξξξ u,χχχu).

Similarly, thanks to the positivity of OhS in (3.28), we obtain

06 OhS(uh; χχχu,χχχu) = O1

S +O2S +O3

S +O4S, (5.10)

with O1S through O4

S given by

O1S = Oh

S(uh;u,χχχu)−OhS(vh;u,χχχu),

O2S = Oh

S(vh;u,χχχu)−OhS(u;u,χχχu),

O3S = Oh

S(u;u,χχχu)−OhS(uh;uh,χχχu),

O4S = −Oh

S(uh;ξξξ u,χχχu).

From the first error equation in Lemma 5.1, it further follows that

A3S +O3

S = D(eθ ,χχχu) = D(ξθ ,χχχu)+D(χθ ,χχχu), (5.11)

where we have used the fact that B(χχχu,ep) = 0 since χχχu ∈ Xh is exactly divergence-free.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 ‖u‖W1,∞(Th)= ‖u‖W1,∞(Ω) 6 M (since u ∈ C1(Ω)). We find that

|A1S|6 Clipνlip‖θh− ψh‖1,Ω‖u‖W1,∞(Th)

‖χχχu‖1,Th 6 MClipνlip‖χθ‖1,Ω‖χχχu‖1,Th ,

|A2S|6 Clipνlip‖θ − ψh‖1,Ω‖u‖W1,∞(Th)

‖χχχu‖1,Th 6 MClipνlip‖ξθ‖1,Ω‖χχχu‖1,Th ,

|A4S|6C‖ξξξ u‖2,Th‖χχχu‖1,Th .

(5.12)

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 ‖u‖1,Ω 6C∞‖u‖W1,∞(Ω) 6C∞M by (1.2). This results in

|O1S|6 CS‖u‖1,Ω‖χχχu‖2

1,Th6 CSC∞M‖χχχu‖2

1,Th,

|O2S|6 CS‖ξξξ u‖1,Th‖u‖1,Ω‖χχχu‖1,Th 6 CSC∞M‖ξξξ u‖2,Th‖χχχu‖1,Th ,

|O4S|6 CS‖uh‖1,Th‖ξξξ u‖1,Th‖χχχu‖1,Th 6 CSCuClift‖θD,h‖H1/2(Γ )‖ξξξ u‖2,Th‖χχχu‖1,Th .

(5.13)

In the bound for |O4S|, we emphasize that ‖θD,h‖H1/2(Γ ) is bounded independently of the mesh size, in

agreement with Lemma 4.3.


Finally, to estimate the terms in (5.11) we employ the continuity of D and the hypothesis that‖g‖0,Ω 6 M. We conclude that

|D(ξθ ,χχχu)|6 MCD‖ξθ‖1,Ω‖χχχu‖1,Th ,

|D(χθ ,χχχu)|6 MCD‖χθ‖1,Ω‖χχχu‖1,Th .(5.14)

Hence, from (5.9), (5.10) and (5.11), and the upper bounds (5.12), (5.13) and (5.14) the assertionfollows.

A corresponding upper bound for ‖χθ‖1,Ω is established in a similar fashion.

LEMMA 5.3 There exists a constant C2 > 0 independent of the mesh size such that

(αT−κlipM)‖χθ‖21,Ω 6C2

(‖ξξξ u‖2,Th +‖ξθ‖1,Ω

)‖χθ‖1,Ω +CTC∞M‖χχχu‖1,Th‖χθ‖1,Ω .

Proof. We proceed similarly to the proof of Lemma 5.2. Indeed, by adding and subtracting suitableterms and noting that χθ ∈ H1

0 (Ω), the ellipticity (2.22) of AT and property (3.29) for OT imply that

αT‖χθ‖21,Ω 6 AT(θh; χθ ,χθ )+OT(uh; χθ ,χθ )

= A1T +A2

T +A3T +A4

T +O1T +O2

T +O3T +O4

T,(5.15)

with

A1T = AT(θh;θ ,χθ )−AT(ψh;θ ,χθ ), A2

T = AT(ψh;θ ,χθ )−AT(θ ;θ ,χθ ),

A3T = AT(θ ;θ ,χθ )−AT(θh;θh,χθ ), A4

T = −AT(θh;ξθ ,χθ ),

and

O1T =−OT(χχχu;θ ,χθ ), O2

T =−OT(ξξξ u;θ ,χθ ),

O3T = OT(u;θ ,χθ )−OT(uh;θh,χθ ), O4

T = −OT(uh;ξθ ,χθ ).

As before, the third error equation in Lemma 5.1 yields

A3T +O3

T = 0.

Now, by the continuity properties of AT in (2.14), (2.17), and since ‖θ‖W 1,∞(Ω) 6 M, we see that

|A1T|6 κlip‖θ‖W 1,∞(Ω)‖θh− ψh‖1,Ω‖χθ‖1,Ω 6 κlipM‖χθ‖2

1,Ω ,

|A2T|6 κlip‖θ − ψh‖1,Ω‖θ‖W 1,∞(Ω)‖χθ‖1,Ω 6 κlipM‖ξθ‖1,Ω‖χθ‖1,Ω ,

|A4T|6 κ2‖ξθ‖1,Ω‖χθ‖1,Ω .

(5.16)

On the other hand, by employing the bound ‖θ‖1,Ω 6C∞‖θ‖W 1,∞(Ω) 6C∞M, the inequality (2.14),and the continuous dependence in (4.11), we obtain

|O1T|6 CT‖χχχu‖1,Th‖θ‖1,Ω‖χθ‖1,Ω 6 CTC∞M‖χχχu‖1,Th‖χθ‖1,Ω ,

|O2T|6 CT‖ξξξ u‖1,Th‖θ‖1,Ω‖χθ‖1,Ω 6 CTC∞CTM‖ξξξ u‖2,Th‖χθ‖1,Ω ,

|O4T|6 CT‖uh‖1,Th‖ξθ‖1,Ω‖χθ‖1,Ω 6 CTCuClift‖θD,h‖H1/2(Γ )‖ξθ‖1,Ω‖1,Ω‖χθ‖1,Ω .

(5.17)


The desired result follows from (5.15) and the estimates in (5.16) and (5.17), noting again that‖θD,h‖H1/2(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.

LEMMA 5.4 There is a constant C > 0 independent of the mesh size such that

‖u−uh‖2,Th +‖θ −θh‖1,Ω 6Chk(‖u‖k+1,Ω +‖θ‖k+1,Ω ).

Proof. Starting from (5.8), it is enough to bound ‖χχχu‖1,Th and ‖χθ‖1,Ω . To this end, we set

L(u,θ) = ‖ξξξ u‖2,Th +‖ξθ‖1,Ω .

Adding the two bounds in Lemma 5.2 and Lemma 5.3 results in

(αS−CSC∞M)‖χχχu‖21,Th

+(αT−κlipM)‖χθ‖21,Ω 6CL(u,θ)

[‖χχχu‖1,Th +‖χθ‖1,Ω

]+ M(Clipνlip +CD)‖χχχu‖1,Th‖χθ‖1,Ω

+CTC∞M‖χχχu‖1,Th‖χθ‖1,Ω .

An application of the inequality |ab|6 a2

2 + b2

2 allows us to bring the last two terms above to the right-hand side. By setting K =

((Clipνlip)+CD +CTC∞

)/2, we obtain

(αS− (CSC∞ + K)M)‖χχχu‖21,Th

+(αT− (κlip + K)M)‖χθ‖21,Ω

6CL(u,θ)[‖χχχu‖1,Th +‖χθ‖1,Ω

].

Hence, if we choose M such that

M < inf

αS

CSC∞ + K,

αT

κlip + K

, (5.18)

we readily obtain

‖χχχu‖1,Th +‖χθ‖1,Ω 6C L(u,θ). (5.19)

From the approximation properties in (5.7), we conclude that

L(u,θ)6C hk(‖u‖k+1,Ω +‖θ‖k+1,Ω),

which implies the desired estimate (5.4).

5.2.3 Error in the pressure. Next, we bound the error in the pressure.

LEMMA 5.5 There is a constant C > 0 independent of the mesh size such that

‖ep‖0,Ω 6Chk(‖u‖k+1,Ω +‖θ‖k+1,Ω +‖p‖k,Ω

).


Proof. From (5.8), it remains to bound ‖χp‖0,Ω . To that end, we invoke the discrete inf-sup condi-tion (3.30) and the boundedness of B in (3.22) to find that

‖χp‖0,Ω 6β−1 sup

v∈Vh\0

B(v,χp)

‖v‖1,Th

6β−1 sup

v∈Vh\0

B(v,ep)

‖v‖1,Th

+ β−1 sup

v∈Vh\0

B(v,−ξp)

‖v‖1,Th

6β−1 sup

v∈Vh\0

B(v,ep)

‖v‖1,Th

+ β−1CB‖ξp‖0,Ω .

(5.20)

Then, from the first error equation in Lemma 5.1, we find that, for any v ∈ Vh,

B(v,ep)6 |D(eθ ,v)|+ |T1|+ |T2|+ |T3|+ |T4|, (5.21)

with

T1 = [AhS(θ ;u,v)−Ah

S(θh;u,v)], T2 = AhS(θh;eu,v),

T3 = [OhS(u;u,v)−Oh

S(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 thetriangle inequality, the continuity bound in Lemma 3.3, and the assumption ‖u‖W1,∞(Th)

= ‖u‖W1,∞(Ω)6

M. We obtain

|T1|6 Clipνlip‖eθ‖1,Ω‖u‖W1,∞(Th)‖v‖1,Th 6 ClipνlipM‖eθ‖1,Ω‖v‖1,Th .

Furthermore, from the bound (3.18),

|T2|6C‖eu‖2,Th‖v‖1,Th .

From the Lipschitz continuity of OhS in (3.24), the stability bound (4.11), and the inequality ‖u‖1,Ω 6

C∞M, we have the estimates

|T3|6CS‖eu‖1,Th‖u‖1,Th‖v‖1,Th 6 CSC∞M‖eu‖2,Th‖v‖1,Th ,

|T4|6CS‖uh‖1,Th‖eu‖1,Th‖v‖1,Th 6 CSCuClift‖θD,h‖H1/2(Γ )‖eu‖2,Th‖v‖1,Th .

Finally, note that, by (3.23) and assumption (5.1),

|D(eθ ,v)|6 CD‖g‖0,Ω‖eθ‖1,Ω‖v‖1,Th 6 CDM‖eθ‖1,Ω‖v‖1,Th .

The above estimates imply

|B(v,ep)|6C3

(‖eθ‖1,Ω +‖eu‖2,Th

)‖v‖1,Th . (5.22)

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.


6. A numerical test

In this section, we present computed errors and orders of convergence for a two-dimensional Boussinesqproblem (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 linearsolver UMFPACK, see Davis (2004). We employ a variant of our method (3.6) adapted to quadrilat-eral meshes. Specifically, for an order k > 1, we employ divergence-conforming Raviart-Thomas (RT)elements RRRTTT k of order k for the velocities, discontinuous tensor product polynomials Qk for the pres-sures, and conforming Qk polynomials for the temperatures. While the velocity-pressure pair is notoptimally matched in terms of approximation properties, the resulting mixed method also yields exactlydivergence-free velocity approximation (since divRRRTTT 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 ofuniformly refined square meshes Thll of mesh size hl = 2−l . We take ggg = (0,1)>, ν = 1, and choosethe 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 thesmooth functions

u1(x,y) = sin(y), u2(x,y) = sin(x)p(x,y) = 1+ sin(xy), θ(x,y) = 1+ cos(xy)

The temperature boundary conditions are enforced as in (4.14). For the velocity field, the inhomo-geneous boundary condition u = uD is essentially enforced in normal direction at the RT degrees offreedom, while standard DG terms are used to incorporate it in tangential direction. The additionalright-hand sides are discretized in a straightforward fashion. Finally, we select the penalty parameter asa0 = (k+1)2, where k is the approximation order.

We use a simple iteration scheme to deal with the non-linearities. Given the velocity un−1h at iteration

level n > 1, we obtain the temperature θ nh by solving the discrete version of the convection-diffusion

problem (2.3), where we take the flow field explicitly as un−1h . Then, we get the updated fluid variables

(unh, pn

h) by solving a discrete Oseen problem of the form (2.1)–(2.2), where the temperature is now takenexplicitly as θ n

h , and the convective term is linearized with un−1h . Proceeding in this way, we obtain a

sequence of iterates (unh, ph

n,θnh ). The iteration is terminated once the difference of the entire coefficient

vectors between two consecutive iterates is sufficiently small, i.e.,

‖coeffn+1− coeffn‖l2 6 tol,

where ‖ · ‖l2 is the standard l2-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 u0

h = 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 H1-norm errors for the temperature, and the DG-norm errors in the velocityconverge of order O(hk), in agreement with Theorem 5.1. In addition, the property that the approximatevelocities are exactly divergence-free is verified by evaluating ‖∇ · uh‖L∞(Ω) over a set of quadraturepoints.

1See www.dealii.com.


In Table 2, we display the computed L2-norm errors and convergence rates for the pressure, thetemperature, 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, theconvergence rate is between O(hk) and O(hk+1). In Cockburn et al. (2007), the same phenomenon hasbeen observed for the Navier-Stokes equations in isolation; this is a reflection of the fact that RRRTTT k−Qkare not optimally matched in terms of approximation properties, in contrast to the simplicial elementsstudied in our theoretical analysis.

k l ‖eθ‖1,Ω ‖eu‖1,h ‖∇ ·uh‖L∞(Ω)

1

1 2.201e-01 - 5.982e-01 - 1.443e-152 9.332e-02 1.24 2.831e-01 1.08 5.329e-153 4.267e-02 1.13 1.315e-01 1.11 1.199e-144 2.079e-02 1.04 6.195e-02 1.09 2.931e-145 1.030e-02 1.01 2.927e-02 1.08 6.584e-14

2

1 4.108e-02 - 1.591e-01 - 4.732e-152 9.169e-03 2.16 4.546e-02 1.81 1.521e-143 2.223e-03 2.04 1.199e-02 1.92 2.429e-144 5.513e-04 2.01 3.060e-03 1.97 5.556e-14

3

1 1.451e-03 - 7.545e-03 - 1.484e-142 1.805e-04 3.01 9.239e-04 3.03 3.040e-143 2.240e-05 3.01 1.077e-04 3.10 6.105e-144 2.791e-06 3.00 1.206e-05 3.16 1.131e-13

41 1.703e-04 - 9.295e-04 - 4.174e-142 9.917e-06 4.10 6.241e-05 3.90 7.412e-143 5.981e-07 4.05 4.010e-06 3.96 1.645e-13

Table 1. H1-norm errors and convergence rates for θh and uh, L∞-norm for ∇ ·uh.

In Figure 1, we plot the residual ‖coeffn+1− coeffn‖l2 against the number of iteration n for the casek = 1. We see that the residuals decrease in a linear fashion. While this is not surprising for the typeof fixed-point iteration applied, our theoretical analysis does not provide any indication of a contractionproperty. For k > 1, in our experiment, almost the same phenomena is observed; for the sake of brevity,these plots have been omitted.


k l ‖ep‖0,Ω ‖eθ‖0,Ω ‖eu‖0,Ω

1

1 9.86e-02 - 7.16e-02 - 6.75e-02 -2 3.50e-02 1.49 1.92e-02 1.90 2.12e-02 1.673 1.32e-02 1.41 4.96e-03 1.95 5.02e-03 2.084 4.62e-03 1.52 1.25e-03 1.98 1.21e-03 2.065 1.53e-03 1.60 3.14e-04 2.00 3.09e-04 1.97

2

1 1.044e-02 - 6.328e-03 - 1.101e-02 -2 1.611e-03 2.70 6.998e-04 3.18 1.306e-03 3.083 2.389e-04 2.75 8.498e-05 3.04 1.600e-04 3.034 3.735e-05 2.68 1.055e-05 3.01 1.975e-05 3.02

3

1 6.910e-04 - 1.520e-04 - 4.713e-04 -2 7.407e-05 3.22 1.009e-05 3.91 3.295e-05 3.843 7.033e-06 3.40 6.455e-07 3.97 2.049e-06 4.014 6.318e-07 3.48 4.083e-08 3.98 1.242e-07 4.04

41 6.456e-05 - 1.201e-05 - 4.212e-05 -2 2.375e-06 4.76 3.402e-07 5.14 1.121e-06 5.233 9.623e-08 4.63 1.057e-08 5.01 3.270e-08 5.10

Table 2. L2-norm errors and convergence rates for ph, θh and uh.

1 2 3 4 5 6 7 8 9 1010−10

10−8

10−6

10−4

10−2

100

102

iteration number n

resi

dual

l=1l=2l=3l=4

FIG. 1. Residuals vs. number of iterations for k = 1, l = 1, ...,4


7. Conclusions

We have introduced a new mixed finite element method for the numerical simulation of a generalizedBoussinesq problem with exactly divergence-free BDM elements of order k for the velocities, discon-tinuous elements of order k− 1 for the pressure, and standard continuous elements of order k for thediscretization of the temperature. The resulting method yields exactly divergence-free velocity approx-imations, and thus it is energy-stable without additional modifications of the convection terms. Undersuitable hypotheses on the data, we have shown the existence and stability of discrete solutions. More-over, we have shown optimal a-priori error estimates with respect to the mesh size h for problems withsmooth and sufficiently small solutions. More precisely, the broken H1-norm errors in the velocity, theH1-norm errors in the temperature, and the L2-norm errors in the pressure are proved to converge withorder 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 dif-ficulties 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 discreteliftings whose actual computation may be expensive. Ongoing research is concerned with finding waysto overcome these issues.

The numerical results shown in this paper are non-exhaustive. Additional testing is necessary to fullyassess the performance of the proposed scheme. This includes tests for physically relevant problemswith realistic parameters and three-dimensional geometries, the development of efficient linearizationstrategies (such as Newton’s methods), and the design of iterative solvers or preconditioners. Some ofthese computational aspects will be addressed in a forthcoming paper.

Finally, we emphasize that using conforming elements for the temperature unknown makes theanalysis 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 aswell. 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 FONDECYTproject 11121347. T.Q. and D.S. were supported in part by the Natural Sciences and Engineering Re-search Council of Canada (NSERC).

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