INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS
Int. J. Numer. Meth. Fluids 192007; 1:1
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Energy norm a-posteriori error estimation for divergence-free
discontinuous Galerkin approximations of the Navier-Stokes
equations
Guido Kanschat1∗ and Dominik Schötzau2
1
Department of Mathematics, Texas A & M University,
College Station, TX 77843-3368, USA
2 Mathematics Department, University of British Columbia,
1984 Mathematics Road, Vancouver, BC V6T 1Z2, Canada
International Journal for Numerical Methods in Fluids,
Volume 57, Pages 1093-1113, 2008
SUMMARY
We develop the energy norm a-posteriori error analysis of exactly divergence-free discontinuous RTk /Qk Galerkin methods for the incompressible Navier-Stokes equations with small data. We
derive upper and local lower bounds for the velocity-pressure error measured in terms of the natural
energy norm of the discretization. Numerical examples illustrate the performance of the error estimator
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within an adaptive refinement strategy. Copyright 1. Introduction
In this paper we derive a residual-based energy norm a-posteriori error estimator for exactly
divergence-free discontinuous Galerkin (DG) methods for the incompressible Navier-Stokes
equations
−ν∆u + (u · ∇)u + ∇p = f
∇·u=0
in Ω ⊂ R2 ,
in Ω,
u=0
on Γ = ∂Ω.
(1.1)
Here, ν > 0 is the kinematic viscosity, u the velocity, p the pressure, and f ∈ L 2 (Ω)2 an
external body force. The domain Ω is assumed to be a Lipschitz polygon in R2 . Throughout,
Contract/grant sponsor: The first author was partially supported by the National Science Foundation (NSF)
under grant DMS-0713829. The second author was partially supported by the National Science and Engineering
Research Council of Canada (NSERC).
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Copyright 2
we assume that ν −2 kf kL2 (Ω) is sufficiently small. Then the Navier-Stokes system has a unique
solution (u, p) ∈ H01 (Ω)2 × L20 (Ω), where H01 (Ω)2 is the usual vector-valued Sobolev space with
zero boundary values and L20 (Ω) is the space of square integrable functions with vanishing
mean value. Moreover, we have the stability bound
k∇ukL2 (Ω) ≤ CP ν −1 kf kL2 (Ω) ,
(1.2)
with CP > 0 denoting the Poincaré constant of Ω.
Exactly divergence-free discontinuous Galerkin methods for the incompressible NavierStokes equations (1.1) were recently introduced in [14]. They are based on divergenceconforming finite element spaces for the approximation of the velocity, such as Brezzi-DouglasMarini (BDM) or Raviart-Thomas (RT) spaces, and on matching discontinuous spaces for
the approximation of the pressure. The H 1 -conformity of the velocity approximation is then
enforced weakly using the discontinuous Galerkin approach. The resulting methods have been
shown to be locally conservative, inf-sup stable, and optimally convergent. They were inspired
by the use of BDM elements for the analysis of DG schemes in [18]. In the lowest-order case,
they have been shown to be closely related to the well-known marker and cell (MAC) scheme;
cf. [22]. The use of element pairs beyond BDM and RT is discussed in [31]. In this article, we
will focus on RT elements but remark here that the analysis applies in exactly the same way
to BDM elements.
The exactly divergence-free methods in [14] originated from the ideas introduced in [13].
There, an element-by-element post-processing procedure was devised to render DG velocity
approximations exactly divergence-free. The divergence-conforming velocity approximations
in [14] can then be understood and analyzed in the setting of [13], with a post-processing
operator that is equal to the identity operator. The work [13], in turn, builds upon the
earlier papers [11, 12, 15], where discontinuous Galerkin methods for linear incompressible
flow problems were proposed and analyzed.
In this paper, we develop the energy-norm a-posteriori error estimation for the RT k /Qk
method proposed in [14] for quadrilateral meshes. Here, the approximation of the velocity is
based on Raviart-Thomas elements of order k while the pressure is discretized using tensor
product polynomials of order k. We derive a residual-based error estimator that is both reliable
and efficient for the error measured in a natural energy norm that includes the broken H 1 norm for the error in the velocity and the L2 -norm for the error in the pressure. Our technique
of proof relies on the approach introduced in [20] for the Stokes problem. Let us also mention
that a-posteriori error analyses for DG methods applied to elliptic problems can be found, e.g.,
in [6, 7, 10, 19, 21, 23, 24] and the references therein.
The outline of the paper is as follows. In Section 2, we describe the RTk /Qk methods
from [14], and establish the stability properties that are crucial for our analysis. For simplicity,
we restrict ourselves to the interior penalty approach for the discretization of the diffusive
terms. In Section 3, we present our energy norm error estimator and state our main result. The
proof of these results is carried out in Section 4. In Section 5 we present a series of numerical
experiments that show the usefulness of our estimator in adaptive refinement strategies. Finally,
we conclude our presentation in Section 6.
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2. Divergence-free DG methods
In this section, we recall the RTk /Qk –DG method for the Navier-Stokes equations. We
follow [13] and [14]. However, instead of the local discontinuous Galerkin (LDG) approach
presented there, we employ the interior penalty (IP) method for the discretization of the
Laplace operator.
2.1. Discretization
We assume that the domain Ω can be partitioned into shape-regular rectangular meshes
Th = {K}. The diameter of an element K is denoted by hK , and the mesh size parameter h
of Th is the piecewise constant function h with h|K = hK . We will assume that each edge of a
mesh cell K is a boundary edge, an edge of a neighboring cell or consists of two equally long
edges of refined neighboring cells. The latter corresponds to so called one-irregular meshes
with one “hanging node” on refined edges. We further assume that the neighboring nodes of
hanging nodes are not hanging themselves. The adaptively generated meshes in our numerical
experiments satisfy these properties, see [4].
Remark 2.1. The inf-sup stability of discretizations with hanging nodes using RaviartThomas elements is in part still an open question. For quadrilaterals with one-irregular meshes,
a stability proof exists only for the pair RTk /Qk defined in (2.1) below with k ≥ 2; see [28].
Nevertheless, we conjecture from our computational results that stability also holds for k = 1.
For triangles with hanging nodes, there is no stability result for the divergence-free elements
proposed in [14]; on the other hand, locally refined triangular meshes without hanging nodes
can be obtained using bisection. The results below are all to be read in view of the restrictions
cited in this remark.
We will use the standard average and jump operators. To do define them, we denote by E(T h )
the set of all edges in Th , by EI (Th ) the set of all interior edges, and by EB (Th ) the set of all
boundary edges. The length of an edge E is denoted by hE .
Let now K + and K − be two adjacent elements of Th that share an interior edge E =
∂K + ∩ ∂K − ∈ EI (Th ). Let ϕ be a any piecewise smooth function (matrix-, vector- or scalarvalued), and let us denote by ϕ± the traces of ϕ on E taken from within the interior of K ± .
We then define the average operator {{·}} across the edge E as
{{ϕ}} =
1 +
(ϕ + ϕ− ).
2
Further, let σ, u, and p be a piecewise smooth matrix-valued, vector-valued and scalar-valued
function, respectively. The jumps [[·]] of these functions across E are defined as
[[σn]] = σ + n+ + σ − n− ,
[[u ⊗ n]] = u+ ⊗ n+ + u− ⊗ n− ,
[[pn]] = p+ n+ + p− n− ,
where n± denote the outward unit normal vectors on the boundary ∂K ± of the elements K ± .
Analogously, for a boundary edge E ∈ EB (Th ), we set {{ϕ}} = ϕ, [[σn]] = σn, [[u ⊗ n]] = u ⊗ n,
and [[pn]] = pn. Here, n is the outward unit normal on Γ. Finally, if the jump appears
quadratically, we abbreviate [[u]] = u+ − u− .
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For an approximation order k ≥ 1, we then seek DG approximations to the Navier-Stokes
equations in the finite element space V h × Qh where
V h = { v ∈ H(div; Ω) : v|K ∈ RTk (K), K ∈ Th , v · n = 0 on Γ },
Qh = { q ∈ L20 (Ω) : q|K ∈ Qk (K), K ∈ Th },
(2.1)
with Qk (K) denoting the tensor product polynomials of degree k on K, RTk (K) ⊃ Qk (K)2
the Raviart-Thomas polynomials of order k on K, see e.g., [9] and the references therein, and
H(div; Ω) = { v ∈ L2 (Ω)2 : ∇ · v ∈ L2 (Ω) }.
A crucial property of the pair V h × Qh is that, on the meshes considered,
∇ · V h ⊂ Qh ;
(2.2)
see the discussion in [14, Section 2.2.2].
We consider the discontinuous Galerkin method: find (uh , ph ) ∈ V h × Qh such that
Z
Ah (uh , v) + Oh (uh ; uh , v) + Bh (v, ph ) =
f · v dx,
Ω
(2.3)
Bh (uh , q) = 0
for all (v, q) ∈ V h × Qh .
The forms Ah , Oh and Bh are associated to the discretizations of the Laplacian, the
convective term, and the incompressibility condition, respectively.
The form Bh is given by
Z
∇ · vq dx.
Bh (v, q) = −
Ω
From the second equation in (2.3), it then follows that ∇ · uh is orthogonal to all pressures
in Qh . Hence, the inclusion (2.2) implies that uh is indeed exactly divergence-free.
For the form Ah , we choose the classical (symmetric) interior penalty discretization defined
by
X Z
X Z
[[v ⊗ n]] : {{ν∇u}} ds
ν∇u : ∇v dx −
Ah (u, v) =
K∈Th
−
X
K
E∈E(Th )
Z
E∈E(Th )
[[u ⊗ n]] : {{ν∇v}} ds +
E
E
X
E∈E(Th )
κE ν
Z
[[u]] · [[v]] ds.
E
The parameter κE is the interior penalty parameter stabilizing the discontinuous Galerkin
form. In order to ensure the coercivity of the form Ah , it has to be chosen sufficiently large.
For a boundary edge E ∈ EB (Th ) of a mesh cell K, its lower limit can be determined by an
inverse estimate and is of the form s/h⊥ , where h⊥ is the length of K orthogonal to E and s
depends on the polynomial degree and the shape of the cell. In particular, for rectangular mesh
cells, stability is obtained for
κE >
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k(k + 1)
,
2h⊥
Int. J. Numer. Meth. Fluids 192007; 1:1–1
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and we usually choose twice that value in our numerical tests. On interior edges, stability is
obtained by taking 1/2 of the mean value of this value from both cells, respectively. We will
always assume that this parameter is chosen such that stability is guaranteed.
We point that, for the discretization of the Laplace operator, several other discontinuous
Galerkin methods can be chosen, for which our results hold true as well; see the discussions
in [3, 28] and in [14, Table I]. For these methods, we have to at least require stability
and consistency; for our purposes, we considered the LDG and (symmetric) interior penalty
methods.
Finally, to define the convective form Oh , let w be a piecewise smooth and divergence-free
flow field in the space
J (Th ) = { v ∈ H(div; Ω) : ∇ · v ≡ 0 in Ω, v|K ∈ H 1 (K)2 , K ∈ Th }.
Clearly, the discrete velocity field uh belongs to this space. We then take Oh to be the standard
upwind form introduced in [26, 27]:
X Z
(w · ∇)v · u dx
Oh (w; u, v) = −
K∈Th
K
K∈Th
∂K
+
X Z
1
e
w · nK {{u}} − |w · nK |(u − u) · v ds.
2
Here, ue denotes the exterior trace of u taken over the edge under consideration and set to
zero on the boundary. Upon integration by parts, we also have
X Z
Oh (w; u, v) =
(w · ∇)u · v dx
+
K∈Th
K
K∈Th
∂K
X Z
1
1
w · nK (ue − u) − |w · nK |(ue − u) · v ds.
2
2
This completes the definition of the DG methods for the incompressible Navier-Stokes
equations. We point out that, since uh is exactly divergence-free, the resulting DG methods
are locally conservative and energy-stable; cf. [13, 14].
2.2. Stability properties
In this section, we recapitulate the main stability properties of the DG forms from [13, 14, 28].
First, we rewrite the interior penalty form Ah in the form
Z
Z
Ah (u, v) =
ν∇u : ∇v dx −
ν∇u : L(v) dx
Ω
ZΩ
Z
X
−
ν∇v : L(u) dx +
κE ν [[u]] · [[v]] ds.
Ω
E∈E(Th )
where the lifting operator L : V h → Σh is given by
Z
X Z
L(v) : τ dx =
[[v ⊗ n]] : {{τ }} ds
Ω
E∈E(Th )
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E
E
∀τ ∈ Σh ,
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with
Σh = { τ ∈ L2 (Ω)2×2 : τ |K ∈ Qk+1 (K)2×2 , K ∈ Th }.
We can now easily extend the form Ah to the space V (h) = H01 (Ω) + V h , using the same
definition of the lifting operator. This space is endowed with the broken H 1 -norm
X
X
kuk21,h =
k∇uk2L2 (K) +
κE νk[[u]]k2L2 (E) .
K∈Th
E∈E(Th )
Then, the form Ah satisfies the following continuity and coercivity properties: If the interior
penalty parameter is chosen as above, then there are constants ca > 0 and α > 0, independent
of the viscosity and the mesh size, such that
Ah (u, v) ≤ νca kuk1,h kvk1,h ,
Ah (u, u) ≥
ναkuk21,h ,
u, v ∈ V (h),
(2.4)
u ∈ V h.
(2.5)
The form Oh is Lipschitz continuous: There is a constant c0 > 0, independent of the viscosity
and the mesh size, such that
|Oh (w 1 ; u, v) − Oh (w2 ; u, v)| ≤ co kw1 − w2 k1,h kuk1,h kvk1,h ,
(2.6)
for all w1 , w 2 ∈ J(Th ), u ∈ V (h) and v ∈ V (h). This statement has been proven in [13,
Proposition 4.2] for v ∈ V h . Using similar arguments, it can be readily seen that it also holds
for v ∈ H01 (Ω)2 . Moreover, there holds
Oh (w; u, u) ≥ 0,
(2.7)
for w ∈ J (Th ) and u ∈ V h or u ∈ H01 (Ω)2 .
Finally, the form Bh is continuous and satisfies the discrete inf-sup condition: There exist
constants cb > 0 and β > 0, independent of the viscosity and the mesh size, such that
|Bh (u, p)| ≤ cb kuk1,hkpkL2 (Ω) ,
sup
u∈V h
Bh (u, p)
≥ βkpkL2 (Ω) ,
kuk1,h
u ∈ V (h), p ∈ L2 (Ω),
(2.8)
p ∈ Qh .
(2.9)
While the continuity of Bh is obvious, the inf-sup condition follows from the results in [28]. It
holds on regular meshes for any k. For k ≥ 2, it holds on the irregular meshes considered in this
paper. As pointed out in Remark 2.1, the stability on irregular meshes for k = 0, 1 remains
an open problem. We further refer the reader to [18] for the stability of BDM elements on
conforming triangular meshes.
With these stability properties at hand and by proceeding as in the analysis presented in [14,
Theorem 3.1] for triangular elements, we readily obtain the following result: If ν −2 kf kL2 (Ω) is
sufficiently small, then the DG discretization (2.3) has a unique solution (uh , ph ) ∈ V h × Qh
and we have the a-priori error estimate
ku − uh k1,h + kp − ph kL2 (Ω) ≤ C khk ukH k+1 (Th ) + khk pkH k (Th ) ,
with a constant C > 0 independent of the mesh size. Here, the k.kH k (Th ) denotes the cellwise
H k -norm. For later use, we also note that the discrete velocity uh satisfies the bound
kuh k1,h ≤ cp ν −1 kf kL2 (Ω) ,
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(2.10)
Int. J. Numer. Meth. Fluids 192007; 1:1–1
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where cp is the constant in the discrete Poicaré inequality: There is cp > 0 independent of the
mesh size such that
kvkL2 (Ω) ≤ cp kvk1,h ,
∀ v ∈ V h,
(2.11)
see [2, Lemma 2.1] and [8].
2.3. Additional stability properties
In this section, we establish two additional stability results that are key in our a-posteriori
error analysis. To that end, we introduce the global form
Ah (w)(u, p; v, q) = Ah (u, v) + Oh (w; u, v) + Bh (v, p) − Bh (u, q),
for any w ∈ J(Th ), (u, p) and (v, q) in V (h) × L2 (Ω). We then rewrite the DG methods in
the form: Find (uh , ph ) ∈ V h × Qh such that
Z
f · v dx
∀ (v, q) ∈ V h × Qh .
(2.12)
Ah (uh )(uh , ph ; v, q) =
Ω
We further introduce the product norm
||| (u, p) |||2 = νkuk21,h + ν −1 kpk2L2 (Ω) ,
and define
c̄p = max{CS , cp α−1 }.
In the following, we consider small data and assume that
co c̄p ν −2 kf kL2 (Ω) < 1.
(2.13)
Then the following continuity and stability properties hold.
Lemma 2.1. Assume (2.13) and let w ∈ J(Th ) satisfy
kwk1,h ≤ c̄p ν −1 kf kL2 (Ω) .
Then there is a continuity constant cA > 0, only depending ca and cb , such that
|Ah (w)(u, p; v, q)| ≤ cA ||| (u, p) |||||| (v, q) |||
for all u, v ∈ V (h) and p, q ∈ L2 (Ω).
Proof. The assertion follows from the continuity of Ah and Bh , the Cauchy-Schwarz inequality
and the fact that
|Oh (w; u, v)| ≤ co kwk1,h kuk1,h kvk1,h
1
1
1
1
≤ co c̄p ν −2 kf kL2 (Ω) ν 2 kuk1,h ν 2 kvk1,h ≤ ν 2 kuk1,h ν 2 kvk1,h ,
which is due to (2.6), the bound on w and the smallness assumption (2.13).
2
Lemma 2.2. Assume (2.13) and let w ∈ J(Th ) satisfy
kwk1,h ≤ c̄p ν −1 kf kL2 (Ω) .
Then there is constant γ > 0, only depending on Ω, such that for any tuple (u, p) ∈
H01 (Ω)2 × L20 (Ω) there is (v, q) ∈ H01 (Ω)2 × L20 (Ω) with ||| (v, q) ||| ≤ 1 and
Ah (w)(u, p; v, q) ≥ γ||| (u, p) |||.
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Proof. Fix (u, p) ∈ H01 (Ω)2 × L20 (Ω). We have
Ah (w)(u, p; u, p) = Ah (u, u) + Oh (w; u, u) ≥ νkuk21,h .
(2.14)
Here, we have used (2.7) and that, for u ∈ H01 (Ω)2 ,
Ah (u, u) = νk∇uk2L2 (Ω) = νkuk21,h .
Further, due to the continuous inf-sup condition, see [9, 17], there is a field v̄ ∈ H01 (Ω)2 that
satisfies
Bh (v̄, p) ≥ CΩ ν −1 kpk2L2 (Ω) ,
1
1
ν 2 kv̄k1,h ≤ ν − 2 kpkL2 (Ω) .
with CΩ > 0 denoting the continuous inf-sup constant, which only depends on Ω. Then,
Ah (w)(u, p; v̄, 0) = Ah (u, v̄) + Oh (w; u, v̄) + Bh (v̄, p)
≥ CΩ ν −1 kpk2L2 (Ω) − |Ah (u, v̄)| − |Oh (w; u, v̄)|.
The H 1 -conformity of the functions u, v̄ and the bound for v̄ yield
|Ah (u, v̄)| ≤ νkuk1,h kv̄k1,h ≤ kuk1,h kpkL2 (Ω) .
Furthermore, employing the continuity of Oh , the bounds for w and v̄ and the smallness
assumption (2.13), we obtain
|Oh (w; u, v̄)| ≤ co ν −1 kwk1,h kuk1,h kpkL2(Ω) ≤ kuk1,h kpkL2 (Ω) .
Hence, the inequality |ab| ≤ 2ε a2 +
1 2
2ε b
now gives
Ah (w)(u, p; v̄, 0) ≥ CΩ ν −1 kpk2L2 (Ω) − 2kuk1,h kpkL2 (Ω)
1
ν −1 kpk2L2 (Ω) − νεkuk21,h ,
≥ CΩ −
ε
(2.15)
for any ε > 0.
From (2.14) and (2.15), we conclude that, for δ > 0,
Ah (w)(u, p; u + δv̄, p) = Ah (w)(u, p; u, p) + δAh (w)(u, p; v̄, 0)
1
≥ (1 − εδ)νkuk21,h + δ CΩ −
ν −1 kpk2L2 (Ω) .
ε
Taking ε =
2
CΩ
and δ =
CΩ
4 ,
we obtain that
1 C2
Ah (w)(u, p; u + δv̄, p) ≥ min{ , Ω }||| (u, p) |||2 .
2 8
(2.16)
On the other hand, using the triangle inequality and the bound for v̄, it can be readily seen
that
CΩ
||| (u + δv̄, p) ||| ≤ 1 +
||| (u, p) |||.
(2.17)
4
The assertion now readily follows from (2.16) and (2.17).
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3. Energy norm a-posteriori error estimation
In this section, we present a reliable and efficient energy norm error estimator for the exactly
divergence-free DG approximations of the Navier-Stokes equation with small data.
3.1. Error indicators
We begin by introducing the error indicators. To that end, let (uh , ph ) be the discontinuous
Galerkin approximation of (2.3). Let f h be a piecewise polynomial approximation of f , possibly
discontinuous across elemental edges. For any element K ∈ Th and interior edge E ∈ EI (Th ),
we introduce the residuals
RK = f h + ν∆uh − (uh · ∇)uh − ∇ph |K ,
RE = [[(ph I − ν∇uh ) n]]|E ,
respectively. Here, the matrix I is the identity matrix in R2×2 . For each K ∈ Th , we then
introduce the local error indicator ηK
2
2
2
+ ηJ2K ,
ηK
= ηR
+ ηE
K
K
(3.1)
where
2
ηR
= ν −1 h2K kRK k2L2 (K) ,
K
2
ηE
=
K
ηJ2K =
1
2
X
ν −1 hE kRE k2L2 (E) ,
E∈∂K\Γ
(3.2)
1 X
κE νk[[uh ]]k2L2 (E) .
2
E∈∂K
Finally, we introduce the data oscillation term
O K = (f − f h )|K ,
K ∈ Th ,
and define
2
ωK
= ν −1 h2K kO K k2L2 (K) .
(3.3)
The error estimator η is now given by
η=
X
K∈Th
2
ηK
! 12
,
(3.4)
! 21
.
(3.5)
while the data error ω is defined by
ω=
X
K∈Th
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ωK
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3.2. Reliability
We now state that the error indicator η gives rise to a reliable energy norm a-posteriori error
estimator, up to the data approximation term ω.
We assume that the data is small and satisfies
1
max{1, γ −1}co c̄p ν −2 kf kL2 (Ω) ≤ .
(3.6)
2
The following result holds.
Theorem 3.1. Assume (3.6). Let (u, p) be the solution of the Navier-Stokes equation (1.1)
and (uh , ph ) ∈ V h × Qh the discontinuous Galerkin approximation obtained by (2.3). Let η
and ω be the error estimator and the data approximation term in (3.4) and (3.5), respectively.
Then the following a-posteriori error bound holds:
||| (u − uh , p − ph ) ||| ≤ C η + ω ,
with a constant C > 0 that is independent of the viscosity and the mesh size.
The proof of this theorem will be presented in Section 4.1.
Next, we state that the local error indicators can be bounded from above by the local energy
error, up to data approximation errors. Hence, the estimator η is also efficient. We first consider
the residuals ηRK and ηEK . To that end, we define δK by
δK = { K 0 ∈ Th : K and K 0 share an edge or a vertex }.
Theorem 3.2. Assume (3.6). Let (u, p) be the solution of the Navier-Stokes equation (1.1)
and (uh , ph ) ∈ V h × Qh the discontinuous Galerkin approximation obtained by (2.3). Let η RK ,
ηEK and ωK be defined in (3.2) and (3.3), respectively. For any element K ∈ Th , there holds
1
1
ηRK ≤ C ν 2 ku − uh kH 1 (K) + ν − 2 kp − ph kL2 (K) + ωK ,
X 1
1
ν 2 ku − uh kH 1 (K) + ν − 2 kp − ph kL2 (K) + ωK ,
η EK ≤ C
K∈δK
with a constant C > 0 that is independent of K, the viscosity and the mesh size.
The proof of this theorem will be given in Sections 4.2 and 4.3.
Finally, let us note that we trivially have efficiency for the jump residuals ηJK . Indeed, since
the jumps of the exact solution vanish, there holds
X
1 X
ηJ2K =
κE νk[[u − uh ]]k2L2 (E) +
κE νk[[u − uh ]]k2L2 (E) .
2
E∈∂K∩Γ
E∈∂K\Γ
As a consequence of the above results, we have the following lower bound of the energy
error.
Corollary 3.1. Assume (3.6). Let (u, p) be the solution of the Navier-Stokes equation (1.1)
and (uh , ph ) ∈ V h × Qh the discontinuous Galerkin approximation obtained by (2.3). Let η
and ω be the error estimator and the data approximation term in (3.4) and (3.5), respectively.
Then the following efficiency bound holds:
η ≤ C ||| (u − uh , p − ph ) ||| + ω),
with a constant C > 0 that is independent of the viscosity and the mesh size.
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4. Proofs
In this section, we present the proofs of Theorems 3.1 and 3.2.
4.1. Proof of Theorem 3.1
We introduce the discontinuous RT space Ve h = { v ∈ L2 (Ω)2 : v|K ∈ RTk (K), K ∈ Th }.
Then, following [20, Section 4], we define V ch = Ve h ∩ H01 (Ω)2 . The orthogonal complement of
c
⊥
e
V ch in Ve h with respect to the norm k · k1,h is denoted by V ⊥
h . Hence, we have V h = V h ⊕ V h .
If uh is the discontinuous Galerkin velocity approximation, we can decompose it uniquely
into
uh = uch + urh ,
(4.1)
c
c
with uch ∈ V ch and urh ∈ V ⊥
h . Then, since uh ∈ V h and uh ∈ V h ⊂ V h , we must also have that
r
c
uh = uh − uh ∈ V h . By employing arguments analogous to those in [20, Section 4] and [24]
for the discontinuous space Ve h , the following result is obtained: there is ce > 0, independent
of the viscosity and the mesh size, such that
ν
1
2
kurh k1,h
≤ ce
X
ηJ2K
K∈Th
! 12
.
(4.2)
Let now (u, p) be the solution of the Navier-Stokes equations, and (uh , ph ) the DG solution.
We denote the error of the DG approximation by
(eu , ep ) = (u − uh , p − ph ),
and also set ecu = u − uch .
Lemma 4.1. Assume (3.6). Then there is (v, q) ∈ H01 (Ω)2 × L20 (Ω) such that ||| (v, q) ||| ≤ 1
and
Z
γ
||| (eu , ep ) ||| ≤
f · (v − v h ) dx − Ah (uh )(uh , ph ; v − v h , q)
2
Ω
! 21
X
2
,
+ (cA + γ)ce
η JK
K∈Th
for any v h ∈ V h .
Here, note that, since uh is exactly divergence-free, an approximation qh of q is not needed
(and is set to zero).
Proof. From the triangle inequality and (4.2), we have
γ||| (eu , ep ) |||
≤ γ||| (ecu , ep ) ||| + γ||| (urh , 0) |||
X
1
≤ γ||| (ecu , ep ) ||| + γce
ηJ2K 2 .
K∈Th
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Furthermore, from the stability estimate (2.10), the fact that (3.6) implies condition (2.13),
the inf-sup condition in Lemma 2.2 is applicable to (ecu , ep ). It follows that there is a test
function (v, q) ∈ H01 (Ω)2 × L20 (Ω) such that ||| (v, q) ||| ≤ 1 and
γ||| (ecu , ep ) ||| ≤ Ah (uh )(ecu , ep ; v, q)
= Ah (uh )(eu , ep ; v, q) + Ah (uh )(urh , 0; v, q)
≤ Ah (uh )(eu , ep ; v, q) + cA ||| (urh , 0) |||
X
1
≤ Ah (uh )(eu , ep ; v, q) + cA ce
ηJ2K 2 .
K∈Th
Here, we have also used the continuity of Ah in Lemma 2.1 and the bound (4.2). Since
A(uh )(u, p; v, q) = Ah (u)(u, p; v, q) − Oh (eu ; u, v),
we conclude that
γ||| (eu , ep ) ||| ≤ Ah (u)(u, p; v, q) − Oh (eu ; u, v)
X
1
ηJ2K 2 .
− Ah (uh )(uh , ph ; v, q) + (cA ce + γce )
(4.3)
K∈Th
Due to the continuity of Oh in (2.6), the stability bound (1.2) and the smallness
assumption (3.6), we have
|Oh (eu ; u, v)|
≤ co keu k1,h kuk1,h kvk1,h
1
1
≤ co c̄p ν −2 kf kL2 (Ω) ν 2 keu k1,h ν 2 kvk1,h
γ 1
≤
ν 2 keu k1,h .
2
Hence, this term can be brought to the left-hand side of (4.3). Then, we have that
Z
Ah (u)(u, p; v, q) =
f · v dx.
Ω
Therefore, we obtain
γ
||| (eu , ep ) ||| ≤
2
Z
f · v dx − Ah (uh )(uh , ph ; v, q) + (cA ce + γce )
Ω
X
K∈Th
ηJ2K
21
.
The assertion now follows from this inequality by noting that
Z
Ah (uh )(uh , ph ; v h , 0) −
f · v h dx = 0,
Ω
for all v h ∈ V h .
2
Lemma 4.2. For (v, q) ∈ H01 (Ω)2 × L20 (Ω) there is v h ∈ V h such that
Z
1
f · (v − v h ) dx − Ah (uh )(uh , ph ; v − v h , q) ≤ Cν 2 k∇vkL2 (Ω) (η + ω),
Ω
with a constant C > 0 that is independent of the viscosity and the mesh size.
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Proof. We set ξ v = v − v h , with v h ∈ V ch to be selected. Then,
Z
T =
f · ξv dx − Ah (uh )(uh , ph ; ξ v , q)
Ω
Z
=
f · ξv dx − Ah (uh , ξ v ) − Oh (uh ; uh , ξv ) − Bh (ξ v , ph ).
Ω
Here, we have used that Bh (uh , q) = 0, thanks to the fact that uh is divergence-free. Integration
by parts yields
Z
−Ah (uh , ξ v ) = −ν
(∇uh − L(uh )) : ∇ξ v dx
Ω
!
Z
X Z
(ν∇uh )nK · ξ v ds
ν∆uh · ξ v dx −
≤
K∈Th
∂K\Γ
K
1
2
1
2
+ν kL(uh )kL2 (Ω) ν k∇ξv kL2 (Ω) ,
with nK denoting the unit outward normal vector on ∂K. There holds
! 12
X
1
2
η JK
ν 2 kL(uh )kL2 (Ω) ≤ cl
,
K∈Th
for a constant cl > 0 that is independent of the viscosity and the mesh size, see [28, Lemma 7.2
and Lemma 7.4]. We further have


XZ
X Z

(ph nK ) · ξ v ds ,
∇ph · ξ v dx −
−Bh (ξ v , ph ) = −
K∈Th
K
K\Γ
and
−Oh (uh ; uh , ξv ) = −
X Z
K∈Th
K
(uh · ∇)uh · ξv dx.
Combining the above relations, we conclude that
X Z
T ≤
(RK + O K ) · ξ v dx +
K
K∈Th
+ cl
X
K∈Th
ηJ2K
! 21
∂K
X
E∈EI (Th )
Z
E
RE · ξ v ds
(4.4)
1
2
ν k∇ξv kL2 (Ω) .
We now choose v h ∈ V ch as the standard Scott-Zhang interpolant of v, see, e.g., [17]. It satisfies
X
2
2
2
(4.5)
h−2
K kv − v h kL2 (K) + k∇(v − v h )kL2 (K) ≤ Ck∇vkL2 (Ω) ,
K∈Th
and
X
2
2
h−1
E kv − v h kL2 (E) ≤ Ck∇vkL2 (Ω) .
(4.6)
E∈EI (Th )
Note that on an edge with a hanging node we use the interpolant from the unrefined side
in order to maintain conformity. The assertion now follows from (4.4), by using the weighted
Cauchy-Schwarz inequality and the approximation properties in (4.5)-(4.6).
2
Theorem 3.1 is an immediate consequence of Lemma 4.1 and Lemma 4.2.
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4.2. Efficiency bound for ηRK
In this section, we prove the efficiency bound for ηRK stated in Theorem 3.2. We do so by
employing the bubble function technique introduced in [29, 30].
Let K be an element of Th . We denote by bK the standard polynomial bubble function on K.
Let now v be a (vector-valued) polynomial function on K; then there exists a constant C > 0
independent of v and K such that
kbK vkL2 (K)
kvkL2 (K)
≤ CkvkL2 (K) ,
(4.7)
1
2
≤ CkbK vkL2 (K) ,
k∇(bK v)kL2 (K)
≤
kbK vkL∞ (K)
≤
(4.8)
Ch−1
K kvkL2 (K) ,
Ch−1
K kvkL2 (K) .
(4.9)
(4.10)
The proof of (4.7) and (4.8) is given in [29, Lemma 4.1]. The proof of (4.9) and (4.10) can be
obtained by similar arguments, see [1, Theorems 2.2 and 2.4].
Fix an element K ∈ Th and set V b = bK RK . By (4.8), the definition of the data
approximation term O K , and the fact that (u, p) solves the Navier-Stokes equations, we have
Z
1
2
RK · V b dx
RK k2L2 (K) =
CkRK k2L2 (K) ≤ kbK
K
Z
Z
O K · V b dx.
(f + ν∆uh − (uh · ∇)uh − ∇ph ) · V b dx −
=
K
K
Hence,
CkRK k2L2 (K) ≤ T1 + T2 + T3 ,
where
T1
T2
=
=
Z
(4.11)
(−ν∆eu + ∇ep ) · V b dx,
ZK
((u · ∇)u − (uh · ∇)uh ) · V b dx,
K
T3
= −
Z
O K · V b dx.
K
To bound T1 , we first integrate by parts over K and use the Cauchy-Schwarz inequality:
1
1
1
T1 ≤ Cν 2 k∇V b kL2 (K) ν 2 k∇eu kL2 (K) + ν − 2 kep kL2 (K) .
Upon application of (4.9) we then obtain
1
1
− 21
2
T1 ≤ Cν 2 h−1
kep kL2 (K) .
K kRK kL2 (K) ν k∇eu kL2 (K) + ν
(4.12)
For the term T2 , we proceed as follows. By the Cauchy-Schwarz inequality and (4.10), we have
Z
T2 =
((eu · ∇)u + (uh · ∇)eu ) · V b dx
K
≤ CkV b kL∞ (K) keu kH 1 (K) k∇ukL2 (K) + kuh kL2 (K)
≤ Ch−1
K kRK kL2 (K) keu kH 1 (K) k∇ukL2 (K) + kuh kL2 (K) .
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The stability bound (1.2) and the smallness assumption (3.6) then yield
−1
k∇ukL2 (K) ≤ c̄p ν −1 kf kL2 (Ω) = c−1
kf kL2 (Ω) ≤ Cν.
o co c̄p ν
From the discrete Poincaré inequality (2.11) and the bound (2.10), we conclude similarly that
kuh kL2 (K) ≤ kuh kL2 (Ω) ≤ cp kuh k1,h ≤ cp c̄p ν −1 kf kL2 (Ω) ≤ Cν.
Hence,
1
1
2
T2 ≤ Cν 2 h−1
K kRK kL2 (K) ν keu kH 1 (K) .
(4.13)
Finally, for the term T3 , we use (4.7) and conclude that
T3 ≤ Ckf − f h kL2 (K) kV b kL2 (K) ≤ Ckf − f h kL2 (K) kRK kL2 (K) .
(4.14)
We now combine (4.11) with the bounds in (4.12), (4.13) and (4.14), divide the resulting
1
estimate by kRK kL2 (K) and multiply it with ν − 2 hK . This yields
1
1
1
ηRK = ν − 2 hK kRK kL2 (K) ≤ C ν 2 keu kH 1 (K) + ν − 2 kep kL2 (K) + ωK .
(4.15)
This is the desired bound for ηRK .
4.3. Efficiency bound for ηEK
Next, we show the efficiency of ηEK stated in Theorem 3.2.
Consider an interior edge E that is shared by two elements K and K 0 . We denote by bE
the standard polynomial bubble function on E. Let δE = {K, K 0 }. If E is a regular edge, we
e = K 0 . If one vertex of E is a hanging node, we may assume without loss of generality
set K
e ⊂ K 0 the largest rectangle contained
that E is an entire edge of K. We then denote by K
0
e
in K so that E is also an entire edge of K.
e If w is a (vector-valued) polynomial function on E, then there exists
Let now δeE = {K, K}.
a constant C > 0 independent of w and hE such that
1
kwkL2 (E) ≤ CkbE2 wkL2 (E) .
(4.16)
e 2 such that wb |E = bE w and
Furthermore, there exists an extension w b ∈ H01 (K ∩ K)
1
kwb kL2 (K)
≤ ChE2 kwkL2 (E)
k∇wb kL2 (K)
≤
kwb kL∞(K)
≤
−1
ChE 2 kwkL2 (E)
−1
ChE 2 kwkL2 (E)
∀K ∈ δeE ,
∀K ∈ δeE ,
∀K ∈ δeE .
(4.17)
(4.18)
(4.19)
with a constant C > 0 independent of w and E; cf. [29, Lemma 4.1] and [1, Theorems 2.2
e
and 2.4]. We extend w b by zero outside the patch formed by the union of K and K.
Let now RE be the edge residual over the edge E. We denote by W b the extension of bE RE
constructed above. By (4.16), we obtain
Z
1
RE · W b ds.
CkRE k2L2 (E) ≤ kbE2 RE k2L2 (E) =
E
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To estimate the latter integral, we first note that the solution (u, p) of the Navier-Stokes
equations satisfies
[[(pI − ν∇u) n]]|E = 0.
Using this and Green’s formula in each element of the patch δeE , we then conclude that
Z
X Z RE · W b ds =
(−ν∆eu + ∇ep ) · W b + (ep I − ν∇eu ) : ∇W b dx.
E
K∈δeE
K
Consequently, using that (u, p) solves the Navier-Stokes equations, we see that
Z
X Z
(f + ν∆uh − ∇ph − (uh · ∇)uh ) · W b dx
RE · W b ds =
E
K∈δeE
K
X Z
+
((uh · ∇)uh − (u · ∇)u) · W b dx
K∈δeE
K
K∈δeE
K
+
X Z
(ep I − ν∇eu ) : ∇W b dx.
Therefore,
CkRE k2L2 (E) ≤ S1 + S2 + S3 ,
(4.20)
where
S1
S2
S3
X Z
=
K∈δeE
K
K∈δeE
K
K∈δeE
K
X Z
=
=
X Z
((f h + ν∆uh − (uh · ∇)uh − ∇ph ) + (f − f h )) · W b dx,
((uh · ∇)uh − (u · ∇)u) · W b dx,
(ep I − ν∇eu ) : ∇W b dx.
To bound S1 , we employ the Cauchy-Schwarz inequality, the bound (4.17), and take into
account the shape-regularity of the mesh. We obtain
X
1
1
−1
ν − 2 hK kf h + ν∆uh − (uh · ∇)uh − ∇ph kL2 (K)
S1 ≤Cν 2 hE 2 kRE kL2 (E)
K∈δeE
+ Cν
1
2
−1
hE 2 kRE kL2 (E)
X
1
ν − 2 hK kf − f h kL2 (K) .
K∈δeE
Obviously, there holds k · kL2 (K)
e ≤ k · kL2 (K 0 ) . Therefore, we conclude that
1
−1
S1 ≤ Cν 2 hE 2 kRE kL2 (E)
X
(ηRK + ωK ) .
(4.21)
K∈δE
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By proceeding as for the bound of T2 in (4.13) and using (4.19), we find that
X 1
1 −1
ν 2 keu kH 1 (K)
S2 ≤ Cν 2 hE 2 kRE kL2 (E)
K∈δf
E
≤ Cν
1
2
−1
hE 2 kRE kL2 (E)
X
1
(4.22)
ν 2 keu kH 1 (K) .
K∈δE
Similarly, employing (4.18), the sum S3 can be readily bounded by
X 1
1
1 −1
ν 2 k∇eu kL2 (K) + ν − 2 kep kL2 (Ω) .
S3 ≤ Cν 2 hE 2 kRE kL2 (E)
(4.23)
K∈δE
By combining (4.20), (4.21), (4.22) and (4.23), by dividing the resulting bound by kR E kL2 (E)
1
1
and by multiplying it by ν − 2 hE2 , we obtain
X 1
1
1
1
ν 2 keu kH 1 (K) + ν − 2 kep kL2 (K)
ν − 2 hE2 kRE kL2 (E) ≤C
K∈δE
+C
X
(4.24)
(ηRK + ωK ) .
K∈δE
The desired bound for ηEK now follows easily from (4.15) and (4.24). This concludes the
proof of Theorem 3.2.
5. Numerical results
In this section, we test our error estimate by reproducing known analytical solutions. All
the results were obtained using the deal.II finite element library (see [4]). Let us consider
the standard singular solution of the Stokes problem with ν = 1 on an L-shaped domain
(see [20, 30]). The singularities at the reentrant corner are of the form r λ and rλ−1 for the
velocity and pressure, respectively, where λ ≈ 0.544. First, we use uniform mesh refinement
and do not expect convergence orders better than λ/2 in the energy norm in terms of the
number of degrees of freedom ndofs . This is confirmed in Table 5.1 for RT3 /Q3 and RT2 /Q2
elements. The a-posteriori error estimate η is of the same order as the actual energy error,
thereby confirming the reliability and efficiency results from Theorem 3.1 and Theorem 3.2,
respectively. The ratio between estimator and error is also shown and is around 8 for k = 2
and close to 14 for k = 3.
In order to use the estimate η for adaptive mesh refinement, we use a heuristic criterion
based on cellwise error indicators only. We simply refine the third of the overall cells with
the highest indicators. Strategies like this have been used successfully in many applications,
see e. g. [5] and references cited therein. While for this strategy, no quantitative convergence
estimates like in [16, 19] are obtained, its implementation is very simple. From the sets η K
and ηE , we obtain cell refinement indicators by adding all or half of the face indicator ηE to
its both neighboring cells for boundary and interior faces, respectively. Then, the fraction Θ
of the total number of cells is refined.
In Figure 1, we present results on adaptively refined meshes for degrees two to four, again
for the Stokes solution. First, this figure shows that the graphs for the estimates are parallel to
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RT2 /Q2
η
L ||| (eu , ep ) ||| η
||| (eu ,ep ) |||
4
0.612
4.95
8.1
5
0.420
3.39
8.1
6
0.288
2.32
8.1
7
0.197
1.59
8.1
8
0.135
1.09
8.1
RT3 /Q3
||| (eu , ep ) ||| η
0.427
5.87
0.293
4.02
0.201
2.76
0.138
1.89
0.094
1.30
η
||| (eu ,ep ) |||
13.7
13.7
13.7
13.7
13.7
Table 5.1. Convergence history for the Stokes problem on an L-shaped domain with uniform meshes.
10
RT2/Q2 errors
RT2/Q2 estimate
RT3/Q3 errors
RT3/Q3 estimate
RT4/Q4 errors
RT4/Q4 estimate
1
0.1
0.01
0.001
1000
10000
100000
Degrees of freedom
1e+06
Figure 1. Errors and estimates for adaptive refinement. Stokes problem on an L-shaped domain.
those of the error, confirming reliability and efficiency of the estimator. Furthermore, adaptive
k/2
refinement allows us to recover the optimal convergence rate of ndofs ; here, the fraction Θ was
chosen as 0.15, 0.1 and 0.05 for degrees 2 to 4, respectively.
As a second example, we reproduce the analytical Navier-Stokes solution by Kovasznay [25]
for a viscosity of ν = 1/10. The estimates and the errors in the norm ||| (u, p) ||| are reported in
Figure 2. Since the solution is smooth, both error graphs are parallel and exhibit first order
convergence. Again, the estimate is by a factor of 5 larger than the error, even if the viscosity
is smaller, confirming the robustness of our estimate. On the first mesh, the nonlinearity is
unresolved and therefore the estimate cannot capture the error.
The two meshes in Figure 3 show why adaptive mesh refinement yields much better results
for the L-shaped domain than for Kovasznay flow: while refinement is concentrated around
the reentrant corner in the former case, it extends to a wide area to the left for the latter.
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100
Uniform error
Uniform estimate
Adaptive error
Adaptive estimate
10
1
0.1
100
1000
10000
Degrees of freedom
100000
1e+06
Figure 2. Energy error and estimates for Kovasznay flow, ν = 0.1, uniform and adaptive refinement
with RT1 /Q1 elements.
Figure 3. Adaptive meshes for the L-shaped domain (RT4 /Q4 ) and for Kovasznay flow (RT1 /Q1 ).
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6. Conclusion
We presented a residual based a posteriori error estimate for DG approximations to solutions
of the Navier–Stokes equations. These approximations are based on Raviart-Thomas elements
on locally refined meshes and yield strongly divergence–free discrete solutions. Therefore, a
Helmholtz decomposition of the error is unnecessary and the estimate takes a simple form.
The estimate is shown to be reliable, efficient and robust with respect to the viscosity, as long
as the data are small. Numerical results show that the constants involved are of moderate size.
Finally, we point out that our analysis applies naturally to exactly divergence-free
BDMk /Pk−1 elements on regular triangular meshes; cf. [14].
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