A robust a-posteriori error estimator for
discontinuous Galerkin methods for
convection-diffusion equations 1
Dominik Schötzau
Liang Zhu
Mathematics Department, University of British Columbia,
Vancouver, BC, Canada, V6T 1Z2
Appl. Numer. Math., Vol. 59, pp. 2236–2255, 2009
Abstract
A robust a-posteriori error estimator for interior penalty discontinuous Galerkin
discretizations of a stationary convection-diffusion equation is derived. The estimator yields global upper and lower bounds of the error measured in terms of the
energy norm and a semi-norm associated with the convective term in the equation. The ratio of the upper and lower bounds is independent of the magnitude
of the Péclet number of the problem, and hence the estimator is fully robust for
convection-dominated problems. Numerical examples are presented that illustrate
the robustness and practical performance of the proposed error estimator in an
adaptive refinement strategy.
Key words: Discontinuous Galerkin methods, robust a-posteriori error estimation,
convection-diffusion equations
Email addresses: schoetzau@math.ubc.ca (Dominik Schötzau),
zhuliang@math.ubc.ca (Liang Zhu).
1 This work was partially supported by Natural Sciences and Engineering Research
Council of Canada (NSERC).
Preprint submitted to Elsevier
13 July 2009
1
Introduction
One of the main difficulties in the finite element approximation of convectiondiffusion equations is that solutions to these problems may have layers of small
width where their gradients change extremely rapidly. Such layers appear as
boundary layers near the outflow boundary of the domain, or as internal layers,
caused by non-smooth data near the inflow boundary. The effective numerical
resolution of these solution features requires adaptive finite element methods
that are capable of locally refining the meshes in the vicinity of the layers and
other singularities.
At the heart of adaptive finite element methods are a-posteriori error estimators that provide information on the local error distribution. While there is
a huge amount of literature available on error estimation for pure diffusion
problems (here we only mention [1,29] and the references therein), much fewer
results can be found for convection-diffusion problems. Here, one is particularly interested in robust a-posteriori error estimators that yield upper and
lower bounds for the error (measured in a suitable norm) that differ by a factor
that is independent of the Péclet number of the convection-diffusion problem
at hand. Important advances in this direction were made in [30] where an
estimator for a conforming SUPG method was derived for which the ratio of
the upper and lower bounds scales with the square root of the Péclet number.
Other estimators that are almost robust can be found in [22], still in a conforming setting, and in [2] for a non-conforming finite element method with
face penalties. In the recent work [31], a fully robust error estimator has been
proposed. There, in addition to the energy norm, the error measure now also
includes a dual norm of the convective derivative. Another approach to robust
error estimation can be found in [27,28], whereby the error in the convective
term is evaluated in an interpolation norm of order 1/2.
In this paper, we propose and analyze a robust a-posteriori error estimator for
discontinuous Galerkin discretizations of convection-diffusion problems. The
estimator yields upper and lower bounds of the error measured in terms of the
natural energy norm and a semi-norm associated with the convective terms.
Our analysis is based on the approach in [18] where energy norm a-posteriori
error estimates were developed for pure diffusion problems. In this approach,
the error is decomposed into a conforming part and a remainder. The conforming contribution can then be dealt with using standard techniques, while the
rest term can be controlled using the stabilizing jump terms. The same techniques were used in the related papers [15–17] on energy norm error estimation
for discontinuous Galerkin discretizations of saddle point problems. The error measure used in our analysis includes a non-local norm similar to the one
in [31]. Our numerical examples indicate that this error contribution is smaller
than the energy error and of higher-order once the mesh is sufficiently refined.
2
To the best of our knowledge, this is the first approach to robust energy norm
error estimation for discontinuous Galerkin methods for convection-diffusion
equations. Other approaches to energy norm error estimation for discontinuous
Galerkin methods applied to pure diffusion problems can be found in [7–9,21]
and the references therein. For L2 -norm and functional error estimation, we
also mention [8,14,20,26] and the references therein.
The discontinuous Galerkin method proposed here is based on the upwind
discretization of the transport terms, as originally introduced in [23,25]. The
diffusive terms are discretized using the classical interior penalty method;
see [3,24]. The resulting scheme is ideally suited for the numerical approximation of convection-diffusion equations. Indeed, it is known to be robust and
stable in the hyperbolic limit, in contrast to standard Galerkin methods. For
recent accounts on the state-of-the-art of discontinuous Galerkin methods we
refer the reader to the articles in [4,11,10,12] and the references therein.
The outline of this article is as follows. In Section 2, we introduce the discontinuous Galerkin method for a convection-diffusion model problem. In Section 3,
our a-posteriori error estimator is presented and discussed. The proof of its
reliability and efficiency is carried out in Section 4. In Section 5, we show
a series of numerical tests. Finally, in Section 6 we present some concluding
remarks.
2
Interior penalty discretization of a convection-diffusion problem
2.1 Model problem
We consider the convection-diffusion model problem:
−ε∆u + a(x) · ∇u + b(x)u = f (x)
u=0
in Ω,
on Γ.
(2.1)
Here, Ω is a bounded Lipschitz polygon in R2 with boundary Γ = ∂Ω. The
right-hand side f is a given function in L2 (Ω). We assume that the diffusion
coefficient satisfies
0 < ε 1.
The coefficient functions a(x) and b(x) are supposed to belong to W 1,∞ (Ω)2
and L∞ (Ω), respectively. Without loss of generality, we may assume that a
and the size of the domain Ω are of order one so that ε−1 is the Péclet number
of problem (2.1). We further assume that there is a constant β ≥ 0 such that
1
− ∇ · a(x) + b(x) ≥ β,
2
3
x ∈ Ω.
(2.2)
Finally, we suppose that there is a second constant c? ≥ 0 such that
k − ∇ · a + bkL∞ (Ω) ≤ c? β.
The weak form of (2.1) is to find u ∈ H01 (Ω) such that
A(u, v) =
Z
Ω
ε∇u · ∇v + a · ∇uv + buv dx =
(2.3)
Z
Ω
f v dx
for all v ∈ H01 (Ω). Under the above assumptions on the coefficients, this variational problem is uniquely solvable. Upon integration by parts of the convective
term, we also have
A(u, v) =
Z
Ω
ε∇u · ∇v − au · ∇v + (b − ∇ · a)uv dx.
Remark 2.1 If β = 0, we obtain from assumption (2.3) that b = ∇·a. Hence,
the convection-diffusion equation (2.1) can be written in the divergence form
−ε∆u + ∇ · (au) = f.
In this case, assumption (2.2) is satisfied provided that ∇ · a ≥ 0.
2.2 Discretization
To discretize (2.1), we consider regular and shape-regular meshes Th = {K}
that partition the computational domain Ω into open triangles and parallelograms. We define E(Th ) to be the set of all edges of the mesh Th . For the
set of all interior edges we write EI (Th ). The diameter of an element K and
the length of an edge E are denoted by hK and hE , respectively. Furthermore,
we write nK for the outward unit normal vector on the boundary ∂K of an
element K.
The jumps and averages of piecewise smooth functions are defined as follows.
Let the edge E be shared by two neighboring elements K and K e . For a
piecewise smooth function v, we denote by v|E its trace on E taken from
inside K, and by v e |E the one taken inside K e . The average and jump of v
across the edge E are then defined as
1
{{v}} = (v|E + v e |E ),
2
[[v]] = v|E nK + v e |E nK e .
Similarly, if q is piecewise smooth vector field, its average and (normal) jump
across E are given by
{{q}} =
1
q|E + q e |E ,
2
[[q]] = q|E · nK + q e |E · nK e .
4
On a boundary edge E shared by Γ and K, we set accordingly {{q}} = q and
[[v]] = vn, with n denoting the unit outward normal vector on Γ.
We denote by Γin and Γout the inflow and outflow parts of Γ:
Γin = {x ∈ Γ : a(x) · n(x) < 0},
Γout = {x ∈ Γ : a(x) · n(x) ≥ 0}.
Similarly, the inflow and outflow boundaries of an element K are defined by
∂Kin = {x ∈ ∂K : a(x) · nK (x) < 0},
∂Kout = {x ∈ ∂K : a(x) · nK (x) ≥ 0}.
For an approximation order p ≥ 1, let now Vh be the finite element space
Vh = { v ∈ L2 (Ω) : v|K ∈ Sp (K), K ∈ Th },
where Sp (K) is the space Pp (K) of polynomials of total degree ≤ p if K is a
triangle, and the space Qp (K) of polynomials of degree ≤ p in each variable
if K is a parallelogram.
We consider the following discontinuous Galerkin method that is based on the
original upwind discretization in [23,25] for the convective term and on the
classical interior penalty discretization in [3,4,24] for the Laplacian. It is given
by: Find uh ∈ Vh such that
Ah (uh , v) =
Z
Ω
f v dx
(2.4)
for all v ∈ Vh , with the bilinear Ah given by
Ah (u, v) =
−
X Z
K∈Th K
X
E∈E(Th )
X
Z
(ε∇u · ∇v + a · ∇uv + buv) dx
E
εγ
+
h
E∈E(Th ) E
+
X Z
K∈Th
{{ε∇u}} · [[v]] ds −
Z
∂Kin \Γ
E
[[u]] · [[v]] ds −
X
E∈E(Th )
X Z
Z
E
{{ε∇v}} · [[u]] ds
K∈Th ∂Kin ∩Γin
a · nK uv ds
a · nK (ue − u)v ds.
Here, for a piecewise smooth function, the gradient operator ∇ is taken elementwise. The constant γ > 0 is the interior penalty parameter. To ensure the
stability of the discontinuous Galerkin discretization, it has to be chosen sufficiently large, independently of the mesh size and the diffusion coefficient ε,
see, e.g., [3,4,18].
5
Upon integration by parts of the convective term, we also have
Ah (u, v) =
−
+
+
3
X Z
K∈Th K
X
E∈E(Th )
X
Z
E
εγ
h
E∈E(Th ) E
ε∇u · ∇v − au · ∇v + (b − ∇ · a)uv dx
{{ε∇u}} · [[v]] ds −
Z
E
X Z
K∈Th ∂Kout \Γ
[[u]] · [[v]] ds +
X
E∈E(Th )
Z
E
{{ε∇v}} · [[u]] ds
X Z
K∈Th ∂Kout ∩Γout
a · nK uv ds
a · nK u(v − v e ) ds.
Robust a-posteriori error estimation
In this section, we present and discuss our main results.
3.1 Norms
We begin by introducing the norm
||| u |||2 =
X εk∇uk2L2 (K) + βkuk2L2 (K) +
K∈Th
X
γε
k[[u]]k2L2 (E) .
h
E
E∈E(Th )
(3.5)
This norm can be viewed as the energy norm associated with the discontinuous
Galerkin discretization of the convection-diffusion problem (2.1).
For q ∈ L2 (Ω)2 , we further define the semi-norm
|q|? =
sup
v∈H01 (Ω)\{0}
R
Ω
q · ∇v dx
.
||| v |||
Remark 3.1 The semi-norm | · |? can be characterized by using a Helmholtz
decomposition similar to the one in [13, Theorem 3.2]. We write q in the form
q = ∇ϕ + q 0 ,
where ϕ ∈ H01 (Ω) solves
Z
Ω
∇ϕ · ∇v dx =
Z
Ω
q · ∇v dx
6
∀ v ∈ H01 (Ω),
and q 0 = q − ∇ϕ is divergence-free in the sense that
Z
Ω
∀ v ∈ H01 (Ω).
q 0 · ∇v dx = 0
This decomposition is unique and orthogonal in L2 (Ω)2 . Thus, we observe
that |q|? = 0 if and only if q = q 0 . Furthermore, if we introduce the norm
kϕk? =
sup
v∈H01 (Ω)\{0}
R
Ω
∇ϕ · ∇v dx
,
||| v |||
we have that |q|? = kϕk? .
We now define
| u |2A
=
|au|2?
+
X
hE
βhE +
ε
E∈E(Th )
!
k[[u]]k2L2 (E) .
(3.6)
The semi-norm |au|2? and the jump terms hE ε−1 k[[u]]k2L2 (E) will be used to
bound the convective derivative, analogously to [31]. Here we note that hE ε−1
is the local mesh Péclet number, see also the discussion in Remark 3.5. Finally,
the jump terms βhE k[[u]]|2L2 (E) are associated with the reaction term in the
equation.
3.2 A robust a-posteriori error estimator
Next, we define our a-posteriori error estimator. To that end, we set
1
1
1
ρK = min{hK ε− 2 , β − 2 },
1
ρE = min{hE ε− 2 , β − 2 }.
1
1
In the case β = 0, we set ρK = ε− 2 hK and ρE = ε− 2 hE .
Let now uh be the discontinuous Galerkin approximation obtained by (2.4).
Moreover, let fh , ah , and bh denote piecewise polynomial approximations in Vh
to the right-hand side and the coefficient functions, respectively.
For each element K ∈ Th , we introduce a local error indicator ηK which is
given by the sum of the three terms
2
ηK
= ηR2 K + ηE2 K + ηJ2K .
The first term ηRK is the interior residual defined by
ηR2 K = ρ2K kfh + ε∆uh − ah · ∇uh − bh uh k2L2 (K) .
7
The second term ηE2 K is the edge residual given by
ηE2 K =
1 X −1
ε 2 ρE k[[ε∇uh ]]k2L2 (E) .
2 E∈∂K\Γ
The last term ηJK measures the jumps of the approximate solution uh and is
defined by
ηJ2K
γε
hE
1 X
+ βhE +
=
2 E∈∂K\Γ hE
ε
+
X
E∈∂K∩Γ
!
hE
γε
+ βhE +
hE
ε
k[[uh ]]k2L2 (E) ,
!
k[[uh ]]k2L2 (E) .
We also introduce a data approximation term by
Θ2K
=
ρ2K
kf −
fh k2L2 (K)
∇uh k2L2 (K)
+ k(a − ah ) ·
+ k(b −
bh )uh k2L2 (K)
.
We then define the a-posteriori error estimator
η=
X
2
ηK
K∈Th
1
2
.
(3.7)
1
(3.8)
The data approximation error is given by
Θ=
X
K∈Th
Θ2K
2
.
3.3 Reliability and efficiency
In the following, we use the symbols . and & to denote bounds that are valid
up to positive constants independent of the local mesh sizes and the diffusion
coefficient ε. The constants will also be independent of γ, provided that γ ≥ 1.
Our first main result states that, up to a constant and to the data approximation error, the estimator (3.7) gives rise to a reliable a-posteriori error
bound.
Theorem 3.2 Let u be the solution of (2.1) and uh ∈ Vh its DG approximation obtained by (2.4). Let the error estimator η be defined by (3.7), and
the data approximation error Θ by (3.8). Then we have the a-posteriori error
bound
||| u − uh ||| + | u − uh |A . η + Θ.
8
Our next theorem presents a lower bound for the error and shows the efficiency
of the error estimator η.
Theorem 3.3 Let u be the solution of (2.1) and uh ∈ Vh its DG approximation obtained by (2.4). Let the local error estimator η be defined by (3.7), and
the data approximation error Θ by (3.8). Then we have the bound
η . ||| u − uh ||| + | u − uh |A + Θ.
Remark 3.4 Both the reliability and efficiency constants in Theorem 3.2 and
Theorem 3.3 are independent of the diffusion coefficient ε. Hence, the ratio of
the upper and lower bounds is independent of the Péclet number ε, up to data
approximation errors. In this sense, the error estimator η in (3.7) is robust in
the diffusion parameter ε.
Remark 3.5 In our numerical tests in Section 5, we show that the nonstandard error | u − uh |A is of at least the same order as the energy error ||| u − uh ||| and even of higher-order, once the local mesh Péclet number
is sufficiently small. Heuristically, this can be explained as follows. We expect
p
the error ||| u − uh ||| to converge with the optimal order O(N − 2 ), where N is
the number of degrees of freedom; cf. [4,19]. We then have the bound
1
|a(u − uh )|? . √ ku − uh kL2 (Ω) .
ε
(3.9)
If we now also assume that the L2 -error ku − uh kL2 (Ω) converges with the
p
1
optimal rate O(N − 2 − 2 ), we obtain
1
N−2 √ −p
|a(u − uh )|? .
εN 2 .
ε
−1
The fraction N ε 2 is the local mesh Péclet number. Hence, |a(u − uh )|? is of
at least the same order as the energy error, once the mesh Péclet number is
sufficiently small. Similar arguments show that
X
1
E∈Th
X
E∈Th
1
hE
N−2 √ −p
k[[u − uh ]]k2L2 (E) 2 .
εN 2 ,
ε
ε
βhE k[[u − uh ]]k2L2 (E)
1
2
.
q
p
1
βN − 2 − 2 ,
where we have used that [[u]] = 0. Thus, the same conclusion as for |a(u − u h )|?
can also be made for the error | u − uh |A .
9
4
Proofs
In this section, we present the proofs of Theorem 3.2 and Theorem 3.3. We
proceed in several steps.
4.1 Auxiliary forms and their properties
The discontinuous Galerkin form Ah (u, v) is not well-defined for functions u, v
in H01 (Ω). In [18], this difficulty has been overcome by the use of a suitable
lifting operator. Here, we present a different and new approach where we
split the discontinuous Galerkin form into several parts. More precisely, we
introduce the auxiliary forms
Dh (u, v) =
X Z
K∈Th K
Oh (u, v) = −
ε∇u · ∇v + (b − ∇ · a)uv dx,
X Z
K∈Th K
X Z
+
K∈Th
Kh (u, v) = −
Jh (u, v) =
X
au · ∇v dx +
∂Kout \Γ
Z
E∈E(Th ) E
X
εγ
h
E∈E(Th ) E
Z
X Z
K∈Th
a · nK uv ds
a · nK u(v − v e ) ds,
{{ε∇u}} · [[v]] ds −
E
∂Kout ∩Γout
X
Z
E∈E(Th ) E
{{ε∇v}} · [[u]] ds,
[[u]] · [[v]] ds.
Then, we set
Aeh (u, v) = Dh (u, v) + Jh (u, v) + Oh (u, v).
This form is well-defined for all u, v ∈ Vh + H01 (Ω). Obviously, we have
Aeh (u, v) = A(u, v),
(4.1)
Ah (u, v) = Aeh (u, v) + Kh (u, v),
(4.2)
for all u, v ∈ H01 (Ω). Furthermore,
for all u, v ∈ Vh .
As a consequence of (2.2) and (4.1), we have the following coercivity result.
Lemma 4.1 For any u ∈ H01 (Ω), we have
Aeh (u, u) ≥ ||| u |||2 .
10
Furthermore, the auxiliary forms are continuous.
Lemma 4.2 There holds
|Dh (u, v)| . ||| u |||||| v |||,
u, v ∈ Vh + H01 (Ω),
|Jh (u, v)| . ||| u |||||| v |||,
u, v ∈ Vh + H01 (Ω),
|Oh (u, v)| . |au|? ||| v |||,
u ∈ Vh + H01 (Ω), v ∈ H01 (Ω).
Proof : The first claim follows from the Cauchy-Schwarz inequality and the
bound in (2.3). The second is a straightforward consequence of the CauchySchwarz inequality. The third one follows immediately from the definition
of |au|? .
2
Lemma 4.3 For u ∈ Vh and v ∈ H01 (Ω) ∩ Vh we have
1
Kh (u, v) . γ − 2
1
εγ
k[[u]]k2L2 (E) 2 ||| v |||.
h
E∈E(Th ) E
X
Proof : Since v ∈ H01 (Ω) ∩ Vh , we have
Kh (u, v) = −
X
E∈E(Th )
Z
E
{{ε∇v}} · [[u]] ds.
Using the Cauchy-Schwarz inequality, the inverse estimate,
−1
kvkL2 (∂K) . hK 2 kvkL2 (K) ,
v ∈ Sp (K),
and the shape-regularity of the mesh, we obtain
Kh (u, v) .
X
Z
E∈E(Th ) E
.γ
− 21
1
. γ− 2
X
|ε∇v||[[u]]| ds
εhE
E∈E(Th )
X
K∈Th
Z
2
E
|∇v| ds
εk∇vk2L2 (K)
1 2
1 2
1
εγ Z
2
|[[u]]| ds 2
h E
E∈E(Th ) E
X
1
εγ
k[[u]]k2L2 (E) 2 .
h
E∈E(Th ) E
X
This yields the assertion.
2
Next, we show the following inf-sup condition.
11
Lemma 4.4 There is a constant C > 0 such that
Aeh (u, v)
≥ C > 0.
u∈H0 (Ω)\{0} v∈H 1 (Ω)\{0} (||| u ||| + |au|? )||| v |||
0
inf
1
sup
Proof : Let u ∈ H01 (Ω) and θ ∈ (0, 1). Then there exists wθ ∈ H01 (Ω) such that
||| wθ ||| = 1,
Oh (u, wθ ) = −
Z
Ω
au · ∇wθ dx ≥ θ|au|? .
From the continuity properties in Lemma 4.2, we obtain
Aeh (u, wθ ) = Dh (u, wθ ) + Jh (u, wθ ) + Oh (u, wθ )
≥ θ|au|? − C1 ||| u |||||| wθ |||
= θ|au|? − C1 ||| u |||,
for a constant C1 > 0. Let us then define
vθ = u +
Obviously,
||| vθ ||| ≤ (1 +
||| u |||
wθ .
1 + C1
1
)||| u |||.
1 + C1
By Lemma 4.1,
A(u, u) ≥ ||| u |||2 ,
so that
Aeh (u, v) Aeh (u, vθ )
≥
||| v |||
||| vθ |||
v∈H01 (Ω)\{0}
sup
||| u |||2 + (1 + C1 )−1 ||| u |||(θ|au|? − C1 ||| u |||)
1
)||| u |||
(1 + 1+C
1
1
=
(||| u ||| + θ|au|? ).
2 + C1
≥
Since θ ∈ (0, 1) and u ∈ H01 (Ω) are arbitrary, we obtain the inf-sup condition.2
4.2 Approximation operators
Let Vhc be the conforming subspace of Vh given by
Vhc = Vh ∩ H01 (Ω).
12
We denote by Ah : Vh → Vhc the approximation operator defined in [21, Theorem 2.2 and Theorem 2.3]; see also [18, Proposition 5.4] for an extension to
the hp-version of the discontinuous Galerkin method. The following approximation result holds.
Lemma 4.5 For any v ∈ Vh , we have
X
K∈Th
X
K∈Th
kv − Ah vk2L2 (K) .
k∇(v − Ah v)k2L2 (K) .
X
Z
E∈E(Th ) E
X
Z
E∈E(Th ) E
hE |[[v]]|2 ds,
2
h−1
E |[[v]]| ds.
Moreover, we will make use of the Clément-type interpolant constructed in [31,
Lemma 3.3] and the references therein.
Lemma 4.6 There exists an interpolation operator
Ih : H01 (Ω) → {ϕ ∈ C(Ω) : ϕ|K ∈ S1 (K), ∀ K ∈ Th , ϕ = 0 on Γ},
that satisfies ||| Ih v ||| . ||| v ||| and
2
ρ−2
K kv − Ih vkL2 (K)
1
. ||| v |||,
2
ε 2 ρ−1
E kv − Ih vkL2 (E)
1
. ||| v |||,
X
K∈Th
X
E∈E(Th )
1
2
2
for any v ∈ H01 (Ω).
4.3 Proof of Theorem 3.2
We are now ready to prove Theorem 3.2.
Following [18], we decompose the discontinuous Galerkin solution into a conforming part and a remainder. That is, we write
uh = uch + urh ,
where uch = Ah uh ∈ Vhc , with Ah the approximation operator from Lemma 4.5.
The rest is then given by urh = uh − uch . By the triangle inequality we obtain
||| u − uh ||| + | u − uh |A ≤ ||| u − uch ||| + | u − uch |A + ||| urh ||| + | urh |A .
(4.3)
Next, we prove that both the continuous error u − uch and the rest term urh
can be bounded by the error estimator. We proceed in several steps.
13
Lemma 4.7 There holds
||| urh ||| + | urh |A . η.
Proof : Since [[urh ]] = [[uh ]], we have
||| urh |||2 + | urh |2A =
X K∈Th
εk∇urh k2L2 (K) + βkurhk2L2 (K) + |aurh |2?
X
+
γε
hE
+ βhE +
hE
ε
E∈E (Th )
.
X K∈Th
!
k[[uh ]]k2L2 (E)
εk∇urhk2L2 (K) + βkurh k2L2 (K) + |aurh |2? +
X
ηJ2K .
K∈Th
Hence, only the volume terms and the expression involving the | · |? semi-norm
need to be bounded further. Lemma 4.5 yields
ε
X
K∈Th
β
k∇urh k2L2 (K) . γ −1
X
K∈Th
kurh k2L2 (K) .
X
εγ
k[[uh ]]k2L2 (E) . γ −1
ηJ2K ,
h
K∈Th
E∈E(Th ) E
X
X
E∈E(Th )
βhE k[[uh ]]k2L2 (E) .
X
ηJ2K .
K∈Th
To estimate |aurh |? , we apply Lemma 4.5 once more, and obtain
X hE
X
1
ηJ2K .
|aurh |2? . kurh k2L2 (Ω) .
k[[uh ]]k2L2 (E) .
ε
ε
K∈Th
E∈E(Th )
This finishes the proof.
2
Lemma 4.8 For any v ∈ H01 (Ω), we have
Z
Ω
f (v − Ih v) dx − Aeh (uh , v − Ih v) . (η + Θ) ||| v |||.
Here, Ih is the interpolant introduced in Lemma 4.6.
Proof : Set
T =
Z
Ω
f (v − Ih v) dx − Aeh (uh , v − Ih v).
14
Integration by parts immediately yields
T =
X Z
K∈Th K
−
(f + ε∆uh − a · ∇uh − buh )(v − Ih v) dx
X Z
K∈Th ∂K
X Z
+
K∈Th
ε∇uh · nK (v − Ih v) ds
∂Kin \Γ
a · nK (uh − ueh )(v − Ih v) ds
= T1 + T2 + T3 .
In the term T1 , we first add and subtract the data approximations terms. This
gives
T1 =
+
X Z
K∈Th K
X Z
K∈Th K
(fh + ε∆uh − ah · ∇uh − bh uh )(v − Ih v) dx
(f − fh ) − (a − ah ) · ∇uh − (b − bh )uh (v − Ih v) dx.
Using the Cauchy-Schwarz inequality and Lemma 4.6 yields
T1 .
X
ηR2 K
X
Θ2K
X
(ηR2 K + Θ2K )
K∈Th
+
K∈Th
.
1 X
2
K∈Th
1 X
2
K∈Th
K∈Th
2
ρ−2
K kv − Ih vkL2 (K)
2
ρ−2
K kv − Ih vkL2 (K)
1
2
1
2
1
2
||| v |||.
Next, if we rewrite the term T2 in terms of the jumps of ε∇uh , we obtain
T2 = −
X
Z
E∈EI (Th ) E
[[ε∇uh ]](v − Ih v) ds.
Hence, the Cauchy-Schwarz inequality and Lemma 4.6 yield
T2 .
.
X
E∈EI (Th )
X
K∈Th
1
ε− 2 ρE k[[ε∇uh ]]k2L2 (E)
ηE2 K
1
2
1 2
X
E∈E(Th )
1
2
ε 2 ρ−1
E kv − Ih vkL2 (E)
1
2
||| v |||.
In order to bound T3 , we use the Cauchy-Schwarz inequality, Lemma 4.6, and
15
1
the fact that ρE ≤ hK ε− 2 :
T3 .
.
X
1
E∈E(Th )
X
ε− 2 ρE k[[uh ]]k2L2 (E)
ηJ2K
K∈Th
1
2
1 2
X
E∈E(Th )
1
2
ε 2 ρ−1
E kv − Ih vkL2 (E)
1
2
||| v |||.
This finishes the proof.
2
Lemma 4.9 There holds:
||| u − uch ||| + | u − uch |A . η + Θ.
Proof : Note that | u − uch |A = |a(u − uch )|? . Then the inf-sup condition yields:
||| u − uch ||| + |a(u − uch )|? .
Aeh (u − uch , v)
.
||| v |||
v∈H01 (Ω)\{0}
sup
(4.4)
The properties (4.1) and (4.2) allow us to conclude that, for any v ∈ H01 (Ω),
Aeh (u − uch , v) =
=
=
Z
Z
Z
Ω
Ω
Ω
f v dx − Aeh (uch , v)
f v dx − Dh (uch , v) − Jh (uch , v) − Oh (uch , v)
f v dx − Aeh (uh , v) + Dh (urh , v) + Jh (urh , v) + Oh (urh , v).
From the discontinuous Galerkin method in (2.4), we have
Z
Ω
f Ih v dx = Ah (uh , Ih v) = Aeh (uh , Ih v) + Kh (uh , Ih v),
where Ih is the operator introduced in Lemma 4.6. Therefore,
e − uc , v) = T + T + T ,
A(u
1
2
3
h
where
T1 =
Z
Ω
f (v − Ih v) dx − Aeh (uh , v − Ih v),
T2 = Dh (urh , v) + Jh (urh , v) + Oh (urh , v),
T3 = −Kh (uh , Ih v).
The estimate in Lemma 4.8 yields
T1 . (η + Θ) ||| v |||.
16
Similarly, the continuity result in Lemma 4.2 and the approximation properties
in Lemma 4.7 give
T2 . (||| urh ||| + |aurh |? )||| v ||| ≤ η||| v |||.
Finally, we use Lemma 4.3 to bound the term T3 . We obtain
T3 . γ
− 21


1
2
X
K∈Th
ηJ2K 
||| Ih v ||| . γ
− 21


X
K∈Th
1
2
ηJ2K 
||| v |||.
Here, we have also used the ||| · |||-stability of the operator Ih from Lemma 4.6.
This completes the proof.
2
The proof of Theorem 3.2 now immediately follows from (4.3), Lemma 4.7 and
Lemma 4.9.
4.4 Proof of Theorem 3.3
To prove Theorem 3.3, we use the same arguments as in [31]. To that end, for
any interior edge E ∈ EI (Th ), we denote by wE the union of the two elements
that share it. Furthermore, we denote by ψK and ψE the bubble functions
constructed and defined in [31, p. 1771]. The function ψK belongs to H01 (K),
while ψE is in H01 (wE ). We have
kψK kL∞ (K) = 1,
kψE kL∞ (E) = 1.
(4.5)
In the following, we denote by (·, ·)K and (·, ·)E the inner products in L2 (K)
and L2 (E), respectively. Furthermore, for a set of elements D, we denote
by ||| · |||D the local energy norm
||| u |||2D =
X K∈D
We also set
kuk2L2 (D) =
and define
Z
D
εk∇uk2L2 (K) + βkuk2L2 (K) .
f (x) dx =
X
K∈D
kuk2L2 (K) ,
X Z
K∈D K
f (x) dx.
The following result holds; cf. [31, Lemma 3.6].
Lemma 4.10 We have
17
kvk2L2 (K) . (v, ψK v)K ,
(4.6)
||| ψK v |||K . ρ−1
K kvkL2 (K) ,
(4.7)
kσk2L2 (E) . (σ, ψE σ)E ,
(4.8)
1
1
1
−1
kψE σkL2 (wE ) . ε 4 ρE2 kσkL2 (E) ,
(4.9)
||| ψE σ |||wE . ε 4 ρE 2 kσkL2 (E) ,
(4.10)
for any element K, edge E, and polynomials v and σ defined on elements and
faces, respectively. In the last two inequalities, the polynomial σ defined on E
is extended to R2 in a canonical fashion.
To prove Theorem 3.3, we first note that, since [[u]] = 0, we have
X
K∈Th
ηJ2K . ||| u − uh ||| + | u − uh |A .
(4.11)
Hence, we only need to show the efficiency of the indicators ηRK and ηEK ,
respectively. This will be done in the next two lemmas.
Lemma 4.11 There holds
X
K∈Th
ηR2 K
1
2
. ||| u − uh ||| + | u − uh |A + Θ.
Proof : Let K be an element in Th . We define
R|K = (fh + ε∆uh − ah · ∇uh − bh uh )|K ,
and set W |K = ρ2K RψK . By inequality (4.6) in Lemma 4.10,
X
ηR2 K =
K∈Th
X
ρ2K kRk2L2 (K) .
X
(R, W )K =
K∈Th
=
K∈Th
X
(R, ρ2K ψK R)K
K∈Th
X
K∈Th
(fh + ε∆uh − ah · ∇uh − bh uh , W )K .
Since the exact solution satisfies (f + ε∆u − a · ∇u − bu)|K = 0, we obtain,
18
by integration by parts and addition and subtraction of the exact data,
X
ηR2 K
.
X K∈Th
K∈Th
+
ε(∇(uh − u), ∇W )K − (a(u − uh ), ∇W )K
X
((b − ∇ · a)(u − uh ), W )K
K∈Th
+
X K∈Th
(fh − f ) + (a − ah ) · ∇uh + (b − bh )uh , W
.
K
Here, we have also used that W |∂K = 0. Then, by the Cauchy-Schwarz inequality, the bound in (2.3), and the definitions of | · |A and the data approximation
error Θ, we obtain
X
K∈Th
ηR2 K . ||| u − uh ||| + | u − uh |A + Θ
X
K∈Th
2
||| W |||2K + ρ−2
K kW kL2 (K)
1
2
.
By (4.7) in Lemma 4.10 and by (4.5), we have the estimates
||| W |||2K . ρ2K kRk2L2 (K) ,
and
2
2
2
ρ−2
K kW kL2 (K) . ρK kRkL2 (K) .
This yields
X
ηR2 K
K∈Th
. ||| u − uh ||| + | u − uh |A + Θ
X
ηR2 K
K∈Th
1
2
,
which shows the assertion.
2
Lemma 4.12 There holds
X
ηE2 K
K∈Th
1
2
Proof : We set
τ=
. ||| u − uh ||| + | u − uh |A + Θ.
X
1
ε− 2 ρE [[ε∇uh ]]ψE .
E∈EI (Th )
By (4.8) in Lemma 4.10 and the fact that [[ε∇u]] = 0 on interior edges, we
obtain
X
K∈Th
ηE2 K .
X
([[ε∇uh ]], τ )E =
X
E∈EI (Th )
E∈EI (Th )
19
([[ε∇(uh − u)]], τ )E .
After integration by parts over each of the two elements of wE , we have
X
E∈EI (Th )
([[ε∇(uh −u)]], τ )E =
X
E∈EI (Th )
Z
wE
ε(∆uh −∆u)τ +ε(∇uh −∇u)·∇τ dx.
Using the differential equation and approximating the data, we obtain
X
ηE2 K .
K∈Th
+
E∈EI (Th ) wE
wE
E∈EI (Th ) wE
Z
X
Z
(fh + ε∆uh − ah · ∇uh − buh )τ dx
X
E∈EI (Th )
+
Z
X
(a · ∇(uh − u) + b(uh − u))τ + ε(∇uh − ∇u) · ∇τ dx
(f − fh ) + (ah − a) · ∇uh + (bh − b)uh τ dx.
Integration by parts over wE of the convection term a · ∇(uh − u) yields
X
ηE2 K . T1 + T2 + T3 + T4 + T5 ,
K∈Th
where
T1 =
X
Z
X
Z
E∈EI (Th )
T2 =
E∈EI (Th ) wE
T3 = −
T4 =
T5 =
wE
(fh + ε∆uh − ah · ∇uh − buh )τ dx,
Z
X
E∈EI (Th ) wE
X
Z
X
Z
E∈EI (Th ) E
E∈EI (Th )
(−∇ · a + b)(uh − u)τ + ε(∇uh − ∇u) · ∇τ dx,
a(uh − u) · ∇τ dx,
a · [[uh ]]τ ds,
wE
(f − fh ) + (ah − a) · ∇uh + (bh − b)uh τ dx.
The Cauchy-Schwarz inequality, shape-regularity of the mesh and Lemma 4.11
yield
T1 . ||| u − uh ||| + | u − uh |A + Θ
20
X
E∈EI (Th )
2
ρ−2
E kτ kL2 (wE )
1
2
.
By inequality (4.9) in Lemma 4.10 we obtain,
so that
X
E∈EI (Th )
2
ρ−2
E kτ kL2 (wE )
1
2
.
X
ηE2 K
K∈Th
T1 . ||| u − uh ||| + | u − uh |A + Θ
X
1
2
,
ηE2 K
K∈Th
1
2
.
Using the shape regularity and inequality (4.10) in Lemma 4.10, the term T2
can be bounded by
T2 . ||| u − uh |||
X
E∈EI (Th )
||| τ |||2wE
1
2
. ||| u − uh |||
X
ηE2 K
K∈Th
1
.
1
.
2
For the term T3 , we use the previous estimate and obtain
T 3 . | u − u h |A
X
E∈EI (Th )
||| τ |||2wE
1
2
. | u − u h |A
X
ηE2 K
K∈Th
2
Since the support of ψE intersects the support of at most two other ψE ’s and
since kψE kL∞ (E) = 1, we have
T4 .
X
E∈EI (Th )
1
ε− 2 ρE k[[uh ]]k2L2 (E)
1 2
X
1
E∈EI (Th )
2
ε 2 ρ−1
E kτ kL2 (E)
1
2
.
1
Then, due to ρE ≤ hE ε− 2 , we obtain
T4 .
X
1 X
1
1
hE
k[[uh ]]k2L2 (E) 2
ηE2 K 2 .
ηE2 K 2 . | u − uh |A
ε
K∈Th
K∈Th
E∈EI (Th )
X
Finally, the data error term T5 can be bounded by
T5 . Θ
X
K∈Th
2
ρ−2
K kτ kL2 (K)
This finishes the proof.
1
2
.Θ
X
K∈Th
ηE2 K
1
2
.
2
The proof of Theorem 3.3 follows from the estimates in (4.11), Lemma 4.11,
and Lemma 4.12, respectively.
5
Numerical experiments
In this section, we present a set of numerical examples where we use η in (3.7)
as an error indicator in an adaptive refinement strategy.
21
Our implementation of the discontinuous Galerkin method (2.4) is based on
the deal.II finite element library [5,6]. In all the examples presented below, we
construct adaptively refined sequences of square meshes by marking elements
for refinement or derefinement according to the size of the local indicators ηK ,
with refinement and derefinement fractions set to 25% and 10%, respectively.
We begin all the tests with a uniform square mesh of 16×16 elements. Refinement and derefinement are done so that the resulting meshes are at least
1-irregular. We set the stabilization parameter to γ = 10p2 , where p is the
polynomial degree. This is the standard choice in hp-version discontinuous
Galerkin methods, see, e.g., [18]. In all the examples, the data approximation
error Θ is of higher order and is neglected.
5.1 Example 1
In this example, we take Ω = (0, 1)2 in R2 , and choose a = (1, 1)> and b ≡ 0.
This implies that β = 0. Further, we set u = 0 on Γ = ∂Ω, and select the
right-hand side f so that the analytical solution to (2.1) is given by
u(x, y) =
e
x−1
ε
1
−1
e− ε − 1
+x−1
e
y−1
ε
1
−1
e− ε − 1
+y−1 .
The solution is smooth, but has boundary layers at x = 1 and y = 1; their
widths are both of order O(ε). This problem is well-suited to test whether the
estimator η is able to pick up the steep gradients near these boundaries.
In Figure 5.1, we show the performance of our estimator η for piecewise linear
elements (p = 1) and for ε = 1, ε = 10−2 , and ε = 10−4 . In the upper row
of subfigures, we plot the ”true” energy error ||| u − uh ||| and the value of the
1
estimator η against N 2 , with N denoting the number of degrees of freedom
in each refinement step. These curves are labelled ”ERR” and ”EST”, respectively. The estimator always overestimates the true energy error, in agreement
with Theorem 3.2. Asymptotically, the curves are straight lines, indicating
1
convergence of the optimal order O(N − 2 ). The asymptotic regime is achieved
immediately for the diffusion-dominated problem where ε = 1. For the smaller
values of ε, it is achieved once the boundary layers are sufficiently resolved.
Motivated by Remark 3.5, in the curve ”DERR”, we also calculate the error
1
ε− 2 ku − uh kL2 (Ω) , which is an upper bound for |a(u − uh )|? , cf. equation (3.9).
We can see that this error is of higher order for ε = 1. For the smaller values
of ε, it is initially of the same order as the energy error, but then decreases
faster once the boundary layers are approximated well enough. The same beP
havior is observed for the error E∈E(Th ) (βhE + hE ε−1 ) k[[uh ]]k2L2 (E) , which is
computed in the curve ”JERR”; see also Remark 3.5. In the second row of
subfigures in Figure 5.1 we show the ratio of the estimator and the true energy error. It stays bounded between 6 and 7, uniformly in ε, as predicted by
22
Theorem 3.2 and Theorem 3.3. Based on the above observations, here we have
neglected the error component | u − uh |A . Finally, we note that, for ε small,
the effectivity ratio is substantially smaller in the pre-asymptotic regime.
−1
10
−2
10
0
10
0
10
−3
−1
−5
10
−4
−1
epsilon=10
−2
10
−4
10
epsilon=10
epsilon=1
10
−2
10
10
−2
10
−3
10
−6
10
EST
ERR
DERR
JERR
−7
10
−8
2
10
3
10
N1/2
10
−4
1
2
10
10
3
10
N1/2
10
2
7
7
7
6
6
6
epsilon=10
5
4
10
3
ratio
2
1
10
3
10
1/2
N
4
3
ratio
2
10
5
4
2
1
10
4
10
N1/2
−4
8
−2
8
3
3
10
8
5
EST
ERR
DERR
JERR
−3
10
−5
1
10
epsilon=10
epsilon=1
10
EST
ERR
DERR
JERR
−4
10
2
ratio
2
2
10
3
10
1/2
N
10
3
4
10
1/2
N
10
Fig. 5.1. Example 1: Convergence behavior for ε = 1, 10 −2 , 10−4 and p = 1.
−3
10
1
10
−4
10
0
10
0
10
−5
10
−1
10
−1
epsilon=10−2
epsilon=1
−7
10
−8
10
epsilon=10−4
−6
10
−2
10
−3
10
−3
10
EST
ERR
DERR
JERR
−10
10
−5
4
10
5
N
6
10
10
10
−5
3
10
4
10
5
N
6
10
10
10
4
10
12
12
11
11
11
10
10
10
8
epsilon=10−4
12
9
9
8
7
4
5
N
10
6
10
N
7
6
10
6
3
10
8
10
10
9
8
ratio
7
10
5
10
ratio
ratio
6
3
10
EST
ERR
DERR
JERR
−4
10
−6
3
10
epsilon=10−2
epsilon=1
EST
ERR
DERR
JERR
10
−11
10
−2
10
10
−4
−9
10
10
7
4
10
5
N
10
6
10
6
4
10
5
10
6
10
N
7
10
8
10
Fig. 5.2. Example 1: Convergence behavior for ε = 1, 10 −2 , 10−4 and p = 2.
23
In Figure 5.2, we show the same plots for piecewise quadratic elements (p = 2).
Qualitatively, we observe the same behavior as before, but obtain convergence
of the order O(N −1 ) in the asymptotic regime. The ratio of the estimator and
the true energy error is independent of ε, and in the range of 11.
In Figure 5.3, we show the adaptive meshes after 7 refinement steps, both
for p = 1 and p = 2 and the same values of ε. Obviously, we observe strong
mesh refinement near the lines y = 1 and x = 1, indicating that the estimator
correctly captures boundary layers and is able to resolve them in convectiondominated regimes. Recall that, in the case where ε = 1, the problem is
diffusion-dominated and no boundary layers are present.
(a) ε = 1, p = 1
(b) ε = 10−2 , p = 1
(c) ε = 10−4 , p = 1
(d) ε = 1, p = 2
(e) ε = 10−2 , p = 2
(f) ε = 10−4 , p = 2
Fig. 5.3. Example 1: Adaptively generated meshes after 7 refinement steps.
5.2 Example 2
Let us next consider an example with an internal layer and with variable
coefficients. In the domain Ω = (−1, 1)2 , we set a(x, y) = (−x, y)> and b ≡ 0.
We choose f and the inhomogeneous Dirichlet boundary conditions such that
the solution to (2.1) is given by
x
u(x, y) = erf( √ )(1 − y 2 ).
2ε
R
2
Here, erf(x) is the error function defined by erf(x) = √2π 0x e−t dt. For small
values of ε, the solution u has an internal layer around x = 0, whose width is
24
0
0
10
10
EST
ERR
DERR
JERR
−1
10
8
EST
ERR
DERR
JERR
−1
10
7
−3
10
−2
10
ratio for p=1
epsilon=10−3, p=1
epsilon=10−2, p=1
6
−2
10
−3
10
5
ε=10−2
4
−3
ε=10
3
−4
10
−4
−5
10
10
2
−5
10
2
10
3
10
1/2
2
1
3
10
1/2
N
10
−1
10
EST
ERR
DERR
JERR
−2
10
11.5
11
−3
10
−4
10
−5
10
10.5
ratio for p=2
epsilon=10−3, p=2
epsilon=10−2, p=2
−3
−4
10
−5
10
10
4
10
5
10
N
6
10
7
10
10
ε=10−3
7.5
−8
3
−2
ε=10
9
8
−7
−7
10
9.5
8.5
−6
10
−6
10
10
12
EST
ERR
DERR
JERR
−2
10
10
10
3
1/2
N
−1
10
10
2
10
N
4
10
5
10
6
N
10
7
10
7
3
10
4
10
5
10
N
6
10
7
10
Fig. 5.4. Example 2: Convergence behavior for ε = 10 −2 , 10−3 and p = 1, 2.
√
of order O( ε). We use standard DG terms to incorporate the inhomogeneous
boundary conditions, and modify the error indicator η correspondingly. For
details we refer the reader to [18].
In Figure 5.4, the numerical results for this example are depicted for the values
ε = 10−2 and ε = 10−3 , and for both linear and quadratic approximations.
In the asymptotic regime, we observe linear and quadratic convergence rates
p
of the optimal order O(N − 2 ) for both the energy error and the estimator.
The additional curves ”DERR” and ”JERR” show the same quantities as in
Example 1 associated with the error | u − uh |A . They are not dominant and
clearly smaller than the energy error. In Figure 5.4 we further show the ratio
of the estimator and the energy error, resulting in values that are bounded
independently of ε. Figure 5.5 shows the adaptive meshes after 7 refinement
steps, both for p = 1 and p = 2. We clearly see strong mesh refinement
along x = 0, indicating that the estimator η is effective in locating the internal
layer there.
5.3 Example 3
Next, we test the estimator for a problem with convection that is not aligned
with the mesh. We take Ω = (−1, 1)2 , a = (− sin π6 , cos π6 )> and b = f = 0,
25
(a) ε = 10−2 , p = 1
(b) ε = 10−3 , p = 1
(c) ε = 10−2 , p = 2
(d) ε = 10−3 , p = 2
Fig. 5.5. Example 2: Adaptively generated meshes after 7 refinement steps.
and consider the boundary conditions
u=0
on x = −1 and y = 1,
1−y
)
ε
1
x
u=
tanh( ) + 1
2
ε
u = tanh(
on x = 1,
on y = −1.
The boundary condition is almost discontinuous √near the point √
(0, −1) and
causes u to have an internal layer of width O( ε) along y + 3x = −1,
with values u = 0 to the left and u = 1 to the right, as well as a boundary
layer along the outflow boundary. There is no exact solution available to this
problem. Again, we incorporate the inhomogeneous boundary conditions as
described in [18].
In Figure 5.6, we plot the values of η for ε = 10−2 , 10−4 and p = 1 and p = 2
p
against N − 2 , respectively. We also indicate the minimum mesh size achieved
through the adaptive refinement. We observe that, when the minimum mesh
size is of order O(ε), i.e., the local mesh Péclet number is of order one, the
p
error estimator converges with the optimal rate O(N − 2 ). Figure 5.7 depicts
the adaptive meshes after 7 refinement steps. The layers are resolved, with
mesh refinement being more pronounced for ε = 10−4 .
26
2
2
10
10
−2
−2
ε=10
ε=10
−4
ε=10−4
ε=10
1
10
1
h
10
h
h
hmin=8.63e−05
hmin=8.63e−05
=2.21e−02
p=2
10
=1.73e−04
min
0
10
min
0
=3.45e−04
min
hmin=1.73e−04
p=1
h
=3.45e−04
min
−1
10
h
hmin=1.10e−2
−2
10
hmin= 5.52e−03
=2.21e−02
min
hmin=1.10e−02
−1
h
10
=5.52e−3
min
−3
10
−2
10
−4
1
2
10
10
3
1/2
10
10
4
10
3
10
N
4
10
5
10
6
N
10
7
10
8
10
Fig. 5.6. Example 3: Convergence behavior with ε = 10 −2 , 10−4 and p = 1, 2.
(a) ε = 10−2 , p = 1
(b) ε = 10−4 , p = 1
(c) ε = 10−2 , p = 2
(d) ε = 10−4 , p = 2
Fig. 5.7. Example 3: The adaptively generated meshes after 7 refinement steps.
5.4 Example 4
This example is known as the double-glazing problem. We take Ω = (−1, 1)2 ,
a = (2y(1 − x2 ), −2x(1 − y 2 ))> and b = f = 0. The boundary conditions are:
u = tanh(
u=0
1−y
)
ε
on x = 1,
on x = −1, y = ±1.
27
Note that ∇ · a = 0, so that the conditions (2.2) and (2.3) are satisfied. Again,
the almost discontinuous boundary conditions lead to boundary layers near
the corners.
Figure 5.8 shows the numerical results for this example. Asymptotically, the
optimal convergence orders are attained for the estimator, once the smallest
mesh size is of order O(ε) and the boundary layers are resolved. The adaptive
meshes are shown in Figure 5.9. Strong mesh refinement near the boundary is
observed.
1
10
ε=10−2
1
10
ε=10−2
ε=10−3
ε=10−3
0
10
0
10
hmin=2.76e−3
−1
10
hmin=1.38e−3
−2
10
hmin=6.91e−4
hmin=2.21e−2
−1
10
h
hmin=2.21e−2
p=2
p=1
hmin=2.76e−3
=1.38e−3
min
hmin=6.91e−4
hmin=1.10e−2
hmin=1.10e−2
hmin=5.52e−3
hmin=5.52e−3
−3
10
−2
10
−4
1
10
2
10
3
N1/2
10
4
10
10
3
10
4
10
5
10
N
6
10
7
10
Fig. 5.8. Example 4: Convergence behavior for ε = 10 −2 , 10−3 and p = 1, 2.
(a) ε = 10−2 , p = 1
(b) ε = 10−3 , p = 1
(c) ε = 10−2 , p = 2
(d) ε = 10−3 , p = 2
Fig. 5.9. Example 4: Adaptively generated meshes after 7 refinement.
28
6
Conclusions
In this paper, we have derived a robust a-posteriori error estimator for a
convection-diffusion equation. The estimator yields upper and lower bounds
for the error measured in terms of the energy norm and a semi-norm associated with the convective term of the equation. The ratio of these bounds is
independent of the Péclet number of the problem; in this sense the estimator is robust in convection-dominated regimes. Our numerical results indicate
that the estimator is effective in locating and resolving interior and boundary
layers. Once the local mesh Péclet number is of order one, the energy error
converges with optimal order, and is dominating the error | u − uh |A related to
convection. In fact, we observe numerically that the error indicator is robust,
reliable and efficient for estimating the error in the energy norm. In all our
examples the effectivity ratio of the estimator and the energy error is around 6
for p = 1, and increases to around 11 for p = 2.
References
[1] M. Ainsworth, J. Oden, A-posteriori Error Estimation in Finite Element
Analysis, Wiley-Interscience Series in Pure and Applied Mathematics, Wiley,
2000.
[2] L. E. Alaoui, A. Ern, E. Burman, A-priori and a-posteriori analysis of nonconforming finite elements with face penalty for advection-diffusion equations,
IMA J. Numer. Anal. 27 (2007) 151–171.
[3] D. Arnold, An interior penalty finite element method with discontinuous
elements, SIAM J. Numer. Anal. 19 (1980) 742–760.
[4] D. Arnold, F. Brezzi, B. Cockburn, L. Marini, Unified analysis of discontinuous
Galerkin methods for elliptic problems, SIAM J. Numer. Anal. 39 (2002) 1749–
1779.
[5] W. Bangerth, R. Hartmann, G. Kanschat, deal.II Differential Equations
Analysis Library, Technical Reference, 5th ed., URL: http://www.dealii.org
(2005).
[6] W. Bangerth, R. Hartmann, G. Kanschat, deal.II — a general purpose object
oriented finite element library, ACM Trans. Math. Software 33.
[7] R. Becker, P. Hansbo, M. Larson, Energy norm a-posteriori error estimation
for discontinuous Galerkin methods, Comput. Methods Appl. Mech. Engrg.
192 (2003) 723–733.
[8] R. Becker, P. Hansbo, R. Stenberg, A finite element method for domain
decomposition with non-matching grids, Modél. Math. Anal. Numér. 37 (2003)
209–225.
29
[9] R. Bustinza, G. Gatica, B. Cockburn, An a-posteriori error estimate for the
local discontinuous Galerkin method applied to linear and nonlinear diffusion
problems, J. Sci. Comput. 22 (2005) 147–185.
[10] B. Cockburn, Discontinuous Galerkin Methods for Convection-Dominated
Problems, in: T. Barth, H. Deconink (eds.), High-Order Methods for
Computational Physics, vol. 9, Springer, 1999, pp. 69–224.
[11] B. Cockburn, G. Karniadakis, C.-W. Shu (eds.), Discontinuous Galerkin
Methods. Theory, Computation and Applications, vol. 11 of Lect. Notes
Comput. Sci. Engrg., Springer, 2000.
[12] B. Cockburn, C.-W. Shu, Runge–Kutta discontinuous Galerkin methods for
convection-dominated problems, J. Sci. Comput. 16 (2001) 173–261.
[13] V. Girault, P.-A. Raviart, Finite Element Methods for Navier-Stokes
Equations: Theory and Algorithms, vol. 5 of Springer Series in Computational
Mathematics, Springer, 1986.
[14] K. Harriman, P. Houston, B. Senior, E. Süli, hp–Version discontinuous Galerkin
methods with interior penalty for partial differential equations with nonnegative
characteristic form, in: C.-W. Shu, T. Tang, S.-Y. Cheng (eds.), Recent
Advances in Scientific Computing and Partial Differential Equations, vol. 330
of Contemporary Mathematics, AMS, 2003.
[15] P. Houston, D. Schötzau, T. Wihler, hp-Adaptive discontinuous Galerkin finite
element methods for the Stokes problem, in: P. Neittaanmäki, T. Rossi,
S. Korotov, E. Oñate, J. Périaux, D. Knörzer (eds.), Proceedings of the
European Congress on Computational Methods in Applied Sciences and
Engineering, Volume II, 2004.
[16] P. Houston, D. Schötzau, T. Wihler, Energy norm a-posteriori error estimation
for mixed discontinuous Galerkin approximations of the Stokes problem, J. Sci.
Comput. 22 (2005) 357–380.
[17] P. Houston, D. Schötzau, T. Wihler, An hp-adaptive discontinuous Galerkin
FEM for nearly incompressible linear elasticity, Comput. Methods Appl. Mech.
Engrg. 195 (2006) 3224–3246.
[18] P. Houston, D. Schötzau, T. P. Wihler, Energy norm a-posteriori error
estimation of hp-adaptive discontinuous Galerkin methods for elliptic problems,
Math. Models Methods Appl. Sci. 17 (2007) 33–62.
[19] C. Johnson, J. Pitkäranta, An analysis of the discontinuous Galerkin method
for a scalar hyperbolic equation, Math. Comp. 46 (1986) 1–26.
[20] G. Kanschat, R. Rannacher, Local error analysis of the interior penalty
discontinuous Galerkin method for second order elliptic problems, J. Numer.
Math. 10 (2002) 249–274.
[21] O. A. Karakashian, F. Pascal, A-posteriori error estimates for a discontinuous
Galerkin approximation of second-order elliptic problems, SIAM J. Numer.
Anal. 41 (2003) 2374–2399.
30
[22] D. Kay, D. Silvester, The reliability of local error estimators for convectiondiffusion equations, IMA J. Numer. Anal. 21 (2001) 107–122.
[23] P. Lesaint, P. A. Raviart, On a finite element method for solving the neutron
transport equation, in: C. de Boor (ed.), Mathematical Aspects of Finite
Elements in Partial Differential Equations, Academic Press, 1974, pp. 89–145.
[24] J. Nitsche, Über ein Variationsprinzip zur Lösung von Dirichlet Problemen bei
Verwendung von Teilräumen, die keinen Randbedingungen unterworfen sind,
Math. Abh. Sem. Univ. Hamburg 36 (1971) 9–15.
[25] W. Reed, T. Hill, Triangular mesh methods for the neutron transport equation,
Report LA-UR-73-479, Los Alamos Scientific Laboratory (1973).
[26] B. Rivière, M. Wheeler, A-posteriori error estimates for a discontinuous
Galerkin method applied to elliptic problems, Comput. Math. Appl. 46 (2003)
141–163.
[27] G. Sangalli, A uniform analysis of nonsymmetric and coercive linear operators,
SIAM J. Math. Anal. 36 (2005) 2033–2048.
[28] G. Sangalli, Robust a-posteriori estimator for advection-diffusion-reaction
problems, Math. Comp. 77 (2008) 41–70.
[29] R. Verfürth, A Review of A-Posteriori Error Estimation and Adaptive MeshRefinement Techniques, Teubner, 1996.
[30] R. Verfürth, A-posteriori error estimators for convection-diffusion equations,
Numer. Math. 80 (1998) 641–643.
[31] R. Verfürth, Robust a-posteriori error estimates for stationary convectiondiffusion equations, SIAM J. Numer. Anal. 43 (2005) 1766–1782.
31