Analysis of Dual Consistency for Discontinuous Galerkin
Discretizations of Source Terms
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Oliver, Todd A., and David L. Darmofal. “Analysis of Dual
Consistency for Discontinuous Galerkin Discretizations of Source
Terms.” SIAM Journal on Numerical Analysis 47.5 (2009): 35073525. ©2009 Society for Industrial and Applied Mathematics
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SIAM J. NUMER. ANAL.
Vol. 47, No. 5, pp. 3507–3525
c 2009 Society for Industrial and Applied Mathematics
ANALYSIS OF DUAL CONSISTENCY FOR DISCONTINUOUS
GALERKIN DISCRETIZATIONS OF SOURCE TERMS∗
TODD A. OLIVER† AND DAVID L. DARMOFAL‡
Abstract. The effects of dual consistency on discontinuous Galerkin discretizations of solution
and solution gradient dependent source terms are examined. Two common discretizations are analyzed: the standard weighting technique for source terms and the mixed formulation. It is shown that
if the source term depends on the first derivative of the solution, the standard weighting technique
leads to a dual inconsistent scheme. A straightforward procedure for correcting this dual inconsistency and arriving at a dual consistent discretization is demonstrated. The mixed formulation,
where the solution gradient in the source term is replaced by an additional variable that is solved for
simultaneously with the state, leads to an asymptotically dual consistent discretization. Numerical
results for a one-dimensional test problem confirm that the dual consistent and asymptotically dual
consistent schemes achieve higher asymptotic convergence rates with grid refinement than a similar
dual inconsistent scheme for both the primal and adjoint solutions as well as a simple functional
output.
Key words. discontinuous Galerkin, dual consistency, high-order
AMS subject classifications. 65N30, 65N15
DOI. 10.1137/080721467
1. Introduction. In recent years, the discontinuous Galerkin (DG) finite element method has become a popular tool in the numerical simulation of many complex physical phenomena. In particular, many researchers have investigated highorder accurate DG discretizations of the Euler and Navier–Stokes equations for use
in computational fluid dynamics [22, 11, 5, 7, 6, 12, 4]. In this context, DG is attractive because it allows the development of high-order accurate discretizations with
elementwise compact stencils. These compact stencils simplify the task of achieving
high-order accuracy for problems involving complex geometries, where unstructured
meshes are often employed, and allow the development of efficient solution methods.
In this paper, high-order accurate DG discretizations of source terms depending
on the state and its gradient are examined. Interest in such terms stems from the
Reynolds-averaged Navier–Stokes (RANS) equations and, specifically, from the turbulence models used to close the RANS equations. For example, the Spalart–Allmaras
turbulence model [25] incorporates state and state derivative dependent source terms
to model the production, destruction, and diffusion of turbulent eddy viscosity, and
state derivative dependent source terms appear in both the k − and k − ω turbulence
models [28].
The focus of the paper is the impact of dual consistency on source term discretizations. Dual consistency provides a connection between the continuous and discrete
dual problems. In particular, if a discretization is dual consistent, then the exact
solution of the strong form of the dual problem satisfies the discrete dual problem
∗ Received by the editors April 16, 2008; accepted for publication (in revised form) August 7,
2009; published electronically November 5, 2009. This work was supported by the U. S. Air Force
Research Laboratory (USAF-3306-03-SC-0001) and The Boeing Company.
http://www.siam.org/journals/sinum/47-5/72146.html
† Institute for Computational Engineering and Sciences, University of Texas at Austin, Austin,
TX 78712 (oliver@ices.utexas.edu).
‡ Department of Aeronautics and Astronautics, Massachusetts Institute of Technology, Cambridge,
MA 02139 (darmofal@mit.edu).
3507
Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.
3508
TODD A. OLIVER AND DAVID L. DARMOFAL
taken about the exact solution of the strong form of the primal problem. A more
precise definition of dual consistency is given in section 2.
For many types of discretizations, algorithms involving the dual problem have
become popular for design optimization, error estimation, and grid adaptation [21, 1,
14, 16, 8, 9, 17, 27]. It is well known that dual consistency can significantly impact
the performance of these algorithms. For example, Collis and Heinkenschloss [13]
showed that when applying a dual inconsistent streamline upwind/Petrov–Galerkin
method for linear advection-diffusion to an optimal control problem, superior results
are obtained using a direct discretization of the continuous dual problem as opposed
to the discrete dual problem derived from the primal discretization. Specifically, both
the control function and the adjoint solution converge at a higher rate when the
continuous dual problem is discretized directly.
For DG discretizations, Harriman and coworkers [19, 18] examined symmetric
and nonsymmetric interior penalty (SIPG and NIPG, respectively) DG methods for
the solution of Poisson’s equation. They showed that to achieve optimal convergence
rates for a linear functional output, the dual consistent method (i.e., SIPG) must be
used. Lu [23] considered the impact of dual consistency on the accuracy of functional
outputs computed using DG discretizations of the Euler and Navier–Stokes equations.
He demonstrated the importance of implementing the boundary conditions on the primal problem in a dual consistent manner. In particular, when using dual consistent
boundary conditions, superconvergent functional output results were obtained, while
when using a dual inconsistent boundary condition treatment, significant degradation
of the output convergence rates was observed. More recently, Hartmann [20] proposed
a framework for analyzing the dual consistency of DG discretizations. He uses the
framework to expand upon the analysis of the SIPG discretization for the Navier–
Stokes equations, proposing a modification of the boundary conditions to make the
scheme dual consistent. Similar to the results shown by Lu, Hartmann’s modification of the SIPG scheme produces superior results to the original, dual inconsistent
boundary condition treatment.
Furthermore, it is well known that dual consistency can impact the convergence
of the L2 norm of the error in the primal solution [3]. For example, for many DG
discretizations of Poisson’s equation, standard proofs of order of accuracy of the solution error in the L2 norm exist. Typically these proofs rely on the Aubin–Nitsche
“duality trick” [26, 24] to obtain an optimal estimate in the L2 norm given an optimal
estimate in the energy norm [10, 3]. The use of this duality argument requires that
the scheme be dual consistent. Thus, some dual inconsistent methods (e.g., NIPG
and the Baumann–Oden method) do not achieve optimal accuracy in the L2 norm,
and dual inconsistent methods that do achieve optimal accuracy in the L2 norm are
typically superpenalized [3].
The paper begins with a brief review of the definition of dual consistency in
section 2. The standard weighting DG discretization of source terms is considered
in section 3. It is shown that, while this treatment of solution derivative dependent
source terms leads to a dual inconsistent DG discretization, dual consistency can be
achieved by adding terms proportional to the jumps in the solution between elements
to the discretization. Mixed formulations for the source term are analyzed in section 4.
The resulting discretizations are shown to be asymptotically dual consistent. Finally,
numerical results for a simple test problem are shown in section 5.
2. Dual consistency definition and preliminaries. Consider the following
primal problem: compute J (u), where J : V → R is a functional of interest, V is an
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DUAL CONSISTENT DISCRETIZATIONS OF SOURCE TERMS
3509
appropriate function space, and u ∈ V solves
R(u, v) = 0 ∀v ∈ V,
where R : V × V → R is the weak form of a PDE of interest. For simplicity, it is
assumed that this primal problem and its dual are well-posed.
Let Vhp be a finite dimensional vector space of piecewise polynomial functions of
degree at most p on a triangulation Th of the domain of interest Ω ⊂ Rn into elements
κ such that Ω̄ = ∪κ∈Th κ̄. In particular,
Vhp ≡ {v ∈ L2 (Ω) | v|κ ∈ P p ∀κ ∈ Th },
where P p denotes the space of polynomial functions of degree at most p.
Let Whp ≡ Vhp + V, where
Vhp + V ≡ {h = f + g | f ∈ Vhp , g ∈ V}.
Then, consider a general DG discretization of the primal problem: find uh ∈ Vhp
such that
Rh (uh , vh ) = 0
∀vh ∈ Vhp ,
where Rh : Whp × Whp → R is a semilinear form derived from the weak form of the
primal problem. For the remainder of the paper, it is assumed that this discrete
problem is also well-posed.
Let Jh : Whp → R be the discrete functional of interest. Then, the discrete dual
problem is given by the following statement: find ψh ∈ Vhp such that
Rh [uh ](vh , ψh ) = Jh [uh ](vh )
∀vh ∈ Vhp ,
where Rh [uh ](·, ψh ) : Whp → R is the linear functional given by evaluating the Frechét
derivative of the function Nψh : Whp → R at uh , where, for fixed ψh ∈ Whp ,
Nψh (wh ) = Rh (wh , ψh ) ∀wh ∈ Whp .
Similarly, Jh [uh ](·) : Whp → R is the linear functional given by evaluating the Frechét
derivative of Jh at uh .
Two concepts of dual consistency are used in this work: dual consistency, defined in Definition 2.1, and asymptotic dual consistency, defined in Definition 2.2.
Dual consistency has been used in the analysis of DG methods by multiple authors.
For example, [19, 18, 3] define dual consistency for linear problems. A general definition of dual consistency for nonlinear problems is provided by both Lu [23] and
Hartmann [20]. Lu also defines asymptotic dual consistency for general nonlinear
problems. The definitions used here follow [23].
Definition 2.1. The discretization defined by the semilinear form Rh and discrete functional Jh is said to be dual consistent if, given exact solutions u ∈ V and
ψ ∈ V of the continuous primal and dual problems, respectively,
Rh [u](v, ψ) = Jh [u](v)
∀v ∈ Whp .
Definition 2.2. The discretization defined by the semilinear form Rh and discrete functional Jh is said to be asymptotically dual consistent if, given exact solutions
u ∈ V and ψ ∈ V of the continuous primal and dual problems, respectively,
⎛
⎞
lim ⎝
h→0
sup
p
v∈Wh
,vW p =1
h
|Rh [u](v, ψ) − Jh [u](v)|⎠ = 0.
Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.
3510
TODD A. OLIVER AND DAVID L. DARMOFAL
Clearly, all dual consistent discretizations are automatically asymptotically dual
consistent. In this work, a discretization will be referred to as asymptotically dual
consistent only if it is not also dual consistent.
3. The standard weighting technique for source terms. This section considers DG discretizations of source terms depending on the state and first derivatives
of the state. As will be shown, the simple approach of weighting by a test function
and integrating leads to a dual inconsistent scheme for source terms that depend on
derivatives of the state. However, a dual consistent discretization can be constructed
by adding terms to the discretization.
Let u ∈ V ≡ H 1 (Ω) be the solution of the following scalar problem:
(3.1)
−∇ · (ν∇u) = f (u, ∇u) for x ∈ Ω ⊂ Rn ,
u = 0 for x ∈ ∂Ω,
where f ∈ C 1 (Rn+1 ) is the source term of interest. As before, it is assumed that this
problem and its dual are well-posed.
Let J : H 1 (Ω) → R be a functional of interest defined as
J (w) =
(3.2)
J(w),
Ω
where J ∈ C(R). For simplicity, the functional chosen here does not include boundary integrals. The inclusion of boundary integrals would slightly alter the boundary
conditions on the dual problem and the analysis of the boundary terms shown here.
However, these modifications are not central to the analysis of the source term discretization. For an analysis of dual consistency for equations without source terms
that includes functionals with boundary integrals, see [23].
For J , the dual problem is given by
(3.3) −∇ · (ν∇ψ) − D1 f (u, ∇u)ψ + ∇ · (D∇u f (u, ∇u)ψ) = J [u] for x ∈ Ω,
where ψ is the adjoint state, D1 f (u, ∇u) is the partial derivative of f with respect to
u evaluated at (u, ∇u), and
T
D∇u f (u, ∇u) = [D2 f (u, ∇u), . . . , Dn+1 f (u, ∇u)] ,
for 2 ≤ i ≤ n + 1,
where Di f (u, ∇u) is the partial derivative of f with respect to ∂x∂u
i−1
evaluated at (u, ∇u). The boundary conditions on the dual problem can be written
in the following weak form:
−
ψq = 0 ∀q ∈ H −1/2 (∂Ω).
∂Ω
Consider the following DG discretization: find uh ∈ Vhp such that
Rh (uh , vh ) ≡ Bh (uh , vh ) −
(3.4)
vh f (uh , ∇uh ) = 0 ∀vh ∈ Vhp ,
κ∈Th
κ
where Bh is a consistent and dual consistent bilinear form for the diffusion operator
(e.g., the second method of Bassi and Rebay (BR2) [6, 4] or local discontinuous
Galerkin method [11, 12]). Furthermore, define the discrete functional of interest
Jh : Whp → R as
(3.5)
J(wh ).
Jh (wh ) =
κ∈Th
κ
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DUAL CONSISTENT DISCRETIZATIONS OF SOURCE TERMS
3511
In the following dual consistency analysis, additional smoothness is assumed for
the exact primal and dual solutions u and ψ, respectively. In particular, it is assumed
that u, ψ ∈ H 2 (Ω). This smoothness assumption is common in the analysis of dual
consistency for discretizations of second-order operators [3, 23].
Proposition 3.1. The discretization defined by the semilinear form Rh , defined
in (3.4), together with the discrete functional Jh , defined in (3.5), is dual inconsistent.
Proof. Linearizing Rh about the exact solution and integrating by parts gives
wh (D1 f (u, ∇u)vh − ∇ · (D∇u f (u, ∇u)vh ))
Rh [u](wh , vh ) = Bh (wh , vh ) −
κ∈Th
−
Γi
κ
(wh · {D∇u f (u, ∇u)vh } + {wh } D∇u f (u, ∇u)vh )
−
∂Ω
wh (D∇u f (u, ∇u)vh ) · n,
where · and {·} denote the standard jump and average operators, respectively, on
interior faces (see, e.g., [3]), n is the outward pointing unit normal vector, and Γi
denotes the union of the interior faces of the triangulation Th .
The assumptions ψ ∈ H 2 (Ω), u ∈ H 2 (Ω) and f ∈ C 1 (Rn+1 ) imply that
{D∇u f (u, ∇u)ψ} = D∇u f (u, ∇u)ψ and D∇u f (u, ∇u)ψ = 0. Thus,
Rh [u](vh , ψ) = Bh (vh , ψ) −
vh (D1 f (u, ∇u)ψ − ∇ · (D∇u f (u, ∇u)ψ))
−
Γi
κ∈Th
κ
vh · (D∇u f (u, ∇u)ψ) −
∂Ω
vh (D∇u f (u, ∇u)ψ) · n.
Evaluating the dual consistency using the discrete functional as defined in (3.5) gives
Rh [u](vh , ψ) − Jh [u](vh ) = (Lh,I (u, ψ))(vh ) + (Lh,B (u, ψ))(vh ),
where
(3.6)
(Lh,I (u, ψ))(vh ) ≡ −
Γi
vh · (D∇u f (u, ∇u)ψ),
(3.7)
(Lh,B (u, ψ))(vh ) ≡ −
∂Ω
vh (D∇u f (u, ∇u)ψ) · n.
In general, there exists vh ∈ Vhp such that at least (Lh,I (u, ψ))(vh ) does not vanish.
Thus, the scheme is dual inconsistent. Due to the boundary condition on the dual
problem, the boundary term (Lh,B (u, ψ))(vh ) will vanish if (D∇u f (u, ∇u)vh · n)|∂Ω ∈
H −1/2 (∂Ω). However, if this condition does not hold, the boundary term will also
contribute to the dual inconsistency.
Remark 3.1. It is possible to construct a dual consistent discretization by adding
terms to the semilinear form Rh . In particular, define a new bilinear form
Rh,DC (wh , vh ) ≡ Rh (wh , vh ) + Ah,I (wh , vh ) + Ah,B (wh , vh ),
where Ah,I will serve to eliminate the interior face dual inconsistency term LI , and
Ah,B will serve to eliminate the boundary face dual inconsistency LB . Furthermore,
to maintain consistency, Ah,I and Ah,B must vanish when evaluated at u:
Ah,I (u, vh ) = 0 ∀vh ∈ Vhp ,
Ah,B (u, vh ) = 0 ∀vh ∈ Vhp .
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3512
TODD A. OLIVER AND DAVID L. DARMOFAL
The interior face and boundary face contributions to the dual inconsistency are
examined separately. To eliminate the dual inconsistency from the interior faces, the
following term is added to the semilinear form Rh :
Ah,I (wh , vh ) =
wh · {βi (wh , vh )},
Γi
where dual consistency requires that {βi (u, ψ)} = D∇u f (u, ∇u)ψ.
To eliminate the boundary dual inconsistency, the following term is added to Rh :
Ah,B (wh , vh ) =
wh βb (wh , vh ) · n,
∂Ω
where dual consistency requires βb (u, ψ) = D∇u f (u, ∇u)ψ.
Proposition 3.2. Let Bh be a dual consistent bilinear form corresponding to the
diffusion operator. Let βi be such that βi (w, v) = D∇u f (w, ∇w)v for all w, v ∈ H 2 (Ω).
Let βb be such that βb (w, v) = D∇u f (w, ∇w)v for all w, v ∈ H 2 (Ω). Then, the
semilinear form given by
Rh,DC (wh , vh ) = Bh (wh , vh ) −
vh f (wh , ∇wh )
+
Γi
κ
κ∈Th
wh · {βi (wh , vh )} +
∂Ω
wh βb (wh , vh ) · n,
together with the discrete functional Jh , defined in (3.5), is dual consistent.
Proof. Linearizing Rh,DC gives
vh · (D∇u f (u, ∇u)ψ)
Rh,DC [u](vh , ψ) = Rh [u](vh , ψ) +
Γi
+
vh (D∇u f (u, ∇u)ψ) · n ∀vh ∈ Whp .
∂Ω
Thus,
Rh,DC
[u](vh , ψ) − Jh [u](vh ) = 0
∀vh ∈ Whp .
Remark 3.2. The choices of βi and βb are not fully determined by requiring
dual consistency. One valid choice is given by
βi (wh , vh ) = {D∇u f (wh , ∇wh )vh };
βb (wh , vh ) = D∇u f (wh , ∇wh )vh .
Then,
(3.8)
Rh,DC (wh , vh ) = Bh (wh , vh ) −
+
Γi
κ∈Th
wh · {D∇u f (wh , ∇wh )vh } +
κ
vh f
∂Ω
wh (D∇u f (wh , ∇wh )vh ) · n.
However, if necessary, one could construct more complex functions that satisfy the
dual consistency requirement as well as add stability to the discretization. For example, given that the dual problem is a convection-diffusion-reaction problem, one
option to add stability is to upwind the dual consistency terms instead of averaging.
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DUAL CONSISTENT DISCRETIZATIONS OF SOURCE TERMS
3513
Remark 3.3. In addition to being dual consistent, if Bh is a consistent bilinear
form for the diffusion operator, the discretization of Proposition 3.2 is consistent for
any choice of βi and βb because, for the exact solution u, u = 0 and u|∂Ω = 0. Thus,
Rh,DC (u, vh ) = Bh (u, vh ) −
vh f (u, ∇u) = 0 ∀vh ∈ Vhp .
κ
κ∈Th
4. The mixed formulation for source terms. In addition to the standard
weighting source term treatment discussed in section 3, another source term treatment
of interest has appeared in the DG literature. In this method, known as the mixed
formulation, the gradient of the state is replaced by a variable that is solved for
simultaneously with the primal state [4]. Variants of this technique are widely used
in DG discretizations of second-order operators. See [3] for a full analysis of those
discretizations. This section provides a brief derivation of the mixed method as applied
to source terms involving the gradient of the state. Furthermore, analysis of the mixed
formulation shows that, in general, it is asymptotically dual consistent.
4.1. Discretization derivation. Consider (3.1) (i.e., the model problem considered in section 3), and consider the following discretization: find uh ∈ Vhp and
n
gh ∈ [Vhp ] such that
(4.1)
vh f (uh , gh ) = 0 ∀vh ∈ Vhp ,
Rh (uh , vh ) ≡ Bh (uh , vh ) −
(4.2)
κ∈Th
κ
τh · gh = −
κ∈Th
κ∈Th
+
∂Ω
κ
κ
uh ∇ · τh +
ub (uh )τh · n
Γi
(û(uh ) · {τh } + {û(uh )} τh )
n
∀τh ∈ [Vhp ] ,
b
where û and u are numerical flux functions on interior and boundary faces, respectively. Integrating by parts on (4.2) gives
(4.3)
τh · gh =
τh · ∇uh +
û(uh ) − uh · {τh }
κ
κ∈Th
κ
κ∈Th
+
Γi
Γi
{û(uh ) − uh } τh +
∂Ω
(ub (uh ) − uh )τh · n
n
∀τh ∈ [Vhp ] .
h (see, e.g., [3]) by the following problems: find
Define the lifting operators rh and p n
p n
rh (uh ) ∈ [Vh ] and h (uh ) ∈ [Vh ] such that
(4.4)
τh · rh (uh ) = −
û(uh ) − uh · {τh }
κ∈Th
κ
Γi
−
(4.5)
κ∈Th
κ
∂Ω
τh · h (uh ) = −
Γi
(ub (uh ) − uh )τh · n
{û(uh ) − uh } τh n
∀τh ∈ [Vhp ] ,
n
∀τh ∈ [Vhp ] .
Then, using (4.3) gives
(4.6)
gh = ∇uh − rh (uh ) − h (uh ).
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3514
TODD A. OLIVER AND DAVID L. DARMOFAL
Substituting (4.6) into (4.1) gives the following discretization: find uh ∈ Vhp such
that
(4.7) Bh (uh , vh ) −
vh f (uh , ∇uh − rh (uh ) − h (uh )) = 0 ∀vh ∈ Vhp ,
κ∈Th
κ
where rh and h are defined in (4.4) and (4.5), respectively. To complete the scheme,
one must define the numerical flux functions û and ub . For the remainder of the
paper, it is assumed that these fluxes have the following properties:
1. for each interior edge e, there exists a constant vector de such that û(wh ) =
{wh } + de · wh ;
2. ub = 0.
Remark 4.1. While the assumptions on the numerical fluxes may seem restrictive, to the best of the authors’ knowledge, all existing mixed method formulations
that are consistent use fluxes satisfying these assumptions for problems with homogeneous Dirichlet boundary conditions [3].
4.2. Dual consistency analysis. In this section, the dual consistency of the
mixed formulation defined in section 4.1 is considered. The dual consistency of this
discretization is analyzed in two parts. First, a restrictive condition is assumed which
implies that the scheme is dual consistent. Then, the proof is extended to the general
case, where only asymptotic dual consistency can be shown.
Proposition 4.1. Let D∇u f (u, ∇u)ψ ∈ [Vhp ]n . Then, the DG formulation defined in (4.7) together with the discrete functional Jh defined in (3.5) forms a dual
consistent discretization.
Proof. Noting that the lifting operators rh and h are linear functionals and that
rh (u) = h (u) = 0, linearizing Rh about the exact solution gives
vh D1 f (u, ∇u)wh
Rh [u](wh , vh ) = Bh (wh , vh ) −
−
κ∈Th
κ
κ∈Th
vh D∇u f (u, ∇u) · (∇wh − rh (wh ) − h (wh )).
Thus, integrating by parts gives
(4.8)
Rh [u](wh , vh )
+
κ∈Th
−
κ
κ
= Bh (wh , vh ) −
κ
κ∈Th
wh D1 f (u, ∇u)vh
wh ∇ · (D∇u f (u, ∇u)vh ) −
∂Ω
wh (D∇u f (u, ∇u)vh ) · n
(wh · {D∇u f (u, ∇u)vh } + {wh } D∇u f (u, ∇u)vh )
+
vh D∇u f (u, ∇u) · (rh (wh ) + h (wh )).
Γi
κ∈Th
κ
Using the assumptions on û and ub combined with the hypothesis that
D∇u f (u, ∇u)ψ ∈ [Vhp ]n ,
(4.9)
(D∇u f (u, ∇u)ψ) · rh (wh ) =
wh · {D∇u f (u, ∇u)ψ}
κ∈Th
κ
Γi
+
∂Ω
wh (D∇u f (u, ∇u)ψ) · n,
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DUAL CONSISTENT DISCRETIZATIONS OF SOURCE TERMS
(4.10)
κ∈Th
κ
3515
(D∇u f (u, ∇u)ψ) · h (wh ) = −
Γi
(de · wh ) D∇u f (u, ∇u)ψ .
Substituting (4.9) and (4.10) into (4.8) gives
wh (D1 f (u, ∇u)ψ − ∇ · (D∇u f (u, ∇u)ψ))
Rh [u](wh , ψ) = Bh (wh , ψ) −
−
κ
κ∈Th
Γi
(û(wh ) D∇u f (u, ∇u)ψ) .
Furthermore, by the assumptions on the smoothness of u, ψ, and f ,
D∇u f (u, ∇u)ψ = 0. Thus,
vh (D1 f (u, ∇u)ψ − ∇ · (D∇u f (u, ∇u)ψ)) .
Rh [u](vh , ψ) = Bh (vh , ψ) −
κ∈Th
κ
Finally,
Rh [u](vh , ψ) − Jh [u](vh ) = 0 ∀vh ∈ Whp ,
which implies that the scheme is dual consistent.
Of course, in general, the assumption of D∇u f (u, ∇u)ψ ∈ [Vhp ]n is not realistic.
For D∇u f (u, ∇u)ψ ∈ [Vhp ]n , Proposition 4.1 does not hold. In this case, the mixed
formulation is only asymptotically dual consistent.
Proposition 4.2. If D∇u f (u, ∇u)ψ ∈ [H k+1 (Ω)]n , where 1 ≤ k ≤ p, then the
DG discretization defined in (4.7) together with the functional Jh defined in (3.5)
forms an asymptotically dual consistent discretization.
To simplify the proof, some preliminary notation and lemmas are required. In
particular, let Eh denote the set of all faces in the triangulation Th . Define the jump
operator · on boundary faces by s = sn for scalar quantities and v = v · n for
vector quantities. Define the average operator {·} on boundary faces by {v } = v .
For the following lemmas, it is assumed that Whp = (Vhp + H 2 (Ω)) ∩ H01 (Ω).
Furthermore, it is assumed that the set of triangulations [Th ]h>0 is quasi-uniform
(see [15, 24] for definition).
Lemma 4.3. There exists a norm ||| · |||∗ : Whp → R and a constant c such that
h−1/2
v 0,e ≤
h−1/2
v 0,e ≤ c|||v|||∗ ∀v ∈ Whp ,
κe
e∈Eh
e∈Eh
where κe is such that e ⊂ ∂κe , h = maxκ∈Th hκ , and hκ = supx,y∈κ |x − y|.
Proof. An example of such a norm is used by Arnold et al. [3]. In particular,
|v|21,κ + h2κ |v|22,κ +
|||v|||2∗ ≡
re (v)20,Ω ,
κ∈Th
where
Ω
e∈Eh
re (v) · τ = −
e
v · {τ } ∀τ ∈ [Vhp ]n .
For the proof of the lemma for this norm, see [3, section 4.1] or [10, Lemma 2].
Note that only the existence of such a norm, not its particular form, is important
here.
Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.
3516
TODD A. OLIVER AND DAVID L. DARMOFAL
Lemma 4.4. For a face e ∈ Eh such that e ⊂ ∂κ, there exists a constant c such
that, for all v ∈ H 1 (κ) and w ∈ L2 (e),
|vw| ≤ ch−1/2
(v0,κ + hκ |v|1,κ )w0,e .
κ
e
Proof. Apply the Cauchy–Schwarz inequality, and then use
v0,e ≤ ch−1/2
(v0,κ + hκ |v|1,κ ) ∀v ∈ H 1 (κ),
κ
which is a standard trace theorem [2].
Lemma 4.5. For all w ∈ H k+1 (Ω), where 1 ≤ k ≤ p, there exists a constant c
such that
1/2
p
2
w − Πh (w)1,κ
≤ chk |w|k+1,Ω ,
κ∈Th
where Πph : L2 (Ω) → Vhp is the L2 (Ω)-orthogonal projection onto Vhp .
Proof. If Πpκ : L2 (κ) → P p is the L2 (κ)-orthogonal projection onto P p , then
Πph (v)|κ = Πpκ (v|κ ) ∀v ∈ L2 (Ω).
To complete the proof, apply Proposition 1.134(iii) from [15] to each element κ and
sum over the elements.
Proof of Proposition 4.2. Define π ∈ [Vhp ]n by
πj = Πph ((D∇u f (u, ∇u)ψ)j ) for j = 1, . . . , n.
Furthermore, define ∈ [L2 (Ω)]n by
j = (D∇u f (u, ∇u)ψ)j − πj
for j = 1, . . . , n.
By assumption, D∇u f (u, ∇u)ψ ∈ [H 1 (Ω)]n . Thus, |κ ∈ [H 1 (κ)]n for all κ ∈ Th .
From the proof of Proposition 4.1, it is clear that
Eh (vh ) ≡ Rh [u](vh , ψ) − Jh [u](vh ) =
(D∇u f (u, ∇u)ψ) · rh (vh ) + h (vh )
κ
κ∈Th
−
where Γ ≡ Γi ∪ ∂Ω. Thus,
(4.11)
Eh (vh ) =
κ∈Th
−
Γ
κ
Γ
vh · (D∇u f (u, ∇u)ψ) ∀vh ∈ Whp ,
(π + ) · rh (vh ) + h (vh )
vh · (π + ) ∀vh ∈ Whp .
From (4.4), (4.5), and the assumptions on û and ub ,
(4.12)
π · rh (vh ) =
vh · {π },
κ∈Th
(4.13)
κ
κ∈Th
κ
Γ
π · h (vh ) = −
Γi
(de · vh ) π .
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DUAL CONSISTENT DISCRETIZATIONS OF SOURCE TERMS
3517
Substituting (4.12) and (4.13) into (4.11) gives
Eh (vh ) =
· (rh (vh ) + h (vh )) − vh · {} −
(de · vh ) π .
κ
κ∈Th
Γ
Γi
By the definition of ,
κ∈Th
κ
· zh = 0
∀zh ∈ [Vhp ]n .
Furthermore, since D∇u f (u, ∇u)ψ = 0, it is clear that π = − . Thus,
Eh (vh ) = − vh · {} +
(de · vh ) .
Γ
Γi
Then, using the assumption on û, Eh (vh ) can be rewritten as
1 1 Eh (vh ) = −
− de · n+ + +
+ de · n+ − ,
vh · n −
vh ·
2
2
∂Ω
Γi
where (·)+ and (·)− refer to trace values taken from opposite sides of an interior face
and n+ is the outward pointing unit normal for κ+ .
Applying Lemma 4.4 to each edge in the triangulation, one can show that
|Eh (vh )| ≤
n ce h−1/2
.
(
+
h
|
|
)
v
j
0,κ
κ
j
1,κ
h
0,e
e
e
e
κe
e∈Eh j=1
Thus,
|Eh (vh )| ≤
n 1/2
ce h−1/2
j 21,κe
vh 0,e
κe
e∈Eh j=1
≤
n
e∈Eh j=1
≤C
n
j=1
⎡
⎣ce h−1/2
κe
⎡
⎣
1/2
⎤
vh 0,e ⎦
j 21,κ
κ∈Th
1/2 ⎤ 2
−1/2
⎦
×
j 1,κ
hκe vh 0,e .
κ∈Th
e∈Eh
Finally, applying Lemmas 4.3 and 4.5 gives
|Eh (vh )| ≤ C|||vh |||∗ h
k
n
|(D∇u f (u, ∇u)ψ)j |k+1,Ω .
j=1
Thus, as h → 0, |Eh (vh )| → 0 for all vh ∈ Whp , which implies that the scheme is
asymptotically dual consistent.
5. Numerical results. As a demonstration of the effects of dual consistency,
a simple test problem based on a nonlinear ODE is considered. The effect of dual
consistency on the convergence rates of the solution and adjoint solution errors as well
as a simple functional output is demonstrated.
Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.
3518
TODD A. OLIVER AND DAVID L. DARMOFAL
Consider the following ODE:
−((ν + u)ux )x − cux ux = g
for x ∈ (0, 1),
u(0) = u(1) = 0,
where ν = 1 and c = 12 . Setting
g(x) = π 2 ((ν + sin(πx)) sin(πx) − (1 + c) cos2 (πx)),
it is easy to show that the exact solution is given by
ue (x) = sin(πx).
This nonlinear problem has been solved using three discretizations: the standard
weighting method as shown in (3.4), a dual consistent method with a penalty term as
shown in (3.8), and an asymptotically dual consistent mixed method with û = {u}.
In all cases, the BR2 scheme is used to discretize the nonlinear diffusion operator.
Figure 5.1 shows the error in the primal solution versus grid refinement. The
error is measured in a broken H 1 norm defined by
(v 2 + vx2 ).
v2H 1 (Ω,Th ) =
κ
κ∈Th
0
0
10
10
1.0
1.0
−2
−2
10
2.0
10
10
h
H (Ω, T )
−4
−4
|| u − u ||
1
10
3.0
−6
10
−8
−10
10
0
3.0
−6
4.0
p=1
p=2
p=3
p=4
10
−8
2
3
10
4
4.0
p=1
p=2
p=3
p=4
10
−10
1
0
1
2
log2(h0/h)
3
log (h /h)
2
(a) Dual consistent
0
(b) Dual inconsistent
0
10
1.0
−2
)
10
2.0
h
|| uh − u ||H1(Ω, T
|| uh − u ||H1(Ω, T
h
2.0
h
)
10
−4
10
3.0
−6
10
−8
−10
10
4.0
p=1
p=2
p=3
p=4
10
0
1
2
3
4
log (h /h)
2
0
(c) Asymptotically dual consistent
Fig. 5.1. Primal error in the broken H 1 norm.
Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.
4
3519
DUAL CONSISTENT DISCRETIZATIONS OF SOURCE TERMS
0
0
10
−2
10
10
−2
10
2.0
2.0
−4
10
2.0
2
L (Ω)
−6
|| u − u ||
3.0
10
−6
10
h
|| uh − u ||L2(Ω)
−4
10
−8
10
4.0
p=1
p=2
p=3
p=4
−10
10
−12
10
0
−8
10
4.0
5.0
−12
1
2
3
10
4
4.0
p=1
p=2
p=3
p=4
−10
10
0
1
2
2
3
4
log2(h0/h)
log (h /h)
0
(a) Dual consistent
(b) Dual inconsistent
0
10
−2
10
2.0
|| uh − u ||L2(Ω)
−4
10
−6
3.0
10
−8
10
4.0
p=1
p=2
p=3
p=4
−10
10
−12
10
0
5.0
1
2
3
4
log (h /h)
2
0
(c) Asymptotically dual consistent
Fig. 5.2. Primal error in the L2 norm.
In this norm, all three schemes produce essentially the same error in the primal
solution. However, as shown in Figure 5.2, the dual consistent and asymptotically
dual consistent schemes produce superior results when the error is measured in the
L2 norm. In particular, for the dual inconsistent discretization, the L2 norm of the
error is proportional to hp for even p and proportional to hp+1 for odd p. It is not
clear why the even and odd p results show different asymptotic rates, but similar
results have been observed for other dual inconsistent discretizations [19].
Alternatively, the dual consistent and asymptotically dual consistent discretizations give O(hp+1 ) for all p tested. Furthermore, it is interesting to note that the
asymptotically dual consistent method produces essentially exactly the same results
as the dual consistent discretization. This fact shows that, for the asymptotically dual
consistent scheme, the contribution of the dual inconsistency error to the total error
is sufficiently high-order that it does not degrade the asymptotic convergence rate of
the L2 error. This result is not surprising given the form of the dual consistency error
derived in section 4.
Examining the adjoint solution error, one can see that the dual consistent and
asymptotically dual consistent schemes are superior for computing the adjoint. Figure 5.3 shows the adjoint error in the broken H 1 norm. The adjoint error is computed
relative to a 40th order solution of a Galerkin spectral discretization of the dual problem. When using the dual inconsistent discretization, the broken H 1 norm of the
Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.
3520
TODD A. OLIVER AND DAVID L. DARMOFAL
0
0
10
10
−2
10
−2
10
h
2.0
−4
−4
10
1
10
−6
10
−8
p=1
p=2
p=3
p=4
−10
0
h
3.0
10
10
|| ψ − ψ ||
|| ψh − ψ ||H1(Ω, T
h
H (Ω, T )
)
1.0
−6
10
−8
p=1
p=2
p=3
p=4
10
4.0
−10
1
2
3
log (h /h)
2
10
4
0
1
2
log (h /h)
2
0
(a) Dual consistent
3
4
0
(b) Dual inconsistent
0
10
−2
1.0
h
H (Ω, T )
10
2.0
−4
h
|| ψ − ψ ||
1
10
−6
10
3.0
−8
p=1
p=2
p=3
p=4
10
−10
10
0
4.0
1
2
log (h /h)
2
3
4
0
(c) Asymptotically dual consistent
Fig. 5.3. Adjoint error in the broken H 1 norm.
adjoint error does not converge to zero with grid refinement. For the dual consistent
and asymptotically dual consistent schemes, this error converges at O(hp ). Similarly,
Figure 5.4 shows that the L2 norm of the adjoint error converges at O(h) when using the dual inconsistent scheme, regardless of p, while for the dual consistent and
asymptotically dual consistent schemes this error converges at O(hp+1 ).
To further understand the differences in the adjoint solutions for the different
discretizations, Figure 5.5 shows the adjoint solution for each scheme, computed with
p = 2 polynomials on an eight element mesh. At the scale shown, it is difficult to see
the difference between the discretizations. However, examining the pointwise error
relative to the 40th order spectral solution, shown in Figure 5.6, it is clear that the
adjoint solution for the dual inconsistent scheme is more oscillatory than that for the
dual consistent or asymptotically dual consistent discretizations.
Finally, let
1
J (u) =
2
1
0
(w − u)2 ,
where w(x) = 2 sin(πx), be the output of interest. Then, computing the exact functional output is trivial, enabling comparison of the computed result with the exact
value J (ue ) = 1/4.
Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.
3521
DUAL CONSISTENT DISCRETIZATIONS OF SOURCE TERMS
−2
−2
10
10
−4
10
−4
−6
L (Ω)
−6
10
10
2
3.0
|| ψ − ψ ||
−8
10
4.0
−10
10
−14
10
0
−10
10
5.0
p=1
p=2
p=3
p=4
−12
10
−8
10
h
|| ψh − ψ ||L2(Ω)
1.0
10
2.0
p=1
p=2
p=3
p=4
−12
10
−14
1
2
3
10
4
0
1
2
log2(h0/h)
3
4
log2(h0/h)
(a) Dual consistent
(b) Dual inconsistent
−2
10
−4
10
2.0
|| ψh − ψ ||L2(Ω)
−6
10
3.0
−8
10
4.0
−10
10
−14
10
5.0
p=1
p=2
p=3
p=4
−12
10
0
1
2
log (h /h)
2
3
4
0
(c) Asymptotically dual consistent
Fig. 5.4. Adjoint error in the L2 norm.
0.14
DC
DinC
ADC
0.12
0.1
ψ
h
0.08
0.06
0.04
0.02
0
0
0.2
0.4
0.6
0.8
1
x
Fig. 5.5. Adjoint solution computed using the p = 2 dual consistent (DC), dual inconsistent
(DinC), and asymptotically dual consistent (ADC) discretizations on an eight element mesh.
Figure 5.7 shows the error in the computed functional for the three discretizations
considered. The figure shows that, as in the state and adjoint results, the performance
of the dual consistent and asymptotically dual consistent schemes is very similar.
Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.
3522
TODD A. OLIVER AND DAVID L. DARMOFAL
−3
−3
x 10
1.5
1
1
0.5
0.5
ψ −ψ
0
h
ψh − ψ
1.5
0
−0.5
−0.5
−1
−1
−1.5
0
0.2
0.4
0.6
0.8
x 10
−1.5
0
1
0.2
0.4
(a) Dual consistent, 8 element
1
0.5
0.5
ψh − ψ
h
ψ −ψ
1.5
1
0
−0.5
−1
−1
0.2
0.4
0.6
0.8
x 10
0
−0.5
−1.5
0
1
x
0.6
0.8
1
−3
x 10
1.5
1
1
0.5
0.5
ψ −ψ
0
h
h
ψ −ψ
0.4
(d) Dual inconsistent, 16 element
−3
−0.5
−1
−1
0.2
0.4
0.6
0.8
x
(e) Asymptotically dual consistent, 8 element
1
x 10
0
−0.5
−1.5
0
0.2
x
(c) Dual inconsistent, 8 element
1.5
1
−3
x 10
−1.5
0
0.8
(b) Dual consistent, 16 element
−3
1.5
0.6
x
x
−1.5
0
0.2
0.4
0.6
0.8
1
x
(f) Asymptotically dual consistent, 16 element
Fig. 5.6. Adjoint error for p = 2 on 8 and 16 element meshes.
Both schemes achieve O(h2p ) for 1 ≤ p ≤ 4. However, for the dual inconsistent
scheme, the convergence rate of the functional is O(hp ) for even p and O(hp+1 ) for
odd p. Thus, the dual consistent and asymptotically dual consistent discretizations
predict the functional with greater accuracy than the dual inconsistent discretization
for similar numbers of degrees of freedom.
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DUAL CONSISTENT DISCRETIZATIONS OF SOURCE TERMS
3523
−1
−1
10
10
2.01
−3
2.02
−3
10
10
1.95
−5
−5
10
4.20
h
|J(u) − J(u )|
|J(u) − J(uh)|
10
−7
10
6.01
−9
10
−9
10
−11
−11
10
10
p=1
p=2
p=3
p=4
−13
10
−15
10
4.05
−7
10
p=1
p=2
p=3
p=4
7.73
7.72
−13
10
−15
0
1
10
2
0
1
2
2
log (h /h)
log (h /h)
2
0
(a) Dual consistent
0
(b) Dual inconsistent
−1
10
2.01
−3
10
−5
h
|J(u) − J(u )|
10
4.27
−7
10
6.01
−9
10
−11
10
−15
10
7.73
p=1
p=2
p=3
p=4
−13
10
0
1
2
log2(h0/h)
(c) Asymptotically dual consistent
Fig. 5.7. Functional output error.
6. Conclusions. The effect of dual consistency on DG discretizations of solution
and solution gradient dependent source terms has been examined. In particular, the
standard weighting DG discretization of source terms depending on the gradient of
the solution has been analyzed and shown to be dual inconsistent. Starting from
this dual inconsistent scheme a dual consistent discretization has been developed.
Furthermore, discretizations derived using the mixed formulation have been shown
to be asymptotically dual consistent. Numerical results from a simple test problem
demonstrate that the dual consistent and asymptotically dual consistent schemes are
superior both in terms of solution accuracy and output accuracy.
Further work is required in many areas. In particular, this work has considered
only the effect of dual consistency. Using the methods presented in sections 3 and 4,
one could construct a consistent and dual consistent but unstable scheme. Thus, while
techniques for constructing a consistent and dual consistent (or asymptotically dual
consistent) discretization have been shown, a method for ensuring that the resulting
scheme is stable is left for future research. Finally, given the extremely similar results
shown for the dual consistent and asymptotically dual consistent schemes considered,
it remains to be determined which of these schemes is most effective for practical
problems.
Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.
3524
TODD A. OLIVER AND DAVID L. DARMOFAL
Acknowledgment. The authors would like to thank the reviewers of the original manuscript for their insightful comments and suggestions, which led to many
improvements in the final version of this paper.
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