c 2005 Society for Industrial and Applied Mathematics
SIAM J. NUMER. ANAL.
Vol. 43, No. 1, pp. 195–219
SYMMETRIC AND NONSYMMETRIC
DISCONTINUOUS GALERKIN METHODS FOR
REACTIVE TRANSPORT IN POROUS MEDIA∗
SHUYU SUN† AND MARY F. WHEELER‡
Abstract. For solving reactive transport problems in porous media, we analyze three primal
discontinuous Galerkin (DG) methods with penalty, namely, symmetric interior penalty Galerkin
(SIPG), nonsymmetric interior penalty Galerkin (NIPG), and incomplete interior penalty Galerkin
(IIPG). A cut-off operator is introduced in DG to treat general kinetic chemistry. Error estimates in
L2 (H 1 ) are established, which are optimal in h and nearly optimal in p. We develop a parabolic lift
technique for SIPG, which leads to h-optimal and nearly p-optimal error estimates in the L2 (L2 ) and
negative norms. Numerical results validate these estimates. We also discuss implementation issues
including penalty parameters and the choice of physical versus reference polynomial spaces.
Key words. error estimates, discontinuous Galerkin methods, reactive transport, porous media,
parabolic partial differential equations, SIPG, NIPG, IIPG
AMS subject classifications. 65M12, 65M15, 65M60, 35K57
DOI. 10.1137/S003614290241708X
1. Introduction. Discontinuous Galerkin (DG) methods employ discontinuous
piecewise polynomials to approximate the solutions of differential equations, with
boundary conditions and interelement continuity weakly imposed through bilinear
forms. Even though they often have larger numbers of degrees of freedom than conforming approaches, DG methods have recently gained popularity for a number of
attractive features [19, 3, 4, 23, 27, 11, 25, 26, 9, 18, 15]: (1) they are element-wise
conservative; (2) they support general nonconforming spaces including unstructured
meshes, nonmatching grids and variable degrees of local approximations, thus allowing efficient h-, p-, and hp-adaptivities; (3) they tend to have localized errors, allowing
sharp a posteriori error indicators and effective adaptivities; (4) they have less numerical diffusion; (5) they treat rough coefficient problems and effectively capture
discontinuities in solutions; (6) they are robust and nonoscillatory in the presence
of high gradients; (7) with appropriate meshing, they are capable of delivering exponential rates of convergences; (8) they have excellent parallel efficiency since data
communications are relatively local; (9) for time-dependent problems in particular,
their mass matrices are block diagonal, providing substantial computational advantages if explicit time integrations are used. In addition, by a simple extension from
the average of the fluxes on element faces, DG can provide a continuous flux field
defined over the entire domain, allowing efficient coupling with conforming methods.
Numerical modeling of reactive transport in porous media has important applications in hydrology, earth sciences, environmental protection, oil recovery, chemical
∗ Received by the editors November 2, 2002; accepted for publication (in revised form) February
25, 2005; published electronically May 27, 2005. This research was partially supported by National
Science Foundation grant DMS-0411413.
http://www.siam.org/journals/sinum/43-1/41708.html
† The Institute for Computational Engineering and Sciences (ICES), The University of Texas at
Austin, 201 E. 24th St. ACE 5.316, Austin, TX 78712 ([email protected]).
‡ ICES, Department of Aerospace Engineering and Engineering Mechanics, Department of
Petroleum and Geosystems Engineering, and Department of Mathematics, The University of Texas
at Austin, 201 E. 24th St. ACE 5.324, Austin, TX 78712 ([email protected]).
195
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SHUYU SUN AND MARY F. WHEELER
industry, and biomedical engineering. Realistic simulations for simultaneous advection, diffusion, and chemical reactions present significant computational challenges
[2, 40, 10, 14, 24, 37, 41, 28, 7, 16, 8]. Recently, it has been shown that adaptive DG can effectively capture moving concentration fronts in reactive transport
[31, 33, 36, 32, 29]. A posteriori error estimates of DG for reactive transport problems have been derived in the L2 (L2 ) [32] and L2 (H 1 ) norms [35]. In addition, DG
has been applied to coupled flow and transport problems in porous media [34, 39, 30].
However, to the best of our knowledge, optimal a priori hp-estimates in the L2 (L2 )
and negative norms have not been established.
The primal DG methods include four members: Oden–Babuška–Baumann DG
(OBB-DG) formulation [19], symmetric interior penalty Galerkin (SIPG) [38], nonsymmetric interior penalty Galerkin (NIPG) [23, 21], and incomplete interior penalty
Galerkin (IIPG) [12, 29]. In this paper, we analyze the three primal DG methods
with penalty, i.e., SIPG, NIPG, and IIPG, for solving reactive transport problems in
porous media. The primal DG method without penalty, i.e., the OBB-DG scheme,
has been analyzed for reactive transport problems elsewhere [22]. In the following
section, we describe the modeling equations. The DG schemes are introduced in section 3. Section 4 contains the L2 (H 1 ) error analysis for SIPG, NIPG, and IIPG. In
section 5, a parabolic lift technique is developed, and an L2 (L2 ) error analysis for
SIPG is conducted. Optimal negative norm estimates are derived in section 6. In
section 7, we present numerical studies of h- and p-convergences for the four primal
DG schemes. In section 8, we discuss choices of penalty parameters as well as DG
implementations using reference versus physical polynomial spaces. Conclusions are
given in the last section.
2. Governing equations. For convenience of presentation, we consider reactive
transport problems of only one species in a single flowing phase in porous media.
Results for systems of multiple species with kinetic reactions can be derived by similar
arguments. We assume that a Darcy velocity field u is given and time-independent,
and satisfies ∇ · u = q, where q is the imposed external total flow rate. In addition,
we assume that Ω is a polygonal and bounded domain in Rd (d = 1, 2, or 3) with
boundary ∂Ω = Γin ∪ Γout . Here we denote by Γin the inflow boundary and by Γout
the outflow/no-flow boundary, i.e.,
Γin := {x ∈ ∂Ω : u · n < 0},
Γout := {x ∈ ∂Ω : u · n ≥ 0},
where n denotes the unit outward normal vector to ∂Ω. Let T be the final simulation
time. The classical advection-diffusion-reaction equation in porous media is given by
(2.1)
∂φc
+ ∇ · (uc − D(u)∇c) = qc∗ + r(c),
∂t
(x, t) ∈ Ω × (0, T ],
where the unknown variable c is the concentration of a species (amount per volume).
Here φ is the effective porosity and is assumed to be time-independent, uniformly
bounded above and below by positive numbers; D(u) is the dispersion-diffusion tensor
and is assumed to be uniformly symmetric positive definite and bounded from above;
r(c) is the reaction term; qc∗ is the source term, where the imposed external total
flow rate q is a sum of sources (injection) and sinks (extraction); c∗ is the injected
concentration cw if q ≥ 0 and is the resident concentration c if q < 0.
DISCONTINUOUS GALERKIN FOR REACTIVE TRANSPORT
197
We consider the following boundary conditions for this problem:
(2.2)
(2.3)
(uc − D(u)∇c) · n = cB u · n,
(−D(u)∇c) · n = 0,
(x, t) ∈ Γin × (0, T ],
(x, t) ∈ Γout × (0, T ],
where cB is the inflow concentration. The initial concentration is specified by
c(x, 0) = c0 (x),
(2.4)
x ∈ Ω.
3. Discontinuous Galerkin schemes.
3.1. Notation. Let Eh be a family of nondegenerate, quasi-uniform and possibly
nonconforming partitions of Ω composed of triangles or quadrilaterals if d = 2, or
tetrahedra, prisms, or hexahedra if d = 3. The nondegeneracy requirement (also
called regularity) is that the element is convex, and that there exists ρ > 0 such
that if hj is the diameter of Ej ∈ Eh , then each of the subtriangles (for d = 2) or
subtetrahedra (for d = 3) of element Ej contains a ball of radius ρhj in its interior.
The quasi-uniformity requirement is that there is τ > 0 such that (h/hj ) ≤ τ for
all Ej ∈ Eh , where h is the maximum diameter of all elements. We assume that no
element crosses the boundaries of Γin or Γout . The set of all interior edges (for d = 2)
or faces (for d = 3) for Eh is denoted by Γh . On each edge or face γ ∈ Γh , a unit
normal vector nγ is chosen. The sets of all edges or faces on Γout and on Γin for Eh
are denoted by Γh,out and Γh,in , respectively, for which the normal vector nγ coincides
with the outward unit normal vector.
We now define the average and jump for φ ∈ H s (Eh ), s > 1/2. Let Ei , Ej ∈ Eh
and γ = ∂Ei ∩ ∂Ej ∈ Γh with nγ exterior to Ei . We denote
{φ} :=
1
((φ|Ei )|γ + (φ|Ej )|γ ),
2
[φ] := (φ|Ei )|γ − (φ|Ej )|γ .
The upwind value of a concentration c∗ |γ is defined as
c|Ei if u · nγ ≥ 0,
∗
c |γ :=
c|Ej if u · nγ < 0.
We denote by ·m,R the usual Sobolev norm over a domain R [1]. The Sobolev
norm ·m,Ω over the entire domain Ω is also denoted simply by ·m . For s ≥ 0, we
define the broken Sobolev space
H s (Eh ) := {φ ∈ L2 (Ω) : φ|E ∈ H s (E), E ∈ Eh }.
One can show that H s (Eh ) is a normed linear space with its norm defined by
1/2
2
φs,E
.
φH s (Eh ) :=
E∈Eh
Following the tradition, we also use the notation ||| · |||s to denote the broken norm
·H s (Eh ) . For a given normed space X and a number p ≥ 1, we define
Lp (0, T ; X) := {φ : φ(t) ∈ X, φX ∈ Lp (0, T )}.
The space Lp (0, T ; X) is also a normed linear space with its norm given by
φLp (0,T ;X) := (φX )Lp (0,T ) .
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SHUYU SUN AND MARY F. WHEELER
The broken norm ·Lp (0,T ;H s (Eh )) is also written as ||| · |||Lp (0,T ;H s ) in the triple bar
notation. We denote by (·, ·)R the inner product in (L2 (R))d or L2 (R) over a domain
R. The inner product (·, ·)Ω over the entire domain Ω is also denoted simply by (·, ·).
r,s
We also need the space W∞
and its norm:
r,s
r,s < ∞},
((0, T ) × Ω) := {f ∈ L2 ((0, T ) × Ω) : f W∞
W∞
α
r,s :=
f W∞
ess sup(0,T )×Ω (|Dx f | + |Dtβ f |).
|α|≤r, β≤s
The discontinuous finite element space is taken to be
(3.1)
Dr (Eh ) := {φ ∈ L2 (Ω) : φ|E ∈ Pr (E), E ∈ Eh },
where Pr (E) denotes the space of polynomials of (total) degree less than or equal to
r on E. Note that we present hp-results in this paper for the local space Pr , but the
results also apply to the local space Qr because Pr (E) ⊂ Qr (E).
We define a cut-off operator as
M(c)(x) := min(c(x), M ),
(3.2)
where M is a large positive constant. By a straightforward algebraic argument, we
can show that the cut-off operator M is uniformly Lipschitz continuous.
Lemma 1 (property of operator M). The cut-off operator M defined in (3.2) is
uniformly Lipschitz continuous with a Lipschitz constant of one; that is,
M(c) − M(w)L∞ (Ω) ≤ c − wL∞ (Ω) .
(3.3)
We use the following hp-approximation results, which can be proved using the
techniques in [6, 5]. Let E ∈ Eh and φ ∈ H s (E). Then there exists a constant K,
independent of φ, r, and hE , and a sequence of zrh ∈ Pr (E), r = 1, 2, . . . , such that
⎧
hμ−q
⎪
h
E
⎪
0 ≤ q < μ,
⎨
φ − zr q,E ≤ K s−q φs,E ,
r
(3.4)
1
μ−q− 2
⎪
hE
⎪
⎩
φ − z h ≤
K
0 ≤ q < μ − 12 ,
1 φs,E ,
r q,∂E
rs−q− 2
where μ = min(r + 1, s) and hE denotes the diameter of E.
We shall also use the following inverse inequalities, which can be derived using
the method in [27]. Let E ∈ Eh and v ∈ Pr (E). Then there exists a constant K,
independent of v, r, and hE , such that
⎧
r
q
q
⎪
q ≥ 0,
⎪
⎨D v0,∂E ≤ K 1/2 D vE ,
hE
(3.5)
r2
⎪
⎪
⎩Dq+1 v0,E ≤ K
Dq v0,E ,
q ≥ 0.
hE
3.2. Continuous-in-time DG schemes. We introduce a bilinear form:
B(c, w; u) :=
(D(u)∇c − cu) · ∇w −
cq − w
E∈Eh
−
E
γ∈Γh
+
{D(u)∇c · nγ }[w] − sform
γ
γ∈Γh
Ω
γ
∗
c u · nγ [w] +
γ∈Γh,out
γ∈Γh
{D(u)∇w · nγ }[c]
γ
cu · nγ w + J0σ (c, w).
γ
DISCONTINUOUS GALERKIN FOR REACTIVE TRANSPORT
199
Here sform = 1 for SIPG; sform = −1 for OBB-DG or NIPG; and sform = 0 for IIPG.
For convenience of presentation, we denote the bilinear form as BS (c, w; u) when it
is symmetric, i.e., sform = 1. We denote by q + the injection source term and by q −
the extraction source term, i.e., q + = max(q, 0) and q − = min(q, 0). By definition,
we have q = q + + q − . To impose interelement continuity weakly, an interior penalty
term J0σ (c, w) is formulated:
(3.6)
J0σ (c, w) :=
r 2 σγ [c][w],
hγ γ
γ∈Γh
where σ is a discrete positive function that takes the constant value σγ on the edge or
face γ. There is no penalty term, i.e., σ = 0, for OBB-DG. In the analysis of SIPG,
NIPG, and IIPG in this paper, we assume 0 < σ0 ≤ σγ ≤ σm . In addition we define
a linear functional:
r(M(c))w +
(3.7)
cw q + w −
cB u · nγ w.
L(w; u, c) :=
Ω
Ω
γ∈Γh,in
γ
The reactive transport problem can be stated in the following equivalent weak
formulation.
Lemma 2 (weak formulation). If c is a solution of (2.1)–(2.3) and c is essentially
bounded, then c satisfies
∂φc
(3.8)
, w + B(c, w; u) = L(w; u, c)
∂t
3
∀t ∈ (0, T ],
∀w ∈ H s (Eh ), s >
2
provided that the constant M for the cut-off operator is sufficiently large.
Proof. Let w ∈ H s (Eh ), s > 3/2 and E ∈ Eh . Multiplying (2.1) by w, integrating
over E, and then integrating by parts, we observe
∂φc
,w
− (uc − D(u)∇c) · ∇w +
(uc − D(u)∇c) · n∂E w
∂t
E
∂E
E
=
qc∗ w + r(c)w.
E
Summing it over all elements in Eh , noting the fact that the traces of the concentration
and its normal flux are continuous across element faces, and applying the boundary
conditions, we obtain the desired result.
The continuous-in-time DG approximation C DG (·, t) ∈ Dr (Eh ) to the solution of
(2.1)–(2.4) is defined by
∂φC DG
, w + B(C DG , w; u) = L(w; u, C DG )
(3.9)
∂t
∀w ∈ Dr (Eh ) ∀t ∈ (0, T ],
(φC DG , w) = (φc0 , w)
(3.10)
∀w ∈ Dr (Eh ), t = 0.
As a valuable property, DG schemes possess element-wise mass conservation.
OBB-DG satisfies local conservation strictly, whereas SIPG, NIPG, and IIPG are
200
SHUYU SUN AND MARY F. WHEELER
locally conservative if the concentration jump term is considered as part of the computed diffusive flux:
Lemma 3 (local mass balance). The approximation of the concentration satisfies
on each element E the following local mass balance equation:
∂φC DG
(3.11)
{D(u)∇C DG · n∂E } +
C DG∗ u · n∂E
−
∂t
E
∂E\∂Ω
∂E
r 2 σγ +
(C DG |E − C DG |Ω\E )
hγ γ
γ⊂∂E\∂Ω
=
C DG∗ q +
r(M(C DG )).
E
E
Proof. The relationship (3.11) follows immediately from the DG schemes by fixing
an element E and letting w ∈ Dr (Eh ) with w|E = 1, w|Ω\E = 0.
It is also important to know that a DG scheme has a solution.
Lemma 4 (existence of a solution). Assume that the reaction rate is a locally
Lipschitz continuous function of the concentration. Then the discontinuous Galerkin
scheme (3.9) and (3.10) has a unique solution for t > 0.
M
DG
Proof. We let {vi }M
= i=1 ζi (t)vi (x).
i=1 be a basis of Dr (Eh ) and write C
Then (3.9) and (3.10) reduce to the following initial value problem:
⎧
⎨ dζ
= −Bζ + R(ζ),
A
dt
⎩
Aζ(0) = b,
where the mass matrix A is block-diagonal, symmetric, and positive definite. From the
properties of the cut-off operator M and the reaction function, we observe that R(ζ)
is (globally) Lipschitz continuous. It follows from the theory of ordinary differential
equations that ζ(t) exists and is unique for t > 0.
4. L2 (H 1 ) and L∞ (L2 ) error estimates. Throughout the paper, we denote
by K a generic positive constant independent of h and r, and by a fixed positive
constant that may be chosen arbitrarily small.
Theorem 1 (L2 (H 1 ) and L∞ (L2 ) error estimates). Let c be the solution to
(2.1)–(2.4), and assume c ∈ L2 (0, T ; H s (Eh )), ∂c/∂t ∈ L2 (0, T ; H s−1 (Eh )), and c0 ∈
H s−1 (Eh ). We further assume that c, u and q are essentially bounded, that the reaction rate is a locally Lipschitz continuous function of c, and that the cut-off constant
M and the penalty parameter σ0 are sufficiently large. Then there exists a constant
K, independent of h and r, such that
1
C DG − cL∞ (0,T ;L2 ) + |||D 2 (u)∇(C DG − c)|||L2 (0,T ;L2 )
12
T
σ
DG
DG
+
J0 (C
− c, C
− c)
0
≤K
hμ−1
hμ−1
2 (0,T ;H s ) + K
|||c|||
(|||∂c/∂t|||L2 (0,T ;H s−1 ) + |||c0 |||s−1 ),
L
rs−1−δ
rs−1
where μ = min(r + 1, s), r ≥ 1, s ≥ 2, δ = 0 for conforming meshes with triangles or
tetrahedra, and δ = 1/2 in general.
Proof. We let c ∈ Dr (Eh ) be an interpolant of concentration c such that the
hp-results (3.4) hold, and define
(4.1)
ξ = C DG − c,
DISCONTINUOUS GALERKIN FOR REACTIVE TRANSPORT
201
c,
ξI = c − A
DG
ξ =C
−
c = ξ + ξI .
(4.2)
(4.3)
Subtracting the weak formulation (3.8) from the DG scheme (3.9), choosing w = ξ A ,
we obtain
∂φξ A A
,ξ
+ B(ξ A , ξ A ; u)
(4.4)
∂t
∂φξ I A
A
DG
A
,ξ
+ B(ξ I , ξ A ; u).
= L(ξ ; u, C ) − L(ξ ; u, c) +
∂t
The first term of the error equation (4.4) may be written in a time derivative of
an L2 norm:
1 d ∂φξ A A
A 2
,ξ
=
φξ .
∂t
2 dt
0,Ω
We expand the second term of (4.4) as
B(ξ A , ξ A ; u) =
(D(u)∇ξ A − ξ A u) · ∇ξ A −
q − (ξ A )2
E∈Eh
E
−(1 + sform )
+
γ∈Γh
γ∈Γh
Ω
{D(u)∇ξ A · nγ }[ξ A ]
γ
ξ A∗ u · nγ [ξ A ] +
γ
γ∈Γh,out
u · nγ (ξ A )2 + J0σ (ξ A , ξ A ).
γ
Integrating the advection term by parts, we observe
−
ξ A u · ∇ξ A
E∈Eh
E
1 1 1 A 2
A 2
=−
u · ∇(ξ ) = −
u · n∂E (ξ ) +
q(ξ A )2
2
2
2
E
∂E
E
E∈Eh
E∈Eh
E∈Eh
1 1
1 A 2
A 2
=−
u · nγ [(ξ ) ] −
u · nγ (ξ ) +
q(ξ A )2 .
2
2
2
γ
γ
E
γ∈Γh,in ∪Γh,out
γ∈Γh
E∈Eh
In addition, noting that [c2 ] = 2{c}[c] and (c∗ − {c})sign(u · n) = [c]/2, we have
1
1
A A
A 2
2
|q|(ξ A )2 − T0 + J0σ (ξ A , ξ A )
B(ξ , ξ ; u) = |||D (u)∇ξ |||0 +
2 Ω
1 1
+
|u · nγ |[ξ A ]2 +
|u · nγ |(ξ A )2 ,
2
2
γ
γ
γ∈Γh,in ∪Γh,out
γ∈Γh
where T0 is defined by
T0 := (1 + sform )
γ∈Γh
γ
{D(u)∇ξ A · nγ }[ξ A ].
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SHUYU SUN AND MARY F. WHEELER
If the penalty parameter σ0 is chosen to be sufficiently large, we may bound T0 by
applying the Cauchy–Schwarz and inverse inequalities:
(4.5)
2
h Kr2 12
A
(u)∇ξ
·
n
+
[ξ A ]20,γ
D
∂E
Kr2
h
0,∂E
T0 ≤
E∈Eh
γ∈Γh
1
1
1
≤ |||D 2 (u)∇ξ A |||20 + J0σ (ξ A , ξ A ).
2
2
The first two terms on the right-hand side of (4.4) may be estimated, by using
Lemma 1, as
L(ξ A ; u, C DG ) − L(ξ A ; u, c) =
≤ K
(r(M(C DG )) − r(M(c)))ξ A
Ω
φξ A 20
+
Kξ I 20
≤ K
φξ A 20 + K
h2μ
|||c|||2s .
r2s
We have a similar result for the third term:
I
∂ξ A ∂φξ I A
≤K
,ξ
∂t φξ 0
∂t
0
I 2
2
2
∂ξ h2μ−2
≤K
≤ K φξ A + K φξ A + K 2s−2 |||ct |||2s−1 .
∂t 0
r
0
0
The fourth term on the right-hand side of (4.4) consists of eight pieces:
B(ξ I , ξ A ; u)
=
D(u)∇ξ I · ∇ξ A −
ξ I u · ∇ξ A −
q− ξI ξA
E∈Eh
−
E
γ∈Γh
+
γ∈Γh
=:
E
{D(u)∇ξ I · nγ }[ξ A ] − sform
γ
8
E∈Eh
ξ I∗ u · nγ [ξ A ] +
γ
γ∈Γh,out
γ∈Γh
Ω
{D(u)∇ξ A · nγ }[ξ I ]
γ
u · nγ ξ I ξ A + J0σ (ξ I , ξ A )
γ
Ti .
i=1
The Cauchy–Schwarz inequality and approximation results yield
h2μ−2
|||c|||2s ,
r2s−2
1
h2μ
T2 ≤ |||D 2 (u)∇ξ A |||20 + K 2s |||c|||2s ,
r
2μ
h
|q − |(ξ A )2 + K 2s |||c|||2s .
T3 ≤ r
Ω
1
T1 ≤ |||D 2 (u)∇ξ A |||20 + K
DISCONTINUOUS GALERKIN FOR REACTIVE TRANSPORT
203
We bound the terms T4 and T5 by hiding a large constant in the penalty term and by
using the inverse inequality, respectively,
σ0 r 2 Kh T4 ≤ [ξ A ]20,γ + 2
∇ξ I · n∂E 20,∂E
h
r
γ∈Γh
≤ J0σ (ξ A , ξ A ) + K
T5 ≤
E∈Eh
2μ−2
h
|||c|||2s ,
r2s−1
1
h Kr2 I 2
2
2 (u)∇ξ A · n
D
+
ξ 0,∂E
∂E
0,∂E
Kr2
h
E∈Eh
E∈Eh
h2μ−2
≤ |||D (u)∇ξ A |||20 + K 2s−3 |||c|||2s .
r
Similar applications of the Cauchy–Schwarz inequality and approximation results give
h2μ−1
T6 ≤ |u · nγ |[ξ A ]2 + K 2s−1 |||c|||2s ,
r
γ∈Γh γ
h2μ−1
T7 ≤ |u · nγ |(ξ A )2 + K 2s−1 |||c|||2s ,
r
γ
1
2
γ∈Γh,out
h2μ−2
|||c|||2s .
r2s−3
For conforming meshes with triangles or tetrahedra, we can choose a continuous approximation c to make the two terms T5 and T8 vanish. Substituting all the
estimates into (4.4), we see that
1
d A 2
φξ 0 + |||D 2 (u)∇ξ A |||20 + J0σ (ξ A , ξ A )
(4.6)
dt
h2μ−2
h2μ−2
≤ K φξ A 20 + K 2s−2−2δ |||c|||2s + K 2s−2 |||ct |||2s−1 ,
r
r
where δ = 0 for conforming meshes with triangles or tetrahedra, and δ = 1/2 in
general. Integrating (4.6) with respect to the time t, noting that
T8 ≤ J0σ (ξ A , ξ A ) + K
hμ−1
φE A 0 (0) ≤ K s−1 |||c0 |||s−1 ,
r
and applying Gronwall’s inequality, we conclude that
T
A
1
1
A
2
J0σ (ξ A , ξ A )) 2
φξ L∞ (0,T ;L2 ) + |||D (u)∇ξ |||L2 (0,T ;L2 ) + (
0
hμ−1
hμ−1
≤ K s−1−δ |||c|||L2 (0,T ;H s ) + K s−1 (|||∂c/∂t|||L2 (0,T ;H s−1 ) + |||c0 |||s−1 ).
r
r
The theorem follows by applying the triangle inequality, the approximation results
and the fact that
(4.7)
|||c|||L∞ (0,T ;H s−1 ) ≤ K|||ct |||L2 (0,T ;H s−1 ) + |||c0 |||s−1 .
We remark that, in [22], L∞ (L2 ) + L2 (H 1 ) error estimates for the OBB-DG
diffusion scheme applied to the transport problem established optimality in h and
suboptimality in p by 3/2. Here for SIPG, NIPG, and IIPG, we obtain optimality in
h and p for conforming meshes with triangles and tetrahedra and a loss of 1/2 in p for
general grids. Obviously, penalty terms improve the provable p-optimality of DGs.
204
SHUYU SUN AND MARY F. WHEELER
5. Optimal L2 (L2 ) error estimates for the symmetric scheme. In this and
following sections, we restrict our attention to SIPG. The derivation in this section
is motivated by the h-optimal L2 result for SIPG applied to an elliptic problem by
Wheeler [38] and the h-optimal L2 (L2 ) result for continuous Galerkin methods applied
to a parabolic problem by Palmer [20]. See also the h-optimal L2 (L2 ) result for
continuous finite element modified methods of characteristics applied to a coupled
system of partial differential equations (PDEs) by Dawson, Russell, and Wheeler
[13] and the h-optimal L∞ (L2 ) result for SIPG applied to a parabolic equation with
diffusion term by Arnold [4, 3]. We first recall a theorem proved in [20, 17].
Theorem 2. Consider the parabolic equation:
∂φΦ
+ ∇ · (uΦ − D∇Φ) + aΦ = f,
∂t
D∇Φ · n = 0,
Φ = 0,
x ∈ Ω, t ∈ (0, T ],
x ∈ ∂Ω, t ∈ (0, T ],
x ∈ Ω, t = 0.
Assume that 0 < φ0 ≤ φ(t, x) ≤ φm , D is uniformly symmetric positive definite
2,1
1,0
and bounded from above, φ ∈ W∞
((0, T ) × Ω), Dij ∈ W∞
((0, T ) × Ω), ui ∈ L∞ (Ω)
2
∞
(u being independent of time), a ∈ L (0, T ; L (Ω)) and f ∈ L2 (0, T ; L2 (Ω)). Then
there exists a unique solution Φ satisfying the above equation and the regularity bounds
given by
ΦL∞ (0,T ;H 1 ) + ΦL2 (0,T ;H 2 ) ≤ Kf L2 (0,T ;L2 ) ,
where K is a constant independent of the input data f .
For simplicity of presentation, we consider problems with no-flow boundary conditions, though the result can be generalized. We make additional assumptions:
2,1
1,0
φ ∈ W∞
((0, T ) × Ω), Dij ∈ W∞
((0, T ) × Ω), and q + ∈ L2 (0, T ; L∞ (Ω)).
5.1. Parabolic lift for SIPG.
Lemma 5 (parabolic lift). Let a ∈ L2 (0, T ; L∞ (Ω)) and e ∈ L2 (0, T ; H 1 (Eh ))
satisfy
∂φe
, w + BS (e, w; u) + (ae, w) = 0
∀w ∈ Dr (Eh ) ∀t ∈ (0, T ],
(5.1)
∂t
(φe, w) = 0
∀w ∈ Dr (Eh ), t = 0.
(5.2)
In addition we let the assumptions in Theorem 1 hold. Then there exists a constant
K, independent of h, r, and e, such that
eL2 (0,T ;L2 )
h
h2
≤ K eL∞ (0,T ;L2 ) + K 2 et L2 (0,T ;L2 )
r
r
T
12
1
h
h
σ
2
+K |||D (u)∇e|||L2 (0,T ;L2 ) + K 3 −2δ
J0 (e, e)
r
r2
0
12
3
h2
2
2
+Kδ 3
(eL2 (0,T ;L2 (∂E)) + ∇e · n∂E L2 (0,T ;L2 (∂E)) )
,
r 2 E∈Eh
where δ = 0 for conforming meshes with triangles or tetrahedra, and δ = 1/2 in
general.
DISCONTINUOUS GALERKIN FOR REACTIVE TRANSPORT
205
Proof. Consider the backward or adjoint parabolic equation:
−
(5.3)
(5.4)
(5.5)
∂φΦ
+ ∇ · (−uΦ − D(u)∇Φ) + (a + q + )Φ = e,
∂t
D(u)∇Φ · n∂Ω = 0,
Φ = 0,
x ∈ Ω, t ∈ [0, T ),
x ∈ ∂Ω, t ∈ [0, T ),
x ∈ Ω, t = T.
Theorem 2 suggests a unique solution Φ for (5.3)–(5.5) satisfying
ΦL∞ (0,T ;H 1 ) + ΦL2 (0,T ;H 2 ) ≤ KeL2 (0,T ;L2 ) .
(5.6)
Observing that D(u)∇Φ · n∂Ω = 0 on ∂Ω, ∇ · u = q, and [D(u)∇Φ · nγ ] = [Φ] = 0,
we multiply both sides of the adjoint equation (5.3) by e, integrate it over the domain
Ω, and then apply integration by parts to conclude that
∂e d 2
e0 = −
φ ,Φ
(e, φΦ)E +
+
((a − q − )e, Φ)E
dt
∂t
E
E∈Eh
E∈Eh
E∈Eh
+
(∇e, D(u)∇Φ)E −
{D(u)∇Φ · nγ }[e] −
(e, u∇ · Φ)E
E∈Eh
=−
γ∈Γh
γ
E∈Eh
∂e
d
(e, φΦ) + φ , Φ + (ae, Φ) + BS (e, Φ; u).
dt
∂t
Applying the orthogonality condition (5.1), we obtain
∂e
d
2
(5.7) e0 = − (e, φΦ) + φ , Φ − Φ̂ + (ae, Φ − Φ̂) + BS (e, Φ − Φ̂; u),
dt
∂t
where Φ̂ ∈ Dr (Eh ) is an interpolant satisfying (3.4) element-wise. The second and
third terms on the right-hand side of (5.7) are bounded, by using the Cauchy–Schwarz
inequality and approximation results, as
∂e
h2
φ , Φ − Φ̂ ≤ Ket 0 Φ − Φ̂0 ≤ K 2 et 0 Φ2 ,
∂t
r
h2
(ae, Φ − Φ̂) ≤ KaL∞ e0 Φ − Φ̂0 ≤ K 2 aL∞ e0 Φ2 .
r
The last term in (5.7) is composed of eight parts:
BS (e, Φ − Φ̂; u)
=
D(u)∇e · ∇(Φ − Φ̂) −
eu · ∇(Φ − Φ̂) −
q − e(Φ − Φ̂)
E∈Eh
−
E
γ∈Γh
+
γ∈Γh
=:
8
i=1
Ti .
{D(u)∇e · nγ }[Φ − Φ̂] −
γ
γ
E
E∈Eh
e∗ u · nγ [Φ − Φ̂] +
γ∈Γh
γ∈Γh,out
Ω
{D(u)∇(Φ − Φ̂) · nγ }[e]
γ
eu · nγ (Φ − Φ̂) + J0σ (e, Φ − Φ̂)
γ
206
SHUYU SUN AND MARY F. WHEELER
Once again, the approximation results and Cauchy–Schwarz inequality yield the estimates for the terms T1 , T2 , and T3 :
1
1
h
T1 ≤ K|||D 2 (u)∇e|||0 ∇(Φ − Φ̂)0 ≤ K |||D 2 (u)∇e|||0 Φ2 ,
r
h
T2 ≤ K e0 Φ2 ,
r
h2
T3 ≤ K 2 e0 Φ2 .
r
The term T7 vanishes because of the assumed no-flow boundary condition. The remaining terms in the bilinear form can be bounded by applying the Cauchy–Schwarz
inequality on element faces:
T4 ≤ K
3
∇e · n∂E 0,∂E Φ − Φ̂0,∂E ≤ K
E∈Eh
T5 ≤
3
h2
r
3
2
3
r2
{D(u)∇(Φ − Φ̂) · nγ }0,γ [e]0,γ ≤ K
γ∈Γh
T6 ≤ K
h2
12
∇e ·
n∂E 20,∂E
Φ2 ,
E∈Eh
h
r
3
2
1
(J0σ (e, e)) 2 Φ2 ,
21
e20,∂E
Φ2 ,
E∈Eh
1
1
T8 ≤ (J0σ (e, e)) 2 (J0σ (Φ − Φ̂, Φ − Φ̂)) 2 ≤ K
h
r
1
2
1
(J0σ (e, e)) 2 Φ2 .
We note that, for conforming meshes with triangles or tetrahedra, terms T4 , T6 ,
and T8 vanish if we choose a continuous interpolant Φ̂. Substituting all the estimates
back into (5.7), we find that
h2
d
h2
(e, φΦ) + K 2 et 0 Φ2 + K 2 aL∞ e0 Φ2
dt
r
r
1
1
h
h
h
+K |||D 2 (u)∇e|||0 Φ2 + K e0 Φ2 + K 3 −2δ (J0σ (e, e)) 2 Φ2
r
r
2
r
12
3
h2
+Kδ 3
(e20,∂E + ∇e · n20,∂E )
Φ2 ,
r 2 E∈Eh
e20,Ω ≤ −
where δ = 0 for conforming meshes with triangles or tetrahedra, and δ = 1/2 in
general.
We complete the proof by integrating (5.8) over the time interval [0, T ], applying
the Cauchy–Schwarz inequality in L2 (0, T ), recalling the regularity bound (5.6), and
observing the fact that
(e, φΦ)(0) = (φe, Φ − Φ̂)(0)
h
h
≤ K e0 (0)Φ1 (0) ≤ K eL∞ (0,T ;L2 ) ΦL∞ (0,T ;H 1 ) .
r
r
5.2. An L2 (L2 ) error estimate for the time derivative of the concentration. To obtain an optimal L2 (L2 ) error estimate for the concentration, we need an
estimate for its time derivative.
DISCONTINUOUS GALERKIN FOR REACTIVE TRANSPORT
207
Theorem 3 (L2 (L2 ) error estimate for ct ). Let the assumptions in Theorem 1
hold. Then there exists a constant K, independent of h and r, such that
∂
1
(C DG − c)
+ |||D 2 (u)∇(C DG − c)|||L∞ (0,T ;L2 )
∂t
2
L (0,T ;L2 )
≤K
hμ−2
hμ−2
hμ−2
|||c|||L2 (0,T ;H s ) + K s−2 |||∂c/∂t|||L2 (0,T ;H s−1 ) + K s−5/2 |||c0 |||s−1 ,
s−3−δ
r
r
r
where μ = min(r + 1, s), r ≥ 1, s ≥ 2, δ = 0 for conforming meshes with triangles or
tetrahedra, and δ = 1/2 in general.
Proof. Let ξ, ξ I , and ξ A be defined by (4.1)–(4.3), respectively. Subtracting (3.8)
from (3.9), choosing w = ∂ξ A /∂t, and integrating the resultant equation over the time
interval [0, t], 0 < t ≤ T , we obtain
t
t
∂φξ A ∂ξ A
∂ξ A
(5.8)
,
+
;u
BS ξ A ,
∂t
∂t
∂t
0
0
A
t A
∂ξ
∂ξ
DG
; u, C
; u, c
=
L
−L
∂t
∂t
0
t
t
A
∂φξ I ∂ξ A
I ∂ξ
,
+
;u .
+
BS ξ ,
∂t
∂t
∂t
0
0
A simple manipulation breaks the bilinear form on the left-hide side of (5.8) into
nine components:
7
A
d
A ∂ξ
BS ξ ,
;u =
Ti + T8 + T9 ,
∂t
dt i=1
where
2
1 1
A
A
A
A
Ti :=
D(u)∇ξ · ∇ξ −
ξ u · ∇ξ −
q− ξA
2
2
Ω
i=1
E∈Eh E
E∈Eh E
−
D(u)∇ξ A · nγ ξ A +
ξ A∗ u · nγ ξ A
7
γ∈Γh
+
1
2
γ
γ∈Γh,out
γ∈Γh
γ
2 1 u · nγ ξ A + J0σ ξ A , ξ A ,
2
γ
∂ξ A
u · ∇ξ A ,
T8 :=
∂t
E
E∈Eh
∂ξ A∗
T9 := −
u · nγ ξ A .
γ ∂t
γ∈Γh
Consequently, the left-hand side of (5.8) may be written as
t
t
∂φξ A ∂ξ A
∂ξ A
,
+
;u
BS ξ A ,
∂t
∂t
∂t
0
0
2
t
t
t
7
7
∂
A
=
φξ
+
T
(t)
−
T
(0)
+
T
+
T9 .
i
i
8
∂t
0
0
0
0
i=1
i=1
208
SHUYU SUN AND MARY F. WHEELER
t ∂ √ A 2
It is easy to see that the terms 0 ∂t
φξ 0 , T1 (t), T3 (t), T6 (t), and T7 (t) are
nonnegative. By applying the Cauchy–Schwarz inequality and Theorem 1, the term
T2 (t) can be bounded as
1
|T2 (t)| ≤ |||D 2 (u)∇ξ A |||20 + Kξ A 20
1
≤ |||D 2 (u)∇ξ A |||20 + Kξ A 2L∞ (0,T ;L2 )
1
≤ |||D 2 (u)∇ξ A |||20 + KRs2 ,
where
Rs :=
hμ−1
hμ−1
2 (0,T ;H s ) +
|||c|||
(|||∂c/∂t|||L2 (0,T ;H s−1 ) + |||c0 |||s−1 ).
L
rs−1−δ
rs−1
Recalling the definition of the penalty term and applying the Cauchy–Schwarz and
inverse inequalities, we may bound the terms T4 and T5 :
|T4 (t)| ≤
1
h
D 2 (u)∇ξ A 20,∂E + J0σ (ξ A , ξ A )
2
K
r
E∈Eh
1
≤ |||D 2 (u)∇ξ A |||20 + J0σ (ξ A , ξ A ),
h
ξ A 20,∂E + J0σ (ξ A , ξ A ) ≤ Kξ A 20 + J0σ (ξ A , ξ A )
|T5 (t)| ≤ K
r2
E∈Eh
≤ KRs2 + J0σ (ξ A , ξ A ).
Applications of the approximation results and the continuity of the L2 projection give
7
i=1
|Ti (0)| ≤ K
h2μ−4
|||c0 |||2s−1 .
r2s−5
The Cauchy–Schwarz inequality and Theorem 1 imply
t ∂ξ A 2
1
T8 ≤ φ
+ |||D 2 (u)∇ξ A |||2L2 (0,T ;L2 )
∂t L2 (0,T ;L2 )
0
∂ξ A 2
≤ φ
+ KRs2 .
∂t L2 (0,T ;L2 )
An application of the Cauchy–Schwarz and inverse inequalities yields
t t
∂ξ A 2
T9 ≤ φ
+K
J0σ ξ A , ξ A
2
∂t
0
0
L (0,T ;L2 )
2
A
∂ξ ≤ + KRs2 .
φ ∂t 2
L (0,T ;L2 )
Collecting the above estimates, we conclude that the left-hide side of (5.8) has the
following lower bound:
DISCONTINUOUS GALERKIN FOR REACTIVE TRANSPORT
209
t
A
∂φξ A ∂ξ A
A ∂ξ
,
+
;u
BS ξ ,
∂t
∂t
∂t
0
0
2
− A 2
1 t
∂ φξ A + 1 |||D 12 (u)∇ξ A |||20 + 1
q ξ
≥
2 0 ∂t
3
2 Ω
0
2 1 1 +
u · nγ ξ A + J0σ ξ A , ξ A
2
3
γ
t
γ∈Γh,out
h2μ−4
|||c0 |||2s−1 .
r2s−5
The first integrand on the right-hand side of (5.8) may be bounded, by using the
Cauchy–Schwarz inequality and the Lipschitz continuity of the cut-off operator, as
A
A
∂ξ A
∂ξ
∂ξ
DG
; u, C
; u, c =
−L
r M(C DG ) − r (M(c))
L
∂t
∂t
∂t
Ω
2
∂ A 2
2
A
2
∂
≤ ∂t φξ + K ξ0 ≤ ∂t φξ + KRs .
−KRs2 − K
0
0
An easy application of the Cauchy–Schwarz inequality and approximation results
yields the following estimate for the second integrand:
∂φξ I ∂ξ A
,
∂t
∂t
2
I 2
2μ−2
∂ A 2
+ K ∂ξ ≤ ∂ φξ A + K h
≤ φξ
|||ct |||2s−1 .
∂t
∂t ∂t
r2s−2
0
0
0
The third integrand may be decomposed into eight parts:
∂ξ A
BS ξ I ,
;u
∂t
∂ξ A
∂ξ A
∂ξ A
−
−
D(u)∇ξ I · ∇
ξI u · ∇
q− ξI
=
∂t
∂t
∂t
Ω
E∈Eh E
E∈Eh E
∂ξ A
∂ξ A
I
−
−
D(u)∇
D(u)∇ξ · nγ
· nγ ξ I
∂t
∂t
γ∈Γh γ
γ∈Γh γ
A
∂ξ
∂ξ A
∂ξ A
+
+ J0σ ξ I ,
+
ξ I∗ u · nγ
u · nγ ξ I
∂t
∂t
∂t
γ
γ
γ∈Γh
=:
8
γ∈Γh,out
Si .
i=1
The terms S3 and S8 are bounded by applying the Cauchy–Schwarz inequality and
approximation results:
2μ
∂ A 2
+ K h |||c|||2s ,
|S3 | ≤ φξ
∂t
r2s
0
|S8 | ≤ J0σ ξ A , ξ A + KJ0σ ξ I , ξ I
h2μ−2
≤ J0σ ξ A , ξ A + K 2s−3 |||c|||2s .
r
210
SHUYU SUN AND MARY F. WHEELER
Applications of the Cauchy–Schwarz and inverse inequalities yield the following estimates for the remaining terms:
2μ−4
∂ A 2
+Kh
φξ
|||c|||2s ,
|S1 | + |S2 | + |S4 | + |S6 | + |S7 | ≤ ∂t
2s−6
r
0
2μ−4
∂ A 2
+Kh
|S5 | ≤ φξ
|||c|||2s .
∂t
r2s−7
0
For conforming meshes with triangles or tetrahedra, we can choose a continuous c to
force S5 = S8 = 0. Combining the bounds for the terms Si , we obtain
∂ξ A
;u
BS ξ ,
∂t
0
2
t
∂
h2μ−4
A
2
2
≤
φξ
∂t
+ RS + K r2s−6−2δ |||c|||L2 (0,T ;H s ) .
t
I
0
0
By back-substituting the estimates into (5.8), we conclude that
2
t
∂
− A 2
1
A
A 2
q ξ
2
∂t φξ + |||D (u)∇ξ |||0 +
0
Ω
0
2
+
u · nγ ξ A + J0σ ξ A , ξ A
γ∈Γh,out
γ
h2μ−4
h2μ−4
|||c0 |||2s−1 + KRs2
r2s−5
h2μ−4
h2μ−4
h2μ−2
≤ K 2s−6−2δ |||c|||2L2 (0,T ;H s ) + K 2s−5 |||c0 |||2s−1 + K 2s−2 |||ct |||2L2 (0,T ;H s−1 ) .
r
r
r
≤K
r2s−6−2δ
|||c|||2L2 (0,T ;H s ) + K
The theorem follows from the triangle inequality, approximation results, and
(4.7).
5.3. Face error estimates. We also need an error estimate on element faces in
order to apply the parabolic lift lemma.
Theorem 4 (face error estimates). Let the assumptions in Theorem 1 hold. Then
there exists a constant K, independent of h and r, such that
C DG − c
2 2
L (0,T ;L2 (∂E))
12
+
E∈Eh
≤K
∇ C DG − c · n∂E 2 2
L (0,T ;L2 (∂E))
12
E∈Eh
h
h
|||c|||L2 (0,T ;H s ) + K s−2 |||∂c/∂t|||L2 (0,T ;H s−1 ) + |||c0 |||s−1 ,
rs−2−δ
r
μ− 32
μ− 32
where μ = min(r + 1, s), r ≥ 1, s ≥ 2, δ = 0 for conforming meshes with triangles or
tetrahedra, and δ = 1/2 in general.
Proof. As the first term can be bounded similarly with even sharper estimates, we
only present the estimation of the second term, which can be obtained by
applying the triangle and inverse inequalities, recalling Theorem 1 and using the
211
DISCONTINUOUS GALERKIN FOR REACTIVE TRANSPORT
approximation results:
∇ C DG − c · n∂E 2 2
L (0,T ;L2 (∂E))
E∈Eh
∇ C DG − ĉ 2 2
L (0,T ;L2 (∂E))
≤
12
12
E∈Eh
≤
≤
r
1
h2
h
E∈Eh
r
1
2
≤K
∇ C DG − ĉ 2 2
L (0,T ;L2 (E))
∇ C DG − c 2 2
L (0,T ;L2 (E))
+
12
2
∇ (ĉ − c)L2 (0,T ;L2 (∂E))
E∈Eh
3
+K
12
E∈Eh
12
+K
hμ− 2
3
rs− 2
|||c|||L2 (0,T ;H s )
r hμ−1
|||c|||L2 (0,T ;H s )
1
h 2 rs−1
3
h
hμ− 2 2 (0,T ;H s ) + K
|||∂c/∂t|||L2 (0,T ;H s−1 ) + |||c0 |||s−1 .
|||c|||
L
rs−2−δ
rs−2
μ− 32
5.4. An L2 (L2 ) error estimate for the concentration.
Theorem 5 (L2 (L2 ) error estimate for c). Let the assumptions in Theorem 1
hold. Then there exists a constant K, independent of h and r, such that
DG
C
(5.9)
− c
L2 (0,T ;L2 )
≤K
hμ
r
|||c|||L2 (0,T ;H s ) + K
s−1−δ
hμ
hμ
2 (0,T ;H s−1 ) + K
|||∂c/∂t|||
|||c0 |||s−1 ,
L
rs−δ
rs−1/2
where μ = min(r + 1, s), r ≥ 1, s ≥ 2, δ = 0 for conforming meshes with triangles or
tetrahedra, and δ = 1/2 in general.
Proof. We recall the concentration error ξ in (4.1), and the error equation:
∂φξ
, w + B(ξ, w; u) = L w; u, C DG − L (w; u, c)
∀w ∈ Dr (Eh ) .
∂t
We define
a(x, t) =
⎧
r M(C DG (x,t)))−r(M(c(x,t)))
⎪
⎨ − (
C DG (x,t)−c(x,t)
⎪
⎩
if C DG (x, t) − c(x, t) = 0,
if C DG (x, t) − c(x, t) = 0.
0
Consequently, we have L(w; u, C DG ) − L(w; u, c) = −(aξ, ω). Noting the fact that
a ∈ L∞ (0, T ; L∞ ) ⊂ L2 (0, T ; L∞ ) and recalling Theorems 1, 3, and 4, we obtain (5.9)
by applying the parabolic lift argument of Lemma 5.
6. Optimal estimates in negative norms for the symmetric scheme.
6.1. Error estimates in terms of linear functionals. We again assume noflow boundary conditions. Given a function f ∈ L2 (0, T ; L2 (Ω)), we consider a linear
functional F (·) of the following form:
T
F (c) =
c(x, t)f (x, t)dx dt.
0
Ω
Lemma 6 (parabolic lift). Let e ∈ L2 (0, T ; H 1 (Eh )) satisfy (5.1)–(5.2) and let the
s1 +2,1
((0, T ) × Ω), Dij ∈
assumptions in Theorem 1 hold. We further assume φ ∈ W∞
212
SHUYU SUN AND MARY F. WHEELER
s1 +1,0
s1
s1 ,0
s1 ,0
W∞
((0, T )×Ω), ui ∈ W∞
(Ω), a ∈ W∞
((0, T )×Ω), and q + ∈ W∞
((0, T )×Ω).
Then there exists a constant K, independent of h, r, e, and f, such that
hμ1 +1
hμ1 +2
|F (e)| ≤ K f L2 (0,T ;H s1 )
e
et L2 (0,T ;L2 )
∞ (0,T ;L2 ) +
L
rs1 +1
rs1 +2
12
T
1
hμ1 +1
hμ1 +1
σ
+ s1 +1 |||D 2 (u)∇e|||L2 (0,T ;L2 ) + s + 3 −2δ
J0 (e, e)
r
r 1 2
0
1 3
2
2
hμ1 + 2
2
,
eL2 (0,T ;L2 (∂E)) + ∇e · n∂E L2 (0,T ;L2 (∂E))
+ s +3 δ
r 1 2
E∈Eh
where μ1 = min(r − 1, s1 ), r ≥ 1, s1 ≥ 0, δ = 0 for conforming meshes with triangles
or tetrahedra, and δ = 1/2 in general.
Proof. We revisit the adjoint parabolic equation (5.3)–(5.5) with e replaced by
f . By applying Theorem 2 repeatedly, we obtain a unique solution Φ for (5.3)–(5.5)
satisfying
ΦL∞ (0,T ;H s1 +1 ) + ΦL2 (0,T ;H s1 +2 ) ≤ K f L2 (0,T ;H s1 ) .
(6.1)
We now consider the L2 (Ω) inner product (e, f ) at t ∈ (0, T ]:
(e, f ) =
(e, f )E
E∈Eh
=
E∈Eh
e, −
∂φΦ
∂t
+
E
E∈Eh
(e, ∇ · (−uΦ − D(u)∇Φ))E +
e, (a + q + )Φ
E
.
E∈Eh
Integrating by parts, applying the orthogonality condition (5.1) and observing that
D(u)∇Φ · n∂Ω = 0 on ∂Ω, ∇ · u = q, and [D(u)∇Φ · nγ ] = [Φ] = 0, we conclude that
d
∂e
(6.2) (e, f ) = − (e, φΦ) + φ , Φ − Φ̂ + ae, Φ − Φ̂ + BS e, Φ − Φ̂; u ,
dt
∂t
where we choose an interpolant Φ̂ ∈ Dr (Eh ) with element-wise optimal approximation
properties (3.4). Applying the Cauchy–Schwarz inequality and approximation results,
we obtain estimates for the second and third terms on the right-hand side of (6.2):
∂e
hμ1 +2
φ , Φ − Φ̂ ≤ K et 0 Φ − Φ̂0 ≤ K s1 +2 et 0 Φs1 +2 ,
∂t
r
hμ1 +2
ae, Φ − Φ̂ ≤ K aL∞ e0 Φ − Φ̂0 ≤ K s1 +2 aL∞ e0 Φs1 +2 .
r
Similar but tedious arguments, together with the inverse inequality and the existence
of continuous interpolants for conforming meshes with triangles or tetrahedra, yield
a bound for the fourth term:
1
hμ1 +1
hμ1 +1
BS e, Φ − Φ̂; u ≤ K s1 +1 |||D 2 (u)∇e|||0 Φs1 +2 + K s1 +1 e0 Φs1 +2
r
r
1
hμ1 +1
+K s + 3 −2δ (J0σ (e, e)) 2 Φs1 +2
r 1 2
1
3
2
2
hμ1 + 2
2
e0,∂E + ∇e · n∂E 0,∂E
+Kδ s + 3
Φs1 +2 .
r 1 2 E∈Eh
DISCONTINUOUS GALERKIN FOR REACTIVE TRANSPORT
213
Observing the fact that
T
d
(e, φΦ) (e, f ) +
|F (e) − (e, φΦ) (0)| = 0
dt
T
(e, f ) + d (e, φΦ)
≤
dt
0
and integrating (6.2) over the time interval [0, T ], we have
|F (e) − (e, φΦ) (0)|
≤ K ΦL2 (0,T ;H s1 +2 )
hμ1 +2
et L2 (0,T ;L2 )
rs1 +2
1
hμ1 +2
hμ1 +1
a
e
+
|||D 2 (u)∇e|||L2 (0,T ;L2 )
2
∞
∞
2
L (0,T ;L )
L (0,T ;L )
s
+2
s
+1
1
1
r
r
12
T
μ1 +1
μ1 +1
h
h
σ
+ s1 +1 eL∞ (0,T ;L2 ) + s + 3 −2δ
J0 (e, e)
r
r 1 2
0
1 3
2
2
hμ1 + 2
2
.
eL2 (0,T ;L2 (∂E)) + ∇e · n∂E L2 (0,T ;L2 (∂E))
+ s +3 δ
r 1 2
E∈Eh
+
The theorem follows from the regularity estimate (6.1) and the fact that
|(e, φΦ) (0)| = φe, Φ − Φ̂ (0)
hmin(r+1,s1 +1)
e(·, 0)0 Φ(·, 0)s1 +1
rs1 +1
hμ1 +1
≤ K s1 +1 eL∞ (0,T ;L2 ) ΦL∞ (0,T ;H s1 +1 ) .
r
≤K
Theorem 6 (linear functional estimates). Let the assumptions in Theorem 1
s1 +2,1
s1 +1,0
hold. In addition, we assume φ ∈ W∞
((0, T ) × Ω), Dij ∈ W∞
((0, T ) × Ω),
s1
+
s1 ,0
ui ∈ W∞ (Ω), q ∈ W∞ ((0, T )×Ω), and that the chemical reaction term has a linear
form r(c) = k0 + k1 c, where k0 = k0 (x, t) and k1 = k1 (x, t) are reaction parameters
s1 ,0
((0, T ) × Ω). Then there exists a constant K, independent of h, r, and
with k1 ∈ W∞
f, such that
F (C DG ) − F (c) ≤ K
hμ1 +μ
rs1 +s−1−δ
f L2 (0,T ;H s1 ) |||c|||L2 (0,T ;H s )
hμ1 +μ
f L2 (0,T ;H s1 ) |||∂c/∂t|||L2 (0,T ;H s−1 )
rs1 +s−δ
hμ1 +μ
+K s +s−1/2 f L2 (0,T ;H s1 ) |||c0 |||s−1 ,
r 1
+K
where μ = min(r + 1, s), μ1 = min(r − 1, s1 ), r ≥ 1, s ≥ 2, s1 ≥ 0, and δ = 0 for
conforming meshes with triangles or tetrahedra, and δ = 1/2 in general.
Proof. Recalling the concentration error ξ in (4.1) and defining a(x, t) = −k1 (x, t),
we obtain the error equation in the following form, provided that the cut-off constant
214
SHUYU SUN AND MARY F. WHEELER
M is chosen to be sufficiently large:
∂φξ
, w + BS (ξ, w; u) + (aξ, w) = 0 ∀w ∈ Dr (Eh )
∂t
∀t ∈ (0, T ].
We obtain the desired estimate by applying the parabolic lift of Lemma 6 together
with estimates in Theorems 1, 3, and 4.
6.2. Error estimates in negative norms. Assuming m is a positive integer,
we define the negative Sobolev norm ·H −m (Ω) in the usual way:
cH −m (Ω) =
sup
v∈C0∞ (Ω)\{0}
|(c, v)|
.
vH m (Ω)
Theorem 7 (estimates in negative norms). Let the assumptions in Theorem 1
m+2,1
m+1,0
hold. In addition, we assume φ ∈ W∞
((0, T ) × Ω), Dij ∈ W∞
((0, T ) × Ω),
m
+
m,0
ui ∈ W∞ (Ω), q ∈ W∞ ((0, T )×Ω), and that the chemical reaction term has a linear
form r(c) = k0 + k1 c, where k0 = k0 (x, t) and k1 = k1 (x, t) are reaction parameters
m,0
((0, T ) × Ω). Then there exists a constant K, independent of h and r,
with k1 ∈ W∞
such that
DG
hmin(r−1,m)+min(r+1,s)
C
− c
L2 (0,T ;H −m (Ω)) ≤ K
|||c|||L2 (0,T ;H s )
rm+s−1−δ
hmin(r−1,m)+min(r+1,s)
+K
|||∂c/∂t|||L2 (0,T ;H s−1 )
rm+s−δ
hmin(r−1,m)+min(r+1,s)
+K
|||c0 |||s−1 ,
rm+s−1/2
where r ≥ 1, s ≥ 2, m ≥ 0, and δ = 0 for conforming meshes with triangles or
tetrahedra, and δ = 1/2 in general.
Proof. The theorem follows directly from Theorem 6 and the definition of negative
norms.
7. Numerical examples. We consider the problem of (2.1)–(2.4) on a domain
Ω = (0, 10)2 without injection or extraction, i.e., q = 0, and with a reaction term
r = r(x, t) independent of the concentration c. The domain is divided into two
disjoint parts: Ω = Ω1 ∪ Ω2 with Ω1 = {(x, y) ∈ Ω : y < 3 + 0.4x}. The porosity
φ has a constant value of 0.1, and the tensor D is a constant diagonal tensor with
Dii = 1.0. We impose the velocities u = (−1, −0.4) in Ω1 and u = (0, 0) in Ω2 .
We choose r(x, t), cB , and c0 such that the equation has an analytical solution of
c = (1 + cos( π5 x) cos( π5 y))2−t/10 . The penalty parameter is chosen according to the
method presented in the next section. The coarsest mesh we take simply consists of
the two quadrilateral elements Ω1 and Ω2 . The simulation time interval is (0, 10],
and we use the backward Euler method for time integration with a uniform time step
Δt = 0.1.
7.1. Convergence of h-refinement. We solve the test case using OBB-DG,
NIPG, IIPG, and SIPG. We use polynomials of degree r = 2 and vary h by uniform refinements starting from the coarsest mesh. The convergence behaviors of h-refinement
in the norms of L2 (L2 ), L∞ (L2 ), and L2 (H 1 ) for NIPG are shown in Figure 7.1. It
is observed that the errors in all norms are O(1/n), where n is the number of degrees
of freedom. As n ∝ 1/h2 for two-dimensional spaces, the experimental convergences
DISCONTINUOUS GALERKIN FOR REACTIVE TRANSPORT
215
0
relative errors in various norms
10
Error in L2H1
Error in LinfL2
Error in L2L2
−1
10
1
−2
10
1
−3
10
1
10
2
3
10
10
4
10
number of degrees of freedom
Fig. 7.1. Convergence of h-refinement for NIPG.
0
relative errors in various norms
10
Error in L2H1
Error in LinfL2
Error in L2L2
−1
10
1
−2
10
1
1.5
−3
10
1
−4
10
1
10
2
10
3
10
4
10
number of degrees of freedom
Fig. 7.2. Convergence of h-refinement for SIPG.
confirm our theoretical estimates in L2 (H 1 ). In addition, the numerical results indicate that the errors in NIPG do not converge optimally in L∞ (L2 ) or L2 (L2 ). The
convergence behaviors of OBB-DG and IIPG (not shown) are nearly identical to those
of NIPG. However, unlike NIPG, OBB-DG, and IIPG, the symmetric scheme (SIPG)
possesses optimal convergence in all norms of L2 (L2 ), L∞ (L2 ), and L2 (H 1 ), as shown
evidently in Figure 7.2, which also validates the predictions from our parabolic lift
arguments.
7.2. Convergence of p-refinement. The test case is solved using the four
primal DGs on the coarsest mesh with polynomials of degrees r=1, 2, 3, . . . , 10.
Figure 7.3 illustrates the convergence behaviors of SIPG in the norms of L2 (L2 ),
L∞ (L2 ), and L2 (H 1 ), where the expected exponential convergence rates are achieved.
The exponential convergence patterns of OBB-DG, NIPG, and IIPG (not shown) are
very similar to those of SIPG. An interesting experimental observation, which is not
covered in previous theoretical sections, is that the DG methods with polynomials
of odd orders have better performance than those of even orders; this is especially
pronounced for OBB-DG.
216
SHUYU SUN AND MARY F. WHEELER
0
relative errors in various norms
10
Error in L2H1
Error in LinfL2
Error in L2L2
−1
10
−2
10
−3
10
−4
10
0
20
40
60
80
100
120
140
number of degrees of freedom
Fig. 7.3. Convergence of p-refinement for SIPG.
8. Discussion.
8.1. Penalty parameters for SIPG. Numerical experiments indicate that
careful implementations of the penalty terms are crucial to SIPG: not only are the
penalty terms necessary for the convergence of SIPG, but also choices of penalty parameters significantly influence the performance of SIPG. Small penalty parameters
might result in divergences of the schemes. On the other hand, very large parameters,
though ensuring the convergence theoretically, lead to a poor condition number for
the resultant linear system, causing numerical difficulties in practice.
Reinvestigating (4.5), we see that it is sufficient to choose σγ = O(|D|1/2 ), where
|D|1/2 and σ
= O(1), we have
| · | is a matrix norm. Letting σγ = σ
J0σ (c, w) =
σ
γ∈Γh
|D|
r2
hγ
[c][w].
γ
For most cases, we recommend σ
= 1. It is found that σ
chosen from (0.1, 10)
works well for many test cases. For cases where aspect ratios are very high and/or
dispersion-diffusion is highly anisotropic, it is found that the following choice generally
gives better results:
r2
J0σ (c, w) =
σ
|Dnγ |
[c][w],
hm,γ γ
γ∈Γh
where hm,γ = minE:γ∈E (meas(E)/meas(γ)).
8.2. Reference versus physical polynomial spaces. In the definition (3.1)
of the DG space Dr (Eh ), the local space Pr (E) is the set of polynomials defined over a
This distinction is unnecessary
physical element E, rather than a reference element E.
to E is affine.
when E is a triangle or tetrahedron because the transformation from E
But for a general quadrilateral or hexahedron, these two spaces are different. We apply
DG methods to the test case in section 7 using the uniform p-refinement in the coarsest mesh. Figure 8.1 provides the error ratio η = er L2 (0,T ;L2 (Ω)) /ef L2 (0,T ;L2 (Ω))
during the p-refinement, where er and ef denote the DG errors based on the reference
and physical spaces, respectively. Clearly, DG solutions based on physical spaces are
more accurate than those of reference spaces for high order approximations; this is
more significant for OBB-DG than for other primal DGs. This observation suggests
DISCONTINUOUS GALERKIN FOR REACTIVE TRANSPORT
217
error ratio (reference over physical)
3.5
OBB−DG
3
NIPG
IIPG
2.5
SIPG
2
1.5
1
0.5
0
20
40
60
80
100
120
number of degrees of freedom
Fig. 8.1. Comparison of reference versus physical polynomial spaces for DG methods (data of
NIPG, SIPG, and IIPG are nearly identical).
that physical polynomial spaces are preferred in p- and hp-implementations of DGs.
It is also noted (not shown) that the improvement of physical over reference spaces
to
is less pronounced on more refined meshes, because the transformation from E
E becomes closer to an affine mapping. Consequently, a choice of physical versus
reference spaces does not significantly impact h-versions of DGs.
9. Conclusions. Three primal DG methods with penalty have been analyzed
for solving reactive transport problems in porous media. The cut-off operator was
introduced in the DG formulations to ensure convergence for general nonlinear kinetic reactions. Error estimates in L2 (H 1 ) for the concentration were derived for
SIPG, NIPG, and IIPG, which are optimal in h and nearly optimal in p. In addition,
we established L2 (H 1 ) concentration error estimates on the element faces as well as
L2 (L2 ) estimates for time derivatives. A parabolic lift technique for SIPG has been
developed, which yields an h-optimal and nearly p-optimal error estimate in L2 (L2 ).
The same lift technique applied to general linear functionals gives optimal estimates
in negative norms. We have also numerically investigated the h- and p-convergence
behaviors of OBB-DG, NIPG, IIPG, and SIPG. It was demonstrated that OBB-DG,
IIPG, and NIPG possess h-optimal convergence rates in L2 (H 1 ), but lack the optimality in L2 (L2 ) and L∞ (L2 ), whereas SIPG performs h-optimally in the three
norms. For smooth problems, exponential convergence rates in p are achieved by the
four primal DG methods. In addition, it was observed that DGs with polynomials
of odd orders perform better than those of even orders. Implementations of penalty
terms are crucial to SIPG and a proper choice of the penalty parameter was proposed.
Another important issue in implementations is the selection of physical versus reference spaces, for which we recommended the physical polynomial spaces for p- and
hp-versions of DGs. As a future extension, we propose to study error estimates of
primal DG methods for transport coupled with kinetic and local-equilibrium reactions
and for multiphase flow in porous media.
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