web.kaust.edu.sa...DISCONTINUOUS GALERKIN FOR REACTIVE TRANSPORT 197 We consider the following boundary conditions for this problem: (2.2) (uc−D(u)∇c)·n = c Bu·n, (x,t) ∈ Γ

SIAM J. NUMER. ANAL. c© 2005 Society for Industrial and Applied MathematicsVol. 43, No. 1, pp. 195–219

SYMMETRIC AND NONSYMMETRICDISCONTINUOUS GALERKIN METHODS FORREACTIVE TRANSPORT IN POROUS MEDIA∗

SHUYU SUN† AND MARY F. WHEELER‡

Abstract. For solving reactive transport problems in porous media, we analyze three primaldiscontinuous 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 inL2(H1) are established, which are optimal in h and nearly optimal in p. We develop a parabolic lifttechnique for SIPG, which leads to h-optimal and nearly p-optimal error estimates in the L2(L2) andnegative norms. Numerical results validate these estimates. We also discuss implementation issuesincluding 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 discontinuouspiecewise polynomials to approximate the solutions of differential equations, withboundary conditions and interelement continuity weakly imposed through bilinearforms. Even though they often have larger numbers of degrees of freedom than con-forming approaches, DG methods have recently gained popularity for a number ofattractive features [19, 3, 4, 23, 27, 11, 25, 26, 9, 18, 15]: (1) they are element-wiseconservative; (2) they support general nonconforming spaces including unstructuredmeshes, nonmatching grids and variable degrees of local approximations, thus allow-ing efficient h-, p-, and hp-adaptivities; (3) they tend to have localized errors, allowingsharp a posteriori error indicators and effective adaptivities; (4) they have less nu-merical diffusion; (5) they treat rough coefficient problems and effectively capturediscontinuities in solutions; (6) they are robust and nonoscillatory in the presenceof high gradients; (7) with appropriate meshing, they are capable of delivering ex-ponential rates of convergences; (8) they have excellent parallel efficiency since datacommunications are relatively local; (9) for time-dependent problems in particular,their mass matrices are block diagonal, providing substantial computational advan-tages if explicit time integrations are used. In addition, by a simple extension fromthe average of the fluxes on element faces, DG can provide a continuous flux fielddefined over the entire domain, allowing efficient coupling with conforming methods.

Numerical modeling of reactive transport in porous media has important appli-cations in hydrology, earth sciences, environmental protection, oil recovery, chemical

∗Received by the editors November 2, 2002; accepted for publication (in revised form) February25, 2005; published electronically May 27, 2005. This research was partially supported by NationalScience 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 Texasat Austin, 201 E. 24th St. ACE 5.324, Austin, TX 78712 ([email protected]).

195

196 SHUYU SUN AND MARY F. WHEELER

industry, and biomedical engineering. Realistic simulations for simultaneous advec-tion, 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 adap-tive DG can effectively capture moving concentration fronts in reactive transport[31, 33, 36, 32, 29]. A posteriori error estimates of DG for reactive transport prob-lems have been derived in the L2(L2) [32] and L2(H1) norms [35]. In addition, DGhas 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–Babuska–Baumann DG(OBB-DG) formulation [19], symmetric interior penalty Galerkin (SIPG) [38], non-symmetric interior penalty Galerkin (NIPG) [23, 21], and incomplete interior penaltyGalerkin (IIPG) [12, 29]. In this paper, we analyze the three primal DG methodswith penalty, i.e., SIPG, NIPG, and IIPG, for solving reactive transport problems inporous media. The primal DG method without penalty, i.e., the OBB-DG scheme,has been analyzed for reactive transport problems elsewhere [22]. In the followingsection, we describe the modeling equations. The DG schemes are introduced in sec-tion 3. Section 4 contains the L2(H1) error analysis for SIPG, NIPG, and IIPG. Insection 5, a parabolic lift technique is developed, and an L2(L2) error analysis forSIPG is conducted. Optimal negative norm estimates are derived in section 6. Insection 7, we present numerical studies of h- and p-convergences for the four primalDG schemes. In section 8, we discuss choices of penalty parameters as well as DGimplementations using reference versus physical polynomial spaces. Conclusions aregiven in the last section.

2. Governing equations. For convenience of presentation, we consider reactivetransport 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 similararguments. 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) withboundary ∂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 simulationtime. The classical advection-diffusion-reaction equation in porous media is given by

∂φc

∂t+ ∇ · (uc− D(u)∇c) = qc∗ + r(c), (x, t) ∈ Ω × (0, T ],(2.1)

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, uniformlybounded above and below by positive numbers; D(u) is the dispersion-diffusion tensorand 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 totalflow rate q is a sum of sources (injection) and sinks (extraction); c∗ is the injectedconcentration 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:

(uc− D(u)∇c) · n = cBu · n, (x, t) ∈ Γin × (0, T ],(2.2)

(−D(u)∇c) · n = 0, (x, t) ∈ Γout × (0, T ],(2.3)

where cB is the inflow concentration. The initial concentration is specified by

c(x, 0) = c0(x), x ∈ Ω.(2.4)

3. Discontinuous Galerkin schemes.

3.1. Notation. Let Eh be a family of nondegenerate, quasi-uniform and possiblynonconforming partitions of Ω composed of triangles or quadrilaterals if d = 2, ortetrahedra, prisms, or hexahedra if d = 3. The nondegeneracy requirement (alsocalled regularity) is that the element is convex, and that there exists ρ > 0 suchthat if hj is the diameter of Ej ∈ Eh, then each of the subtriangles (for d = 2) orsubtetrahedra (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 forall Ej ∈ Eh, where h is the maximum diameter of all elements. We assume that noelement 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 unitnormal vector nγ is chosen. The sets of all edges or faces on Γout and on Γin for Ehare denoted by Γh,out and Γh,in, respectively, for which the normal vector nγ coincideswith the outward unit normal vector.

We now define the average and jump for φ ∈ Hs(Eh), s > 1/2. Let Ei, Ej ∈ Ehand γ = ∂Ei ∩ ∂Ej ∈ Γh with nγ exterior to Ei. We denote

{φ} :=1

2((φ|Ei

)|γ + (φ|Ej )|γ), [φ] := (φ|Ei)|γ − (φ|Ej )|γ .

The upwind value of a concentration c∗|γ is defined as

c∗|γ :=

{c|Ei if u · nγ ≥ 0,

c|Ejif u · nγ < 0.

We denote by ‖·‖m,R the usual Sobolev norm over a domain R [1]. The Sobolevnorm ‖·‖m,Ω over the entire domain Ω is also denoted simply by ‖·‖m. For s ≥ 0, wedefine the broken Sobolev space

Hs(Eh) := {φ ∈ L2(Ω) : φ|E ∈ Hs(E), E ∈ Eh}.

One can show that Hs(Eh) is a normed linear space with its norm defined by

‖φ‖Hs(Eh) :=

( ∑E∈Eh

‖φ‖2s,E

)1/2

.

Following the tradition, we also use the notation ||| · |||s to denote the broken norm‖·‖Hs(Eh). For a given normed space X and a number p ≥ 1, we define

Lp(0, T ;X) := {φ : φ(t) ∈ X, ‖φ‖X ∈ Lp(0, T )}.

The space Lp(0, T ;X) is also a normed linear space with its norm given by

‖φ‖Lp(0,T ;X) := ‖(‖φ‖X)‖Lp(0,T ).


The broken norm ‖·‖Lp(0,T ;Hs(Eh)) is also written as ||| · |||Lp(0,T ;Hs) in the triple bar

notation. We denote by (·, ·)R the inner product in (L2(R))d or L2(R) over a domainR. The inner product (·, ·)Ω over the entire domain Ω is also denoted simply by (·, ·).We also need the space W r,s

∞ and its norm:

W r,s∞ ((0, T ) × Ω) := {f ∈ L2((0, T ) × Ω) : ‖f‖W r,s

∞ < ∞},

‖f‖W r,s∞ :=

∑|α|≤r, β≤s

ess sup(0,T )×Ω(|Dαxf | + |Dβ

t f |).

The discontinuous finite element space is taken to be

Dr(Eh) := {φ ∈ L2(Ω) : φ|E ∈ Pr(E), E ∈ Eh},(3.1)

where Pr(E) denotes the space of polynomials of (total) degree less than or equal tor on E. Note that we present hp-results in this paper for the local space Pr, but theresults 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, wecan 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) isuniformly Lipschitz continuous with a Lipschitz constant of one; that is,

‖M(c) −M(w)‖L∞(Ω) ≤ ‖c− w‖L∞(Ω).(3.3)

We use the following hp-approximation results, which can be proved using thetechniques in [6, 5]. Let E ∈ Eh and φ ∈ Hs(E). Then there exists a constant K,independent of φ, r, and hE , and a sequence of zhr ∈ Pr(E), r = 1, 2, . . . , such that⎧⎪⎪⎨⎪⎪⎩

∥∥φ− zhr∥∥q,E

≤ Khμ−qE

rs−q‖φ‖s,E , 0 ≤ q < μ,

∥∥φ− zhr∥∥q,∂E

≤ Khμ−q− 1

2

E

rs−q− 12

‖φ‖s,E , 0 ≤ q < μ− 12 ,

(3.4)

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⎧⎪⎪⎨⎪⎪⎩

‖Dqv‖0,∂E ≤ Kr

h1/2E

‖Dqv‖E , q ≥ 0,

‖Dq+1v‖0,E ≤ Kr2

hE‖Dqv‖0,E , q ≥ 0.

(3.5)

3.2. Continuous-in-time DG schemes. We introduce a bilinear form:

B(c, w;u) :=∑E∈Eh

∫E

(D(u)∇c− cu) · ∇w −∫

Ω

cq−w

−∑γ∈Γh

∫γ

{D(u)∇c · nγ}[w] − sform

∑γ∈Γh

∫γ

{D(u)∇w · nγ}[c]

+∑γ∈Γh

∫γ

c∗u · nγ [w] +∑

γ∈Γh,out

∫γ

cu · nγw + Jσ0 (c, w).


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 itis 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 penaltyterm Jσ

0 (c, w) is formulated:

Jσ0 (c, w) :=

∑γ∈Γh

r2σγ

hγ

∫γ

[c][w],(3.6)

where σ is a discrete positive function that takes the constant value σγ on the edge orface 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 definea linear functional:

L(w;u, c) :=

∫Ω

r(M(c))w +

∫Ω

cwq+w −

∑γ∈Γh,in

∫γ

cBu · nγw.(3.7)

The reactive transport problem can be stated in the following equivalent weakformulation.

Lemma 2 (weak formulation). If c is a solution of (2.1)–(2.3) and c is essentiallybounded, then c satisfies(

∂φc

∂t, w

)+ B(c, w;u) = L(w;u, c)(3.8)

∀w ∈ Hs(Eh), s >3

2∀t ∈ (0, T ],

provided that the constant M for the cut-off operator is sufficiently large.Proof. Let w ∈ Hs(Eh), s > 3/2 and E ∈ Eh. Multiplying (2.1) by w, integrating

over E, and then integrating by parts, we observe(∂φc

∂t, w

)E

−∫E

(uc− D(u)∇c) · ∇w +

∫∂E

(uc− D(u)∇c) · n∂Ew

=

∫E

qc∗w + r(c)w.

Summing it over all elements in Eh, noting the fact that the traces of the concentrationand its normal flux are continuous across element faces, and applying the boundaryconditions, we obtain the desired result.

The continuous-in-time DG approximation CDG(·, t) ∈ Dr(Eh) to the solution of(2.1)–(2.4) is defined by(

∂φCDG

∂t, w

)+ B(CDG, w;u) = L(w;u, CDG)(3.9)

∀w ∈ Dr(Eh) ∀t ∈ (0, T ],

(φCDG, w) = (φc0, w) ∀w ∈ Dr(Eh), t = 0.(3.10)

As a valuable property, DG schemes possess element-wise mass conservation.OBB-DG satisfies local conservation strictly, whereas SIPG, NIPG, and IIPG are


locally conservative if the concentration jump term is considered as part of the com-puted diffusive flux:

Lemma 3 (local mass balance). The approximation of the concentration satisfieson each element E the following local mass balance equation:∫

E

∂φCDG

∂t−∫∂E\∂Ω

{D(u)∇CDG · n∂E} +

∫∂E

CDG∗u · n∂E(3.11)

+∑

γ⊂∂E\∂Ω

r2σγ

hγ

∫γ

(CDG|E − CDG|Ω\E)

=

∫E

CDG∗q +

∫E

r(M(CDG)).

Proof. The relationship (3.11) follows immediately from the DG schemes by fixingan 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 Galerkinscheme (3.9) and (3.10) has a unique solution for t > 0.

Proof. We let {vi}Mi=1 be a basis of Dr(Eh) and write CDG =∑M

i=1 ζi(t)vi(x).Then (3.9) and (3.10) reduce to the following initial value problem:⎧⎨⎩A

dζ

dt= −Bζ + R(ζ),

Aζ(0) = b,

where the mass matrix A is block-diagonal, symmetric, and positive definite. From theproperties 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 differentialequations that ζ(t) exists and is unique for t > 0.

4. L2(H1) and L∞(L2) error estimates. Throughout the paper, we denoteby K a generic positive constant independent of h and r, and by ε a fixed positiveconstant that may be chosen arbitrarily small.

Theorem 1 (L2(H1) and L∞(L2) error estimates). Let c be the solution to(2.1)–(2.4), and assume c ∈ L2(0, T ;Hs(Eh)), ∂c/∂t ∈ L2(0, T ;Hs−1(Eh)), and c0 ∈Hs−1(Eh). We further assume that c, u and q are essentially bounded, that the reac-tion rate is a locally Lipschitz continuous function of c, and that the cut-off constantM and the penalty parameter σ0 are sufficiently large. Then there exists a constantK, independent of h and r, such that

‖CDG − c‖L∞(0,T ;L2) + |||D 12 (u)∇(CDG − c)|||L2(0,T ;L2)

+

(∫ T

0

Jσ0 (CDG − c, CDG − c)

) 12

≤ Khμ−1

rs−1−δ|||c|||L2(0,T ;Hs) + K

hμ−1

rs−1(|||∂c/∂t|||L2(0,T ;Hs−1) + |||c0|||s−1),

where μ = min(r + 1, s), r ≥ 1, s ≥ 2, δ = 0 for conforming meshes with triangles ortetrahedra, and δ = 1/2 in general.

Proof. We let c ∈ Dr(Eh) be an interpolant of concentration c such that thehp-results (3.4) hold, and define

ξ = CDG − c,(4.1)


ξI = c− c,(4.2)

ξA = CDG − c = ξ + ξI .(4.3)

Subtracting the weak formulation (3.8) from the DG scheme (3.9), choosing w = ξA,we obtain (

∂φξA

∂t, ξA

)+ B(ξA, ξA;u)(4.4)

= L(ξA;u, CDG) − L(ξA;u, c) +

(∂φξI

∂t, ξA

)+ B(ξI , ξA;u).

The first term of the error equation (4.4) may be written in a time derivative ofan L2 norm: (

∂φξA

∂t, ξA

)=

1

2

d

dt

∥∥∥√φξA∥∥∥2

0,Ω.

We expand the second term of (4.4) as

B(ξA, ξA;u) =∑E∈Eh

∫E

(D(u)∇ξA − ξAu) · ∇ξA −∫

Ω

q−(ξA)2

−(1 + sform)∑γ∈Γh

∫γ

{D(u)∇ξA · nγ}[ξA]

+∑γ∈Γh

∫γ

ξA∗u · nγ [ξA] +∑

γ∈Γh,out

∫γ

u · nγ(ξA)2 + Jσ0 (ξA, ξA).

Integrating the advection term by parts, we observe

−∑E∈Eh

∫E

ξAu · ∇ξA

= −1

2

∑E∈Eh

∫E

u · ∇(ξA)2 = −1

2

∑E∈Eh

∫∂E

u · n∂E(ξA)2 +1

2

∑E∈Eh

∫E

q(ξA)2

= −1

2

∑γ∈Γh

∫γ

u · nγ [(ξA)2] − 1

2

∑γ∈Γh,in∪Γh,out

∫γ

u · nγ(ξA)2 +1

2

∑E∈Eh

∫E

q(ξA)2.

In addition, noting that [c2] = 2{c}[c] and (c∗ − {c})sign(u · n) = [c]/2, we have

B(ξA, ξA;u) = |||D 12 (u)∇ξA|||20 +

1

2

∫Ω

|q|(ξA)2 − T0 + Jσ0 (ξA, ξA)

+1

2

∑γ∈Γh

∫γ

|u · nγ |[ξA]2 +1

2

∑γ∈Γh,in∪Γh,out

∫γ

|u · nγ |(ξA)2,

where T0 is defined by

T0 := (1 + sform)∑γ∈Γh

∫γ

{D(u)∇ξA · nγ}[ξA].


If the penalty parameter σ0 is chosen to be sufficiently large, we may bound T0 byapplying the Cauchy–Schwarz and inverse inequalities:

T0 ≤ h

Kr2

∑E∈Eh

∥∥∥D 12 (u)∇ξA · n∂E

∥∥∥2

0,∂E+

Kr2

h

∑γ∈Γh

‖[ξA]‖20,γ(4.5)

≤ 1

2|||D 1

2 (u)∇ξA|||20 +1

2Jσ

0 (ξA, ξA).

The first two terms on the right-hand side of (4.4) may be estimated, by usingLemma 1, as

L(ξA;u, CDG) − L(ξA;u, c) =

∫Ω

(r(M(CDG)) − r(M(c)))ξA

≤ K‖√φξA‖2

0 + K‖ξI‖20 ≤ K‖

√φξA‖2

0 + Kh2μ

r2s|||c|||2s.

We have a similar result for the third term:(∂φξI

∂t, ξA

)≤ K

∥∥∥∥∂ξI∂t

∥∥∥∥0

∥∥∥√φξA∥∥∥

0

≤ K∥∥∥√φξA

∥∥∥2

0+ K

∥∥∥∥∂ξI∂t

∥∥∥∥2

0

≤ K∥∥∥√φξA

∥∥∥2

0+ K

h2μ−2

r2s−2|||ct|||2s−1.

The fourth term on the right-hand side of (4.4) consists of eight pieces:

B(ξI , ξA;u)

=∑E∈Eh

∫E

D(u)∇ξI · ∇ξA −∑E∈Eh

∫E

ξIu · ∇ξA −∫

Ω

q−ξIξA

−∑γ∈Γh

∫γ

{D(u)∇ξI · nγ}[ξA] − sform

∑γ∈Γh

∫γ

{D(u)∇ξA · nγ}[ξI ]

+∑γ∈Γh

∫γ

ξI∗u · nγ [ξA] +∑

γ∈Γh,out

∫γ

u · nγξIξA + Jσ

0 (ξI , ξA)

=:

8∑i=1

Ti.

The Cauchy–Schwarz inequality and approximation results yield

T1 ≤ ε|||D 12 (u)∇ξA|||20 + K

h2μ−2

r2s−2|||c|||2s,

T2 ≤ ε|||D 12 (u)∇ξA|||20 + K

h2μ

r2s|||c|||2s,

T3 ≤ ε

∫Ω

|q−|(ξA)2 + Kh2μ

r2s|||c|||2s.


We bound the terms T4 and T5 by hiding a large constant in the penalty term and byusing the inverse inequality, respectively,

T4 ≤ εσ0r

2

h

∑γ∈Γh

‖[ξA]‖20,γ +

Kh

r2

∑E∈Eh

‖∇ξI · n∂E‖20,∂E

≤ εJσ0 (ξA, ξA) + K

h2μ−2

r2s−1|||c|||2s,

T5 ≤ εh

Kr2

∑E∈Eh

‖D 12 (u)∇ξA · n∂E‖2

0,∂E +Kr2

h

∑E∈Eh

‖ξI‖20,∂E

≤ ε|||D 12 (u)∇ξA|||20 + K

h2μ−2

r2s−3|||c|||2s.

Similar applications of the Cauchy–Schwarz inequality and approximation results give

T6 ≤ ε∑γ∈Γh

∫γ

|u · nγ |[ξA]2 + Kh2μ−1

r2s−1|||c|||2s,

T7 ≤ ε∑

γ∈Γh,out

∫γ

|u · nγ |(ξA)2 + Kh2μ−1

r2s−1|||c|||2s,

T8 ≤ εJσ0 (ξA, ξA) + K

h2μ−2

r2s−3|||c|||2s.

For conforming meshes with triangles or tetrahedra, we can choose a continu-ous approximation c to make the two terms T5 and T8 vanish. Substituting all theestimates into (4.4), we see that

d

dt‖√φξA‖2

0 + |||D 12 (u)∇ξA|||20 + Jσ

0 (ξA, ξA)(4.6)

≤ K‖√φξA‖2

0 + Kh2μ−2

r2s−2−2δ|||c|||2s + K

h2μ−2

r2s−2|||ct|||2s−1,

where δ = 0 for conforming meshes with triangles or tetrahedra, and δ = 1/2 ingeneral. Integrating (4.6) with respect to the time t, noting that

‖√φEA‖0(0) ≤ K

hμ−1

rs−1|||c0|||s−1,

and applying Gronwall’s inequality, we conclude that

‖√φξA‖L∞(0,T ;L2) + |||D 1

2 (u)∇ξA|||L2(0,T ;L2) + (

∫ T

0

Jσ0 (ξA, ξA))

12

≤ Khμ−1

rs−1−δ|||c|||L2(0,T ;Hs) + K

hμ−1

rs−1(|||∂c/∂t|||L2(0,T ;Hs−1) + |||c0|||s−1).

The theorem follows by applying the triangle inequality, the approximation resultsand the fact that

|||c|||L∞(0,T ;Hs−1) ≤ K|||ct|||L2(0,T ;Hs−1) + |||c0|||s−1.(4.7)

We remark that, in [22], L∞(L2) + L2(H1) error estimates for the OBB-DGdiffusion scheme applied to the transport problem established optimality in h andsuboptimality in p by 3/2. Here for SIPG, NIPG, and IIPG, we obtain optimality inh and p for conforming meshes with triangles and tetrahedra and a loss of 1/2 in p forgeneral grids. Obviously, penalty terms improve the provable p-optimality of DGs.


5. Optimal L2(L2) error estimates for the symmetric scheme. In this andfollowing sections, we restrict our attention to SIPG. The derivation in this sectionis motivated by the h-optimal L2 result for SIPG applied to an elliptic problem byWheeler [38] and the h-optimal L2(L2) result for continuous Galerkin methods appliedto a parabolic problem by Palmer [20]. See also the h-optimal L2(L2) result forcontinuous finite element modified methods of characteristics applied to a coupledsystem 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 withdiffusion term by Arnold [4, 3]. We first recall a theorem proved in [20, 17].

Theorem 2. Consider the parabolic equation:

∂φΦ

∂t+ ∇ · (uΦ − D∇Φ) + aΦ = f, x ∈ Ω, t ∈ (0, T ],

D∇Φ · n = 0, x ∈ ∂Ω, t ∈ (0, T ],

Φ = 0, x ∈ Ω, t = 0.

Assume that 0 < φ0 ≤ φ(t, x) ≤ φm, D is uniformly symmetric positive definiteand bounded from above, φ ∈ W 2,1

∞ ((0, T ) × Ω), Dij ∈ W 1,0∞ ((0, T ) × Ω), ui ∈ L∞(Ω)

(u being independent of time), a ∈ L2(0, T ;L∞(Ω)) and f ∈ L2(0, T ;L2(Ω)). Thenthere exists a unique solution Φ satisfying the above equation and the regularity boundsgiven by

‖Φ‖L∞(0,T ;H1) + ‖Φ‖L2(0,T ;H2) ≤ K‖f‖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 con-

ditions, though the result can be generalized. We make additional assumptions:φ ∈ W 2,1

∞ ((0, T ) × Ω), Dij ∈ W 1,0∞ ((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 ;H1(Eh))

satisfy (∂φe

∂t, w

)+ BS(e, w;u) + (ae, w) = 0 ∀w ∈ Dr(Eh) ∀t ∈ (0, T ],(5.1)

(φe,w) = 0 ∀w ∈ Dr(Eh), t = 0.(5.2)

In addition we let the assumptions in Theorem 1 hold. Then there exists a constantK, independent of h, r, and e, such that

‖e‖L2(0,T ;L2)

≤ Kh

r‖e‖L∞(0,T ;L2) + K

h2

r2‖et‖L2(0,T ;L2)

+Kh

r|||D 1

2 (u)∇e|||L2(0,T ;L2) + Kh

r32−2δ

(∫ T

0

Jσ0 (e, e)

) 12

+Kδh

32

r32

( ∑E∈Eh

(‖e‖2L2(0,T ;L2(∂E)) + ‖∇e · n∂E‖2

L2(0,T ;L2(∂E)))

) 12

,

where δ = 0 for conforming meshes with triangles or tetrahedra, and δ = 1/2 ingeneral.


Proof. Consider the backward or adjoint parabolic equation:

−∂φΦ

∂t+ ∇ · (−uΦ − D(u)∇Φ) + (a + q+)Φ = e, x ∈ Ω, t ∈ [0, T ),(5.3)

D(u)∇Φ · n∂Ω = 0, x ∈ ∂Ω, t ∈ [0, T ),(5.4)

Φ = 0, x ∈ Ω, t = T.(5.5)

Theorem 2 suggests a unique solution Φ for (5.3)–(5.5) satisfying

‖Φ‖L∞(0,T ;H1) + ‖Φ‖L2(0,T ;H2) ≤ K‖e‖L2(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 domainΩ, and then apply integration by parts to conclude that

‖e‖20 = − d

dt

∑E∈Eh

(e, φΦ)E +∑E∈Eh

(φ∂e

∂t,Φ

)E

+∑E∈Eh

((a− q−)e,Φ)E

+∑E∈Eh

(∇e,D(u)∇Φ)E −∑γ∈Γh

∫γ

{D(u)∇Φ · nγ}[e] −∑E∈Eh

(e,u∇ · Φ)E

= − d

dt(e, φΦ) +

(φ∂e

∂t,Φ

)+ (ae,Φ) + BS(e,Φ;u).

Applying the orthogonality condition (5.1), we obtain

‖e‖20 = − d

dt(e, φΦ) +

(φ∂e

∂t,Φ − Φ

)+ (ae,Φ − Φ) + BS(e,Φ − Φ;u),(5.7)

where Φ ∈ Dr(Eh) is an interpolant satisfying (3.4) element-wise. The second andthird terms on the right-hand side of (5.7) are bounded, by using the Cauchy–Schwarzinequality and approximation results, as(

φ∂e

∂t,Φ − Φ

)≤ K‖et‖0‖Φ − Φ‖0 ≤ K

h2

r2‖et‖0‖Φ‖2,

(ae,Φ − Φ) ≤ K‖a‖L∞‖e‖0‖Φ − Φ‖0 ≤ Kh2

r2‖a‖L∞‖e‖0‖Φ‖2.

The last term in (5.7) is composed of eight parts:

BS(e,Φ − Φ;u)

=∑E∈Eh

∫E

D(u)∇e · ∇(Φ − Φ) −∑E∈Eh

∫E

eu · ∇(Φ − Φ) −∫

Ω

q−e(Φ − Φ)

−∑γ∈Γh

∫γ

{D(u)∇e · nγ}[Φ − Φ] −∑γ∈Γh

∫γ

{D(u)∇(Φ − Φ) · nγ}[e]

+∑γ∈Γh

∫γ

e∗u · nγ [Φ − Φ] +∑

γ∈Γh,out

∫γ

eu · nγ(Φ − Φ) + Jσ0 (e,Φ − Φ)

=:8∑

i=1

Ti.


Once again, the approximation results and Cauchy–Schwarz inequality yield the esti-mates for the terms T1, T2, and T3:

T1 ≤ K|||D 12 (u)∇e|||0‖∇(Φ − Φ)‖0 ≤ K

h

r|||D 1

2 (u)∇e|||0‖Φ‖2,

T2 ≤ Kh

r‖e‖0‖Φ‖2,

T3 ≤ Kh2

r2‖e‖0‖Φ‖2.

The term T7 vanishes because of the assumed no-flow boundary condition. The re-maining terms in the bilinear form can be bounded by applying the Cauchy–Schwarzinequality on element faces:

T4 ≤ K∑E∈Eh

‖∇e · n∂E‖0,∂E‖Φ − Φ‖0,∂E ≤ Kh

32

r32

( ∑E∈Eh

‖∇e · n∂E‖20,∂E

) 12

‖Φ‖2,

T5 ≤∑γ∈Γh

‖{D(u)∇(Φ − Φ) · nγ}‖0,γ‖[e]‖0,γ ≤ Kh

r32

(Jσ0 (e, e))

12 ‖Φ‖2,

T6 ≤ Kh

32

r32

( ∑E∈Eh

‖e‖20,∂E

) 12

‖Φ‖2,

T8 ≤ (Jσ0 (e, e))

12 (Jσ

0 (Φ − Φ,Φ − Φ))12 ≤ K

h

r12

(Jσ0 (e, e))

12 ‖Φ‖2.

We note that, for conforming meshes with triangles or tetrahedra, terms T4, T6,and T8 vanish if we choose a continuous interpolant Φ. Substituting all the estimatesback into (5.7), we find that

‖e‖20,Ω ≤ − d

dt(e, φΦ) + K

h2

r2‖et‖0‖Φ‖2 + K

h2

r2‖a‖L∞‖e‖0‖Φ‖2

+Kh

r|||D 1

2 (u)∇e|||0‖Φ‖2 + Kh

r‖e‖0‖Φ‖2 + K

h

r32−2δ

(Jσ0 (e, e))

12 ‖Φ‖2

+Kδh

32

r32

( ∑E∈Eh

(‖e‖20,∂E + ‖∇e · n‖2

0,∂E)

) 12

‖Φ‖2,

where δ = 0 for conforming meshes with triangles or tetrahedra, and δ = 1/2 ingeneral.

We complete the proof by integrating (5.8) over the time interval [0, T ], applyingthe Cauchy–Schwarz inequality in L2(0, T ), recalling the regularity bound (5.6), andobserving the fact that

(e, φΦ)(0) = (φe,Φ − Φ)(0)

≤ Kh

r‖e‖0(0)‖Φ‖1(0) ≤ K

h

r‖e‖L∞(0,T ;L2)‖Φ‖L∞(0,T ;H1).

5.2. An L2(L2) error estimate for the time derivative of the concentra-tion. To obtain an optimal L2(L2) error estimate for the concentration, we need anestimate for its time derivative.


Theorem 3 (L2(L2) error estimate for ct). Let the assumptions in Theorem 1hold. Then there exists a constant K, independent of h and r, such that∥∥∥∥ ∂

∂t(CDG − c)

∥∥∥∥L2(0,T ;L2)

+ |||D 12 (u)∇(CDG − c)|||L∞(0,T ;L2)

≤ Khμ−2

rs−3−δ|||c|||L2(0,T ;Hs) + K

hμ−2

rs−2|||∂c/∂t|||L2(0,T ;Hs−1) + K

hμ−2

rs−5/2|||c0|||s−1,


Proof. Let ξ, ξI , and ξA be defined by (4.1)–(4.3), respectively. Subtracting (3.8)from (3.9), choosing w = ∂ξA/∂t, and integrating the resultant equation over the timeinterval [0, t], 0 < t ≤ T , we obtain∫ t

0

(∂φξA

∂t,∂ξA

∂t

)+

∫ t

0

BS

(ξA,

∂ξA

∂t;u

)(5.8)

=

∫ t

0

(L

(∂ξA

∂t;u, CDG

)− L

(∂ξA

∂t;u, c

))+

∫ t

0

(∂φξI

∂t,∂ξA

∂t

)+

∫ t

0

BS

(ξI ,

∂ξA

∂t;u

).

A simple manipulation breaks the bilinear form on the left-hide side of (5.8) intonine components:

BS

(ξA,

∂ξA

∂t;u

)=

(d

dt

7∑i=1

Ti

)+ T8 + T9,

where

7∑i=1

Ti :=1

2

∑E∈Eh

∫E

D(u)∇ξA · ∇ξA −∑E∈Eh

∫E

ξAu · ∇ξA − 1

2

∫Ω

q−(ξA

)2−

∑γ∈Γh

∫γ

{D(u)∇ξA · nγ

} [ξA

]+

∑γ∈Γh

∫γ

ξA∗u · nγ

[ξA

]+

1

2

∑γ∈Γh,out

∫γ

u · nγ

(ξA

)2+

1

2Jσ

0

(ξA, ξA

),

T8 :=∑E∈Eh

∫E

∂ξA

∂tu · ∇ξA,

T9 := −∑γ∈Γh

∫γ

∂ξA∗

∂tu · nγ

[ξA

].

Consequently, the left-hand side of (5.8) may be written as∫ t

0

(∂φξA

∂t,∂ξA

∂t

)+

∫ t

0

BS

(ξA,

∂ξA

∂t;u

)=

∫ t

0

∥∥∥∥ ∂

∂t

√φξA

∥∥∥∥2

0

+

7∑i=1

Ti(t) −7∑

i=1

Ti(0) +

∫ t

0

T8 +

∫ t

0

T9.


It is easy to see that the terms∫ t

0‖ ∂∂t

√φξA‖2

0, T1(t), T3(t), T6(t), and T7(t) arenonnegative. By applying the Cauchy–Schwarz inequality and Theorem 1, the termT2(t) can be bounded as

|T2(t)| ≤ ε|||D 12 (u)∇ξA|||20 + K‖ξA‖2

0

≤ ε|||D 12 (u)∇ξA|||20 + K‖ξA‖2

L∞(0,T ;L2)

≤ ε|||D 12 (u)∇ξA|||20 + KR2

s,

where

Rs :=hμ−1

rs−1−δ|||c|||L2(0,T ;Hs) +

hμ−1

rs−1(|||∂c/∂t|||L2(0,T ;Hs−1) + |||c0|||s−1).

Recalling the definition of the penalty term and applying the Cauchy–Schwarz andinverse inequalities, we may bound the terms T4 and T5:

|T4(t)| ≤ε

K

∑E∈Eh

h

r2‖D 1

2 (u)∇ξA‖20,∂E + εJσ

0 (ξA, ξA)

≤ ε|||D 12 (u)∇ξA|||20 + εJσ

0 (ξA, ξA),

|T5(t)| ≤ K∑E∈Eh

h

r2‖ξA‖2

0,∂E + εJσ0 (ξA, ξA) ≤ K‖ξA‖2

0 + εJσ0 (ξA, ξA)

≤ KR2s + εJσ

0 (ξA, ξA).

Applications of the approximation results and the continuity of the L2 projection give

7∑i=1

|Ti(0)| ≤ Kh2μ−4

r2s−5|||c0|||2s−1.

The Cauchy–Schwarz inequality and Theorem 1 imply

∣∣∣∣∫ t

0

T8

∣∣∣∣ ≤ ε

∥∥∥∥√φ∂ξA

∂t

∥∥∥∥2

L2(0,T ;L2)

+ |||D 12 (u)∇ξA|||2L2(0,T ;L2)

≤ ε

∥∥∥∥√φ∂ξA

∂t

∥∥∥∥2

L2(0,T ;L2)

+ KR2s.

An application of the Cauchy–Schwarz and inverse inequalities yields

∣∣∣∣∫ t

0

T9

∣∣∣∣ ≤ ε

∥∥∥∥√φ∂ξA

∂t

∥∥∥∥2

L2(0,T ;L2)

+ K

∫ t

0

Jσ0

(ξA, ξA

)≤ ε

∥∥∥∥√φ∂ξA

∂t

∥∥∥∥2

L2(0,T ;L2)

+ KR2s.

Collecting the above estimates, we conclude that the left-hide side of (5.8) has thefollowing lower bound:


∫ t

0

(∂φξA

∂t,∂ξA

∂t

)+

∫ t

0

BS

(ξA,

∂ξA

∂t;u

)≥ 1

2

∫ t

0

∥∥∥∥ ∂

∂t

√φξA

∥∥∥∥2

0

+1

3|||D 1

2 (u)∇ξA|||20 +1

2

∫Ω

∣∣q−∣∣ (ξA)2+

1

2

∑γ∈Γh,out

∫γ

u · nγ

(ξA

)2+

1

3Jσ

0

(ξA, ξA

)−KR2

s −Kh2μ−4

r2s−5|||c0|||2s−1.

The first integrand on the right-hand side of (5.8) may be bounded, by using theCauchy–Schwarz inequality and the Lipschitz continuity of the cut-off operator, as

L

(∂ξA

∂t;u, CDG

)− L

(∂ξA

∂t;u, c

)=

∫Ω

(r(M(CDG)

)− r (M(c))

) ∂ξA∂t

≤ ε

∥∥∥∥ ∂

∂t

√φξA

∥∥∥∥2

0

+ K ‖ξ‖20 ≤ ε

∥∥∥∥ ∂

∂t

√φξA

∥∥∥∥2

0

+ KR2s.

An easy application of the Cauchy–Schwarz inequality and approximation resultsyields the following estimate for the second integrand:(

∂φξI

∂t,∂ξA

∂t

)≤ ε

∥∥∥∥ ∂

∂t

√φξA

∥∥∥∥2

0

+ K

∥∥∥∥∂ξI∂t

∥∥∥∥2

0

≤ ε

∥∥∥∥ ∂

∂t

√φξA

∥∥∥∥2

0

+ Kh2μ−2

r2s−2|||ct|||2s−1.

The third integrand may be decomposed into eight parts:

BS

(ξI ,

∂ξA

∂t;u

)=

∑E∈Eh

∫E

D(u)∇ξI · ∇∂ξA

∂t−

∑E∈Eh

∫E

ξIu · ∇∂ξA

∂t−∫

Ω

q−ξI∂ξA

∂t

−∑γ∈Γh

∫γ

{D(u)∇ξI · nγ

} [∂ξA∂t

]−

∑γ∈Γh

∫γ

{D(u)∇∂ξA

∂t· nγ

}[ξI]

+∑γ∈Γh

∫γ

ξI∗u · nγ

[∂ξA

∂t

]+

∑γ∈Γh,out

∫γ

u · nγξI ∂ξ

A

∂t+ Jσ

0

(ξI ,

∂ξA

∂t

)

=:

8∑i=1

Si.

The terms S3 and S8 are bounded by applying the Cauchy–Schwarz inequality andapproximation results:

|S3| ≤ ε

∥∥∥∥ ∂

∂t

√φξA

∥∥∥∥2

0

+ Kh2μ

r2s|||c|||2s,

|S8| ≤ εJσ0

(ξA, ξA

)+ KJσ

0

(ξI , ξI

)≤ εJσ

0

(ξA, ξA

)+ K

h2μ−2

r2s−3|||c|||2s.


Applications of the Cauchy–Schwarz and inverse inequalities yield the following esti-mates for the remaining terms:

|S1| + |S2| + |S4| + |S6| + |S7| ≤ ε

∥∥∥∥ ∂

∂t

√φξA

∥∥∥∥2

0

+ Kh2μ−4

r2s−6|||c|||2s,

|S5| ≤ ε

∥∥∥∥ ∂

∂t

√φξA

∥∥∥∥2

0

+ Kh2μ−4

r2s−7|||c|||2s.

For conforming meshes with triangles or tetrahedra, we can choose a continuous c toforce S5 = S8 = 0. Combining the bounds for the terms Si, we obtain∫ t

0

BS

(ξI ,

∂ξA

∂t;u

)≤ ε

∫ t

0

∥∥∥∥ ∂

∂t

√φξA

∥∥∥∥2

0

+ εR2S + K

h2μ−4

r2s−6−2δ|||c|||2L2(0,T ;Hs).

By back-substituting the estimates into (5.8), we conclude that

∫ t

0

∥∥∥∥ ∂

∂t

√φξA

∥∥∥∥2

0

+ |||D 12 (u)∇ξA|||20 +

∫Ω

∣∣q−∣∣ (ξA)2+

∑γ∈Γh,out

∫γ

u · nγ

(ξA

)2+ Jσ

0

(ξA, ξA

)≤ K

h2μ−4

r2s−6−2δ|||c|||2L2(0,T ;Hs) + K

h2μ−4

r2s−5|||c0|||2s−1 + KR2

s

≤ Kh2μ−4

r2s−6−2δ|||c|||2L2(0,T ;Hs) + K

h2μ−4

r2s−5|||c0|||2s−1 + K

h2μ−2

r2s−2|||ct|||2L2(0,T ;Hs−1).

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 inorder to apply the parabolic lift lemma.

Theorem 4 (face error estimates). Let the assumptions in Theorem 1 hold. Thenthere exists a constant K, independent of h and r, such that

( ∑E∈Eh

∥∥CDG − c∥∥2

L2(0,T ;L2(∂E))

) 12

+

( ∑E∈Eh

∥∥∇ (CDG − c

)· n∂E

∥∥2

L2(0,T ;L2(∂E))

) 12

≤ Khμ− 3

2

rs−2−δ|||c|||L2(0,T ;Hs) + K

hμ− 32

rs−2

(|||∂c/∂t|||L2(0,T ;Hs−1) + |||c0|||s−1

),


Proof. As the first term can be bounded similarly with even sharper estimates, weonly present the estimation of the second term, which can be obtained byapplying the triangle and inverse inequalities, recalling Theorem 1 and using the


approximation results:( ∑E∈Eh

∥∥∇ (CDG − c

)· n∂E

∥∥2

L2(0,T ;L2(∂E))

) 12

≤( ∑

E∈Eh

∥∥∇ (CDG − c

)∥∥2

L2(0,T ;L2(∂E))

) 12

+

( ∑E∈Eh

‖∇ (c− c)‖2L2(0,T ;L2(∂E))

) 12

≤ r

h12

( ∑E∈Eh

∥∥∇ (CDG − c

)∥∥2

L2(0,T ;L2(E))

) 12

+ Khμ− 3

2

rs−32

|||c|||L2(0,T ;Hs)

≤ r

h12

( ∑E∈Eh

∥∥∇ (CDG − c

)∥∥2

L2(0,T ;L2(E))

) 12

+ Kr

h12

hμ−1

rs−1|||c|||L2(0,T ;Hs)

≤ Khμ− 3

2

rs−2−δ|||c|||L2(0,T ;Hs) + K

hμ− 32

rs−2

(|||∂c/∂t|||L2(0,T ;Hs−1) + |||c0|||s−1

).

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∥∥CDG − c∥∥L2(0,T ;L2)

(5.9)

≤ Khμ

rs−1−δ|||c|||L2(0,T ;Hs) + K

hμ

rs−δ|||∂c/∂t|||L2(0,T ;Hs−1) + K

hμ

rs−1/2|||c0|||s−1,


Proof. We recall the concentration error ξ in (4.1), and the error equation:(∂φξ

∂t, w

)+ B(ξ, w;u) = L

(w;u, CDG

)− L (w;u, c) ∀w ∈ Dr (Eh) .

We define

a(x, t) =

⎧⎪⎨⎪⎩− r(M(CDG(x,t)))−r(M(c(x,t)))

CDG(x,t)−c(x,t)if CDG(x, t) − c(x, t) = 0,

0 if CDG(x, t) − c(x, t) = 0.

Consequently, we have L(w;u, CDG) − L(w;u, c) = −(aξ, ω). Noting the fact thata ∈ 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 no-flow boundary conditions. Given a function f ∈ L2(0, T ;L2(Ω)), we consider a linearfunctional F (·) of the following form:

F (c) =

∫ T

0

∫Ω

c(x, t)f(x, t)dx dt.

Lemma 6 (parabolic lift). Let e ∈ L2(0, T ;H1(Eh)) satisfy (5.1)–(5.2) and let theassumptions in Theorem 1 hold. We further assume φ ∈ W s1+2,1

∞ ((0, T ) × Ω), Dij ∈


W s1+1,0∞ ((0, T )×Ω), ui ∈ W s1

∞ (Ω), a ∈ W s1,0∞ ((0, T )×Ω), and q+ ∈ W s1,0

∞ ((0, T )×Ω).Then there exists a constant K, independent of h, r, e, and f, such that

|F (e)| ≤ K ‖f‖L2(0,T ;Hs1 )

(hμ1+1

rs1+1‖e‖L∞(0,T ;L2) +

hμ1+2

rs1+2‖et‖L2(0,T ;L2)

+hμ1+1

rs1+1|||D 1

2 (u)∇e|||L2(0,T ;L2) +hμ1+1

rs1+32−2δ

(∫ T

0

Jσ0 (e, e)

) 12

+hμ1+

32

rs1+32

δ

( ∑E∈Eh

(‖e‖2

L2(0,T ;L2(∂E)) + ‖∇e · n∂E‖2L2(0,T ;L2(∂E))

)) 12),

where μ1 = min(r − 1, s1), r ≥ 1, s1 ≥ 0, δ = 0 for conforming meshes with trianglesor tetrahedra, and δ = 1/2 in general.

Proof. We revisit the adjoint parabolic equation (5.3)–(5.5) with e replaced byf . By applying Theorem 2 repeatedly, we obtain a unique solution Φ for (5.3)–(5.5)satisfying

‖Φ‖L∞(0,T ;Hs1+1) + ‖Φ‖L2(0,T ;Hs1+2) ≤ K ‖f‖L2(0,T ;Hs1 ) .(6.1)

We now consider the L2(Ω) inner product (e, f) at t ∈ (0, T ]:

(e, f) =∑E∈Eh

(e, f)E

=∑E∈Eh

(e,−∂φΦ

∂t

)E

+∑E∈Eh

(e,∇ · (−uΦ − D(u)∇Φ))E +∑E∈Eh

(e, (a + q+)Φ

)E.

Integrating by parts, applying the orthogonality condition (5.1) and observing thatD(u)∇Φ · n∂Ω = 0 on ∂Ω, ∇ · u = q, and [D(u)∇Φ · nγ ] = [Φ] = 0, we conclude that

(e, f) = − d

dt(e, φΦ) +

(φ∂e

∂t,Φ − Φ

)+(ae,Φ − Φ

)+ BS

(e,Φ − Φ;u

),(6.2)

where we choose an interpolant Φ ∈ Dr(Eh) with element-wise optimal approximationproperties (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):(

φ∂e

∂t,Φ − Φ

)≤ K ‖et‖0 ‖Φ − Φ‖0 ≤ K

hμ1+2

rs1+2‖et‖0 ‖Φ‖s1+2 ,(

ae,Φ − Φ)≤ K ‖a‖L∞ ‖e‖0 ‖Φ − Φ‖0 ≤ K

hμ1+2

rs1+2‖a‖L∞ ‖e‖0 ‖Φ‖s1+2 .

Similar but tedious arguments, together with the inverse inequality and the existenceof continuous interpolants for conforming meshes with triangles or tetrahedra, yielda bound for the fourth term:∣∣∣BS

(e,Φ − Φ;u

)∣∣∣ ≤ Khμ1+1

rs1+1|||D 1

2 (u)∇e|||0 ‖Φ‖s1+2 + Khμ1+1

rs1+1‖e‖0 ‖Φ‖s1+2

+Khμ1+1

rs1+32−2δ

(Jσ0 (e, e))

12 ‖Φ‖s1+2

+Kδhμ1+

32

rs1+32

( ∑E∈Eh

(‖e‖2

0,∂E + ‖∇e · n∂E‖20,∂E

)) 12

‖Φ‖s1+2 .


Observing the fact that

|F (e) − (e, φΦ) (0)| =

∣∣∣∣∣∫ T

0

((e, f) +

d

dt(e, φΦ)

)∣∣∣∣∣≤

∫ T

0

∣∣∣∣(e, f) +d

dt(e, φΦ)

∣∣∣∣and integrating (6.2) over the time interval [0, T ], we have

|F (e) − (e, φΦ) (0)|

≤ K ‖Φ‖L2(0,T ;Hs1+2)

(hμ1+2

rs1+2‖et‖L2(0,T ;L2)

+hμ1+2

rs1+2‖a‖L2(0,T ;L∞) ‖e‖L∞(0,T ;L2) +

hμ1+1

rs1+1|||D 1

2 (u)∇e|||L2(0,T ;L2)

+hμ1+1

rs1+1‖e‖L∞(0,T ;L2) +

hμ1+1

rs1+32−2δ

(∫ T

0

Jσ0 (e, e)

) 12

+hμ1+

32

rs1+32

δ

( ∑E∈Eh

(‖e‖2

L2(0,T ;L2(∂E)) + ‖∇e · n∂E‖2L2(0,T ;L2(∂E))

)) 12).

The theorem follows from the regularity estimate (6.1) and the fact that

|(e, φΦ) (0)| =∣∣∣(φe,Φ − Φ

)(0)

∣∣∣≤ K

hmin(r+1,s1+1)

rs1+1‖e(·, 0)‖0 ‖Φ(·, 0)‖s1+1

≤ Khμ1+1

rs1+1‖e‖L∞(0,T ;L2) ‖Φ‖L∞(0,T ;Hs1+1) .

Theorem 6 (linear functional estimates). Let the assumptions in Theorem 1hold. In addition, we assume φ ∈ W s1+2,1

∞ ((0, T ) × Ω), Dij ∈ W s1+1,0∞ ((0, T ) × Ω),

ui ∈ W s1∞ (Ω), q+ ∈ W s1,0

∞ ((0, T )×Ω), and that the chemical reaction term has a linearform r(c) = k0 + k1c, where k0 = k0(x, t) and k1 = k1(x, t) are reaction parameterswith k1 ∈ W s1,0

∞ ((0, T )×Ω). Then there exists a constant K, independent of h, r, andf, such that

∣∣F (CDG) − F (c)∣∣ ≤ K

hμ1+μ

rs1+s−1−δ‖f‖L2(0,T ;Hs1 ) |||c|||L2(0,T ;Hs)

+Khμ1+μ

rs1+s−δ‖f‖L2(0,T ;Hs1 ) |||∂c/∂t|||L2(0,T ;Hs−1)

+Khμ1+μ

rs1+s−1/2‖f‖L2(0,T ;Hs1 ) |||c0|||s−1,

where μ = min(r + 1, s), μ1 = min(r − 1, s1), r ≥ 1, s ≥ 2, s1 ≥ 0, and δ = 0 forconforming 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


M is chosen to be sufficiently large:(∂φξ

∂t, w

)+ BS(ξ, w;u) + (aξ, w) = 0 ∀w ∈ Dr (Eh) ∀t ∈ (0, T ].

We obtain the desired estimate by applying the parabolic lift of Lemma 6 togetherwith 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:

‖c‖H−m(Ω) = supv∈C∞

0 (Ω)\{0}

|(c, v)|‖v‖Hm(Ω)

.

Theorem 7 (estimates in negative norms). Let the assumptions in Theorem 1hold. In addition, we assume φ ∈ Wm+2,1

∞ ((0, T ) × Ω), Dij ∈ Wm+1,0∞ ((0, T ) × Ω),

ui ∈ Wm∞(Ω), q+ ∈ Wm,0

∞ ((0, T )×Ω), and that the chemical reaction term has a linearform r(c) = k0 + k1c, where k0 = k0(x, t) and k1 = k1(x, t) are reaction parameterswith k1 ∈ Wm,0

∞ ((0, T )×Ω). Then there exists a constant K, independent of h and r,such that∥∥CDG − c

∥∥L2(0,T ;H−m(Ω))

≤ Khmin(r−1,m)+min(r+1,s)

rm+s−1−δ|||c|||L2(0,T ;Hs)

+Khmin(r−1,m)+min(r+1,s)

rm+s−δ|||∂c/∂t|||L2(0,T ;Hs−1)

+Khmin(r−1,m)+min(r+1,s)

rm+s−1/2|||c0|||s−1,

where r ≥ 1, s ≥ 2, m ≥ 0, and δ = 0 for conforming meshes with triangles ortetrahedra, and δ = 1/2 in general.

Proof. The theorem follows directly from Theorem 6 and the definition of negativenorms.

7. Numerical examples. We consider the problem of (2.1)–(2.4) on a domainΩ = (0, 10)2 without injection or extraction, i.e., q = 0, and with a reaction termr = r(x, t) independent of the concentration c. The domain is divided into twodisjoint parts: Ω = Ω1 ∪ Ω2 with Ω1 = {(x, y) ∈ Ω : y < 3 + 0.4x}. The porosityφ has a constant value of 0.1, and the tensor D is a constant diagonal tensor withDii = 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 ofc = (1 + cos(π5x) cos(π5 y))2

−t/10. The penalty parameter is chosen according to themethod presented in the next section. The coarsest mesh we take simply consists ofthe 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 stepΔt = 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 re-finements starting from the coarsest mesh. The convergence behaviors of h-refinementin the norms of L2(L2), L∞(L2), and L2(H1) for NIPG are shown in Figure 7.1. Itis observed that the errors in all norms are O(1/n), where n is the number of degreesof freedom. As n ∝ 1/h2 for two-dimensional spaces, the experimental convergences


101

102

103

104

10−3

10−2

10−1

100

number of degrees of freedom

rela

tive

err

ors

in v

ario

us

no

rms

1

1

Error in L2H1Error in LinfL2Error in L2L2

Fig. 7.1. Convergence of h-refinement for NIPG.

101

102

103

104

10−4

10−3

10−2

10−1

100


rela

tive

err

ors

in v

ario

us

no

rms

1

1

1.5

1

Error in L2H1Error in LinfL2Error in L2L2

Fig. 7.2. Convergence of h-refinement for SIPG.

confirm our theoretical estimates in L2(H1). In addition, the numerical results indi-cate that the errors in NIPG do not converge optimally in L∞(L2) or L2(L2). Theconvergence behaviors of OBB-DG and IIPG (not shown) are nearly identical to thoseof 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(H1), as shownevidently in Figure 7.2, which also validates the predictions from our parabolic liftarguments.

7.2. Convergence of p-refinement. The test case is solved using the fourprimal 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(H1), where the expected exponential convergence rates are achieved.The exponential convergence patterns of OBB-DG, NIPG, and IIPG (not shown) arevery similar to those of SIPG. An interesting experimental observation, which is notcovered in previous theoretical sections, is that the DG methods with polynomialsof odd orders have better performance than those of even orders; this is especiallypronounced for OBB-DG.


0 20 40 60 80 100 120 14010

−4

10−3

10−2

10−1

100


rela

tive

err

ors

in v

ario

us

no

rms Error in L2H1

Error in LinfL2Error in L2L2

Fig. 7.3. Convergence of p-refinement for SIPG.

8. Discussion.

8.1. Penalty parameters for SIPG. Numerical experiments indicate thatcareful implementations of the penalty terms are crucial to SIPG: not only are thepenalty terms necessary for the convergence of SIPG, but also choices of penalty pa-rameters significantly influence the performance of SIPG. Small penalty parametersmight result in divergences of the schemes. On the other hand, very large parameters,though ensuring the convergence theoretically, lead to a poor condition number forthe 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| · | is a matrix norm. Letting σγ = σ|D|1/2 and σ = O(1), we have

Jσ0 (c, w) =

∑γ∈Γh

σ√|D| r

2

hγ

∫γ

[c][w].

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/ordispersion-diffusion is highly anisotropic, it is found that the following choice generallygives better results:

Jσ0 (c, w) =

∑γ∈Γh

σ√|Dnγ |

r2

hm,γ

∫γ

[c][w],

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

physical element E, rather than a reference element E. This distinction is unnecessarywhen E is a triangle or tetrahedron because the transformation from E to E is affine.But for a general quadrilateral or hexahedron, these two spaces are different. We applyDG methods to the test case in section 7 using the uniform p-refinement in the coars-est 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 referenceand physical spaces, respectively. Clearly, DG solutions based on physical spaces aremore accurate than those of reference spaces for high order approximations; this ismore significant for OBB-DG than for other primal DGs. This observation suggests


0 20 40 60 80 100 1200.5

1

1.5

2

2.5

3

3.5


erro

r ra

tio

(re

fere

nce

ove

r p

hys

ical

)

OBB−DG

NIPG

IIPG

SIPG

Fig. 8.1. Comparison of reference versus physical polynomial spaces for DG methods (data ofNIPG, 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

is less pronounced on more refined meshes, because the transformation from E toE becomes closer to an affine mapping. Consequently, a choice of physical versusreference spaces does not significantly impact h-versions of DGs.

9. Conclusions. Three primal DG methods with penalty have been analyzedfor solving reactive transport problems in porous media. The cut-off operator wasintroduced in the DG formulations to ensure convergence for general nonlinear ki-netic reactions. Error estimates in L2(H1) for the concentration were derived forSIPG, NIPG, and IIPG, which are optimal in h and nearly optimal in p. In addition,we established L2(H1) concentration error estimates on the element faces as well asL2(L2) estimates for time derivatives. A parabolic lift technique for SIPG has beendeveloped, 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 estimatesin negative norms. We have also numerically investigated the h- and p-convergencebehaviors of OBB-DG, NIPG, IIPG, and SIPG. It was demonstrated that OBB-DG,IIPG, and NIPG possess h-optimal convergence rates in L2(H1), but lack the op-timality in L2(L2) and L∞(L2), whereas SIPG performs h-optimally in the threenorms. For smooth problems, exponential convergence rates in p are achieved by thefour primal DG methods. In addition, it was observed that DGs with polynomialsof odd orders perform better than those of even orders. Implementations of penaltyterms 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 refer-ence spaces, for which we recommended the physical polynomial spaces for p- andhp-versions of DGs. As a future extension, we propose to study error estimates ofprimal DG methods for transport coupled with kinetic and local-equilibrium reactionsand for multiphase flow in porous media.

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Documents

web.kaust.edu.sa...DISCONTINUOUS GALERKIN FOR REACTIVE TRANSPORT 197 We consider the following boundary conditions for this problem: (2.2) (uc−D(u)∇c)·n = c Bu·n, (x,t) ∈ Γ