Anisotropic and Dynamic Mesh Adaptation for Discontinuous ...element, and establish dynamic mesh adaptation strategies. Section 4 is devoted to numerical studies. It is demonstrated

Anisotropic and Dynamic Mesh Adaptation forDiscontinuous Galerkin Methods Applied to

Reactive Transport

Shuyu Sun and Mary F. Wheeler

The Institute for Computational Engineering and Sciences (ICES)

The University of Texas at Austin, Austin, Texas 78712, USA

Abstract: Mesh adaptation strategies are proposed for discon-tinuous Galerkin methods applied to reactive transport problems,with emphasis on dynamic and anisotropic adaptation. They in-clude an anisotropic mesh adaptation scheme and two isotropicmethods using an L2(L2) norm error estimator and a hierarchicerror indicator. These dynamic mesh adaptation approaches areinvestigated using benchmark cases. Numerical results demon-strate that the three approaches resolve time-dependent transportadequately without slope limiting for both long-term and short-term simulations. It is shown that these results apply to problemswhere either diffusion or advection dominates in different subdo-mains. Moreover, for these schemes, mass conservation is retainedlocally during dynamic mesh modification. Comparison studiesindicate that the anisotropic mesh adaptation provides the mostefficient meshes and has the least numerical diffusion among thethree adaptation approaches.

Keywords: Discontinuous Galerkin methods; A posteriori errorestimators; Anisotropic mesh adaptation; Dynamic mesh adapta-tion; Reactive transport

1. Introduction

Discontinuous Galerkin (DG) methods employ non-conforming piecewise poly-nomial spaces to approximate the solutions of differential equations. The conceptof discontinuous space approximations first appeared in the early 1970’s (see, forexample, [49, 25, 22, 6, 7, 4, 16, 24]) when discontinuous basis functions were usedto approximate second-order elliptic equations. But only recently have DG methodsbeen investigated intensively and applied to wide collections of problems. Theseschemes have many attractive properties [10, 23, 27, 50, 29, 30, 5]. The flexibility ofDG allows for general non-conforming meshes with variable degrees of approxima-tion. This makes the implementation of h-p adaptivity for DG substantially easierthan conventional approaches. Moreover, DG methods are locally mass conserva-tive at the element level. In addition, they have less numerical diffusion than mostconventional algorithms. They can treat rough coefficient problems and can effec-tively capture discontinuities in solutions. DG can naturally handle inhomogeneousboundary conditions and curved boundaries. The average of the trace of fluxes from

1

ANISOTROPIC AND DYNAMIC MESH ADAPTATION FOR DG 2

a DG solution along an element edge is continuous and may be extended so thata continuous flux is defined over the entire domain. Consequently, DG may beeasily coupled with conforming methods. Furthermore, with appropriate meshing,DG with varying p can yield exponential convergence rates. For time-dependentproblems in particular, the mass matrices of DG are block diagonal, which is nottrue for conforming methods. This provides a substantial computational advantage,especially if explicit time integrations are used.

Reactive transport is a fundamental process arising in many important engi-neering and scientific fields such as petroleum, aerospace, environmental, chemicaland biomedical engineering, and earth and life sciences. Realistic simulations for si-multaneous transport and chemical reaction present significant computational chal-lenges [3, 33, 53, 13, 17, 18, 21, 31, 32, 34, 46, 48, 47, 54, 35, 11, 20, 51, 19, 44, 45, 43].DG has recently been applied for flow and transport problems in porous media[37, 28, 52, 41, 43], and an optimal convergence in L2(H1) was demonstrated forflow and transport problems [28, 30, 36, 38]. The hp-convergence behaviors in theL2(L2) and negative norms have been analyzed [36, 38]. Explicit a posteriori errorestimates of DG for reactive transport have also been studied [39, 42, 40].

In this paper, we formulate and study three error indicator-guided dynamic meshadaptation strategies; namely, the hierarchic anisotropic (HA), the L2(L2) isotropic(LI), and the hierarchic isotropic (HI) approaches. In Section 2, we formulate thegeneral scalar reactive transport equation and corresponding primal DG schemes.In addition, we review properties and error estimators, both a priori and a poste-riori, for this model. In Section 3, we derive a new a posteriori error estimator,using hierarchic bases, that provides guidance on the anisotropic refinement of anelement, and establish dynamic mesh adaptation strategies. Section 4 is devotedto numerical studies. It is demonstrated that effective mesh adaptations eliminatethe need of slope limiters for DG. Finally, in Section 5, our results are summarized,and future work is described.

2. Governing Equations and Discontinuous Galerkin Schemes

2.1. Governing Equations. We consider reactive transport for a single flowingphase in porous media. We assume that a time-independent Darcy velocity field uis given and that it satisfies ∇ · u = q, where q is the imposed external total flowrate.

For simplicity, only a single advection-diffusion-reaction equation is considered.However, results may be extended to systems with kinetic reactions. In addition,for convenience, we assume Ω is a bounded polygonal domain in Rd (d = 1, 2 or 3)with boundary ∂Ω = Γin ∪ Γout. Here we denote by Γin the inflow boundary andby 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. The classical advection-diffusion-reaction equation for a single flowing phase 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, T is the final simulation time. φ denotes porosity and is assumed to be time-independent and uniformly bounded, above and below, by positive numbers. D(u)is a dispersion-diffusion tensor and is assumed to be uniformly symmetric positivedefinite and bounded from above. r(c) is a reaction term, and qc∗ is a source term,where the imposed external total flow rate q is a sum of sources (injection) andsinks (extraction). c∗ is an injected concentration cw if q ≥ 0 and is a residentconcentration c if q < 0.

The following boundary conditions are imposed 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 in thefollowing way:

(2.4) c(x, 0) = c0(x), x ∈ Ω.

2.2. Notation. Let Eh be a family of non-degenerate and possibly non-conformingpartitions of Ω composed of triangles or quadrilaterals if d = 2, or tetrahedra,prisms or hexahedra if d = 3. Here h is the maximum element diameter for themesh. The non-degeneracy requirement (also called regularity) is that the elementis convex, and that there exists ρ > 0 such that if hj is the diameter of Ej ∈ Eh, theneach of the sub-triangles (for d = 2) or sub-tetrahedra (for d = 3) of element Ej

contains a ball of radius ρhj in its interior. We assume that no element crosses theboundaries in Γin or Γout. The set of all interior edges (for 2-dimensional domain)or faces (for 3-dimensional domain) for Eh is denoted by Γh. On each edge or faceγ ∈ Γh, a unit normal vector nγ is chosen. The sets of all edges or faces on Γout andon Γin for Eh are denoted by Γh,out and Γh,in, respectively, for which the normalvector nγ coincides with the outward unit normal vector.

For s ≥ 0, we define

(2.5) Hs(Eh) :=

φ ∈ L2(Ω) : φ|E ∈ Hs(E), E ∈ Eh

.

The discontinuous finite element space is taken to be

(2.6) 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 equalto r on E. Note that the results in this paper are based on the local space Pr, butmay be easily extended to the local space Qr.

We now define the average and jump for φ ∈ Hs(Eh), s > 1/2. Let Ei, Ej ∈ Eh

and γ = ∂Ei ∩ ∂Ej ∈ Γh with nγ exterior to Ei. Denote

φ :=1

2

(

(

φ|Ei

)∣

∣

γ+(

φ|Ej

)∣

∣

∣

γ

)

,(2.7)

[φ] :=(

φ|Ei

)∣

∣

γ−(

φ|Ej

)∣

∣

∣

γ.(2.8)

Denote the upwind value of the concentration c∗|γ as follows:

c∗|γ :=

c|Eiif u · nγ ≥ 0

c|Ejif u · nγ < 0.


The usual Sobolev norm on a domain R ⊂ Ω is denoted by ‖·‖m,R [1]. Thebroken norms are defined, for a positive integer m, as

(2.9) |||φ|||2m :=∑

E∈Eh

‖φ‖2m,E .

The inner product in(

L2(R))d

or L2(R) is indicated by (·, ·)R, and the inner

product in the boundary function space L2(γ) is denoted by (·, ·)γ . The innerproduct over the entire domain (·, ·)Ω is also written simply as (·, ·). We define acut-off operator M as

(2.10) M(c)(x) := min (c(x), M) ,

where M is a large positive constant.

2.3. Continuous-in-Time Schemes. We introduce the bilinear form B(c, w;u)defined as

B(c, w;u)(2.11)

=∑

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) ,

where sform = −1 for NIPG (the Nonsymmetric Interior Penalty Galerkin method[30]) or OBB-DG (the Oden-Babuska-Baumann formulation of DG [23]), sform =1 for SIPG (the Symmetric Interior Penalty Galerkin method [49, 36, 38]), andsform = 0 for IIPG (the Incomplete Interior Penalty Galerkin method [36, 14, 38]).Here q+ is the injection source term, and q− is the extraction source term, i.e.,

q+ = max (q, 0) , q− = min (q, 0) .

By definition, we have q = q+ + q−. In addition, we define the interior penaltyterm Jσ

0 (c, w) as

Jσ0 (c, w) =

∑

γ∈Γh

r2σγ

hγ

∫

γ

[c] [w] ,

where σ is a discrete positive function that takes the constant value σγ on the edgeor face γ. We let σγ ≡ 0 for OBB-DG and assume 0 < σ0 ≤ σγ ≤ σm for SIPG,NIPG and IIPG.

The linear functional L(w;u, c) is defined as

L(w;u, c) =

∫

Ω

r (M(c)) w +

∫

Ω

cwq+w −∑

γ∈Γh,in

∫

γ

cBu · nγw.(2.12)

We state the weak formulation of the reactive transport problem, for which theproof may be found in [36, 38].

Lemma 2.1. (Weak formulation) If c is a solution of (2.1)-(2.3) and c is es-sentially bounded, then c satisfies

(

∂φc

∂t, w

)

+ B(c, w;u) = L(w;u, c),(2.13)


∀w ∈ Hs (Eh) , s >3

2, ∀t ∈ (0, T ],

provided that the constant M for the cut-off operator is sufficiently large.

The continuous-in-time DG approximation CDG(·, t) ∈ Dr (Eh) of (2.1)-(2.4) isdefined by

(

∂φCDG

∂t, w

)

+ B(CDG, w;u) = L(w;u, CDG),(2.14)

∀w ∈ Dr (Eh) , t ∈ (0, T ],(

φCDG, w)

= (φc0, w) , ∀w ∈ Dr (Eh) , t = 0.(2.15)

A solution for the above DG scheme always exists [36].

Lemma 2.2. (Existence of a solution) Assume that the reaction rate is a locallyLipschitz continuous function of concentration. The discontinuous Galerkin scheme(2.14) and (2.15) has a unique solution for all t > 0.

The element-wise mass conservation of DG schemes is stated in the followingLemma [36]. We note that the concentration jump term, i.e. the fourth term in(2.16), is considered as part of the computed diffusive flux for SIPG, NIPG andIIPG.

Lemma 2.3. (Local mass balance) The approximation of the concentrationsatisfies, on each element E, the following local mass balance property:

∫

E

∂φCDG

∂t−

∫

∂E\∂Ω

D(u)∇CDG · n∂E

+

∫

∂E

CDG∗u · n∂E(2.16)

+∑

γ⊂∂E\∂Ω

r2σγ

hγ

∫

γ

(

CDG∣

∣

E− CDG

∣

∣

Ω\E

)

=

∫

E

(

CDGq− + cwq+)

+

∫

E

r(

M(CDG))

.

2.4. Convergence Results of DG. In [36, 38], the following a priori error esti-mate was established for NIPG, SIPG and IIPG:

Theorem 2.4. (L2(H1) and L∞(L2) error estimate for NIPG, SIPG andIIPG) 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, uand q are essentially bounded and that the reaction rate is a locally Lipschitz con-tinuous function of c. If the constant M for the cut-off operator and the penaltyparameter σ0 are sufficiently large, then there exists a constant K, independent ofh 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− δ2

|||c|||L2(0,T ;Hs) + Khµ−1

rs−1

(

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

)

,


where µ = min (r + 1, s) , r ≥ 1, s ≥ 2, and δ = 0 in the case of conforming mesheswith triangles or tetrahedra. In general cases, δ = 1.

The following a priori error estimate for OBB-DG was derived in [28, 36].

Theorem 2.5. (L2(H1) and L∞(L2) error estimate for OBB-DG) Let allassumptions in Theorem 2.4 hold except sform = −1 and σ ≡ 0. If the constantM for the cut-off operator is sufficiently large, then there exists a constant K,independent of h and r, such that

∥

∥CDG − c∥

∥

L∞(0,T ;L2)+ |||D

12 (u)∇

(

CDG − c)

|||L2(0,T ;L2)

≤ Khµ−1

rs− 52

(

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

,

where µ = min (r + 1, s) , r ≥ 2, and s ≥ 3.

2.5. Residual-Based Explicit A Posteriori Error Estimators. We introduceresidual quantities that only depend on the approximate solution and data. Theresiduals consist of the interior residual RI , the zeroth order boundary residualRB0, and the first order boundary residual RB1 as defined below:

RI = q−CDG + q+cw + r(

M(CDG))

(2.17)

−φ∂CDG

∂t−∇ ·

(

CDGu−D(u)∇CDG)

,

RB0 =

[

CDG]

, x ∈ γ, γ ∈ Γh,0, x ∈ ∂Ω,

(2.18)

RB1 =

[

D(u)∇CDG · n]

, x ∈ γ, γ ∈ Γh,(

cBu − CDGu + D(u)∇CDG)

· n, x ∈ Γin,D(u)∇CDG · n, x ∈ Γout.

(2.19)

We recall two explicit a posteriori error estimators [42, 39]. The L2(L2) normerror estimator will be used in Section 4.

Theorem 2.6. (Explicit a posteriori error estimator in L2(L2) for SIPG)Let the assumptions in Theorem 2.4 hold. In addition, we assume φ ∈ W 2,1

∞ ((0, T )×Ω), Dij ∈ W 1,0

∞ ((0, T )×Ω), u ∈ L∞(Ω) (u being independent of time), u · n|∂Ω = 0,and c0 ∈ Dr (Eh). Then there exists a constant K, independent of h and r, suchthat

∥

∥CDG − c∥

∥

L2(0,T ;L2)≤ K

(

∑

E∈Eh

η2E

)1/2

,

where

η2E =

h4E

r4‖RI‖

2L2(0,T ;L2(E)) +

1

2

∑

γ∈∂E\∂Ω

(

hγ

r+ δrhγ

)

‖RB0‖2L2(0,T ;L2(γ))

+1

2

∑

γ∈∂E\∂Ω

h3γ

r3‖RB1‖

2L2(0,T ;L2(γ)) +

∑

γ∈∂E∩∂Ω

h3γ

r3‖RB1‖

2L2(0,T ;L2(γ)) .


Here, hγ = max(hE1, hE2

) for γ = ∂E1 ∩ ∂E2, r ≥ 1, and δ = 0 in the case ofconforming meshes with triangles or tetrahedra. In general cases, δ = 1.

Theorem 2.7. (Explicit a posteriori error estimator for OBB-DG, NIPG,SIPG or IIPG) Let the assumptions in Theorem 2.4 (for NIPG, SIPG or IIPG)or in Theorem 2.5 (for OBB-DG) hold . In addition, we assume c0 ∈ Dr (Eh).There exists a constant K, independent of h, such that

∥

∥

∥

√

φ(CDG − c)∥

∥

∥

L∞(0,T ;L2(Ω))+ |||D

12 (u)∇(CDG − c)|||L2(0,T ;L2(Ω))

≤ K

(

∑

E∈Eh

η2E

)1/2

,

where

η2E = h2

E ‖RI‖2L2(0,T ;L2(E)) +

∑

γ∈∂E∩∂Ω

hγ ‖RB1‖2L2(0,T ;L2(γ))

+1

2

∑

γ∈∂E\∂Ω

(

hγ ‖RB1‖2L2(0,T ;L2(γ)) + h−1

γ ‖RB0‖2L2(0,T ;L2(γ))

+hγ ‖RB0‖2L∞(0,T ;L2(γ)) + hγ ‖∂RB0/∂t‖2

L2(0,T ;L2(γ))

)

.

Here, hγ = max(hE1, hE2

) for γ = ∂E1 ∩ ∂E2 and r ≥ 1.

We note that, in implementation, time derivative terms may be approximatedby finite differences. For example, we may estimate the time derivative of the DGsolution used in the interior residual RI by

∂CDG

∂t≈

(

CDG)

tn+1−(

CDG)

tn

tn+1 − tnfor t ∈ (tn, tn+1) .

3. Anisotropic Mesh Adaptation

3.1. A Hierarchic A Posteriori Error Indicator. Residual-based explicit errorestimators are efficient to compute and may be used to indicate the subset ofelements that need to be refined, thus guiding adaptivity. However, these residual-based estimators yield only one piece of information for each element. Consequently,they do not provide guidance on anisotropic refinements. In this section, we deriveerror estimators using hierarchic bases. Unlike residual-based error estimators,hierarchic error estimators give point-wise information on the error, and thus maybe used to guide fully anisotropic hp-adaptation. Here, for simplicity, we consideronly the anisotropic h-adaptation.

Hierarchic error estimators consist of solving the problem of interest by em-ploying two discretization schemes of different accuracy and using the differencebetween the approximations as an estimate for the error. The advantages of thisapproach include their applicability to many classes of problems and the simplicityand ease of their implementation. The reader is referred to [2, 8, 9, 15, 26] forfurther information.

For a given mesh Eh, we construct the mesh Eh/2 by isotropically refining each

element of Eh. We denote by CDG the DG solution in the coarse mesh Eh and by


CDG,F the DG solution in the fine mesh Eh/2. We make the following saturationassumption:

(3.1)∥

∥CDG,F − c∥

∥

L2 (t) ≤ β∥

∥CDG − c∥

∥

L2 (t), 0 ≤ β < 1.

Using Theorems 2.4 and 2.5, we observe, for all the four primal DGs, i.e. OBB-DG, SIPG, NIPG and IIPG, that β is less than or equal to the following valueasymptotically:

β ≤1

2µ−1=

1

2min(r,s−1).

Theorem 3.1. (A posteriori error estimator for OBB-DG, SIPG, NIPGor IIPG) Let the saturation assumption (3.1) be satisfied. Then we have

1

1 + β

(

∑

E∈Eh

η2E

)1/2

(t) ≤∥

∥CDG − c∥

∥

L2 (t) ≤1

1 − β

(

∑

E∈Eh

η2E

)1/2

(t),

where ηE(t) =∥

∥CDG,F − CDG∥

∥

L2(E)(t).

Proof. The theorem follows by the saturation assumption together with the triangleinequality:

(

∑

E∈Eh

η2E

)1/2

(t) =∥

∥CDG,F − CDG∥

∥

L2(Ω)(t)

≤∥

∥CDG − c∥

∥

L2(Ω)(t) +

∥

∥CDG,F − c∥

∥

L2(Ω)(t)

≤ (1 + β)∥

∥CDG − c∥

∥

L2(Ω)(t),

(

∑

E∈Eh

η2E

)1/2

(t) =∥

∥CDG,F − CDG∥

∥

L2(Ω)(t)

≥∥

∥CDG − c∥

∥

L2(Ω)(t) −

∥

∥CDG,F − c∥

∥

L2(Ω)(t)

≥ (1 − β)∥

∥CDG − c∥

∥

L2(Ω)(t).

In the above error indicator, the fine grid solution is used as a replacement forthe true solution to estimate the error of the coarse grid solution. We now usethe same principle to select a proper anisotropic refinement. A uniform degree rof polynomial space is assumed over the entire domain. The local space for eachelement E is denoted by Pr(E). We further assume that, for an individual elementE, there are n different refinement options r1, r2, ... , rn. In Figure 3.1, we illustratetwo anisotropic refinements for a quadrilateral element. We denote by ri(E) theset of sub-elements obtained from the refinement ri applied to the element E andby ΠE the L2 projection operator onto Pr(E). We now compute the error indicatorfor each refinement option as follows:

η2E,ri

(t) =∑

bE∈ri(E)

∥

∥CDG,F − Π bECDG,F∥

∥

2

L2( bE)(t).


AB

CD

A

B

CD

E

G

(a) x-refinement

AB

CD

AB

CD

FH

(b) y-refinement

Figure 3.1. Illustration of anisotropic refinements to a quadri-lateral element.

Obviously, the smaller the error indicator ηE,ri(t), the better the anisotropically

refined mesh resolves the higher order accurate solution CDG,F , and the better theanisotropically refined mesh is likely to resolve the true solution c. Therefore, wechoose the refinement ri at the time t such that ηE,ri

(t) is minimized among allavailable options.

3.2. Dynamic and Anisotropic Mesh Adaptation. For transient problems in-volving a long period of simulation time, the location of biogeochemical phenomenagenerally moves with time. Most error indicators for transient problems, includingthe L2(L2) and L2(H1) error indicators reviewed in the previous section, provideglobal spatial and temporal estimates. It is desirable that error indicators accountfor local physics only at the current time, and thus it is preferable to compute errorindicators only for a short time interval. The hierarchic error indicator presentedin this section is point-wise in time and provides guidance on time-dependent meshmodifications. Because it is expensive to change the mesh at each time step, wedivide the entire simulation period into a collection of time slices, each of which mayin turn contain a certain number of time steps. We adopt a non-growing dynamicadaptive strategy [40], where the mesh is adaptively adjusted while the number


of elements remains constant. The maintenance of a constant number of elementsprevents the computational workload from increasing with time steps. The initialmesh is chosen to be a uniform fine grid. We first present the dynamic and isotropicmesh adaptation strategies. We denote by #(S) the number of elements in a set S.

Algorithm 3.2. (dynamic and isotropic mesh adaptation)Given an initial mesh E0, a modification factor α ∈ (0, 1), time slices (T0, T1),

(T1, T2), · · · , (TN−1, TN), and iteration numbers M1, M2, · · · , MN.1. Let n = 1;2. Let m = 1;3. Compute the initial concentration for the time slice (Tn−1, Tn) using either

the initial condition (if n = 1) or the concentration at the end of last time slice (ifn > 1) by local projections;

4. Let Em,n = E0 if n = 1 and m = 1; or Em,n = EMn−1+1,n−1 if n > 1 andm = 1;

5. Compute the DG approximation of the PDE for the time slice (Tn−1, Tn) basedon the mesh Em,n, and compute the error indicator ηE for each element E ∈ Em,n;

6. Select Er ∈ Em,n such that #(Er) = dα#(Em,n)e and minηE : E ∈ Er ≥maxηE : E ∈ Em,n \ Er;

7. Select Ec ∈ Em,n to minimize maxηE : E ∈ Ec subject to that #(Ec) =dα#(Em,n)e and that Ec satisfies the coarsening-compatible condition with regard toEm,n and Er;

8. Refine all E ∈ Er and coarsen all E ∈ Ec to form a new mesh Em+1,n;9. Let m = m + 1. If m ≤ Mn, go to step 3;10. Let n = n + 1. If n ≤ N , go to step 2;11. Report the solution and stop.

We have used a coarsening-compatible condition in the above algorithm. We notethat, because DG allows for an arbitrary degree of nonconformity, each element maybe refined. However, not every element is available for coarsening; for instance, theelement without a father cannot be further coarsened. The coarsening-compatiblecondition is defined as below.

Definition 3.3. (Coarsening-compatible condition) A coarsening element setEc is said to satisfy the coarsening-compatible condition with regard to a mesh Eand a refinement element set Er if and only if:

1. Each element in Ec has a father;2. Brothers of an element in Ec are active, that is, they sit in E ;3. None of the elements in Ec and their brothers are in Er;4. Brothers of an element in Ec are not in Ec.

Some error indicators, including the hierarchic error indicators presented previ-ously, do not only provide guidance on the selection of elements to be coarsenedor refined, but they also yield information on how to anisotropically refine a givenelement. If we employ these error indicators, we obtain a more efficient mesh adap-tation algorithm, i.e. the dynamic and anisotropic mesh adaptation method:

Algorithm 3.4. (dynamic and anisotropic mesh adaptation)Given an initial mesh E0, a modification factor α ∈ (0, 1), time slices (T0, T1),

(T1, T2), · · · , (TN−1, TN), and iteration numbers M1, M2, · · · , MN.1-7. Same as steps 1-7 in Algorithm 3.2;


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10CONC1.000.900.800.700.600.500.400.300.200.100.00

Figure 4.1. The initial fluid concentration and velocity for themoving contaminant case.

8. Refine all E ∈ Er anisotropically, guided by the local error indicator, andcoarsen all E ∈ Ec to form a new mesh Em+1,n;

9-11. Same as steps 9-11 in Algorithm 3.2.

4. Numerical Examples

In this section, we investigate three dynamic mesh adaptation strategies usingnumerical experiments. These adaptation strategies are listed as follows:

• LI approach: the dynamic and isotropic mesh adaptation using the L2(L2)norm error indicator ηE in Theorem 2.6;

• HI approach: the dynamic and isotropic mesh adaptation using the errorindicator ηE of hierarchic bases in Theorem 3.1;

• HA approach: the dynamic and anisotropic mesh adaptation using the errorindicator ηE of hierarchic bases in Theorem 3.1.

4.1. A Moving Contaminant Case. We consider the following problem with thedomain Ω = (0, 10)2:

∂φec

∂t+ ∇ · (uc −D∇c) = 0, (x, t) ∈ Ω × (0, T ],(4.1)

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

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

c(x, 0) = c0(x), x ∈ Ω.

Here, the porosity φ is a constant 0.1, the diffusion-dispersion tensor D is a diagonalmatrix with Dii = 0.01, and the velocity is u = (−0.2, 0). The domain is dividedinto two parts, i.e. the lower half Ωl = (0, 10) × (0, 5) and the upper half Ωu =(0, 10) × (5, 10). Adsorption occurs only in the lower part of the domain, whichresults in an effective porosity φe of 0.2 in Ωl. The effective porosity φe in Ωu

remains 0.1. Due to contamination, the initial total concentration is 0.1 inside thesquare centered at (5,5) with a side of length 0.3125, and 0.0 elsewhere (shown inFigure 4.1). The total concentration is defined as the product of the concentrationin fluid and the effective porosity.


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Figure 4.2. DG with the LI mesh adaptation applied to themoving contaminant case (left column: the concentration in fluid;right column: the mesh structure; from top row to bottom: t=0.5,1, 1.5 and 2, respectively).


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Figure 4.3. DG with the HI mesh adaptation applied to themoving contaminant case (left column: the concentration in fluid;right column: the mesh structure; from top row to bottom: t=0.5,1, 1.5 and 2, respectively).


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Figure 4.4. DG with the HA mesh adaptation applied to themoving contaminant case (left column: the concentration in fluid;right column: the mesh structure; from top row to bottom: t=0.5,1, 1.5 and 2, respectively).


X

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0 0.25 0.5 0.75 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Figure 4.5. The velocity and contaminant source rate for theboundary layer case.

SIPG with the implicit Euler time integration is employed to solve this problemusing the three dynamic strategies with a modification factor α = 0.05. The numberof iterations is chosen to be 5 initially and set to be 2 for all other time slices. Thepenalty parameter is chosen to be 0.1, which is based on an error bound establishedin [38]. The simulation time interval is (0, 2) with a uniform time step ∆t = 0.01.Each time slice contains 10 time steps. The complete quadratic basis function isused for each element. The initial mesh is a 16× 16 uniform rectangular grid.

Results of the concentration profiles and the mesh structures as functions of timeare shown in Figures 4.2, 4.3 and 4.4 for the three mesh adaptation approaches,respectively. Due to the retardation effect arising from adsorption, the advectionof the contaminant is slower in the lower part of the domain. A continuous con-centration profile is formed because of non-zero diffusion-dispersion over the entiredomain. Clearly, the meshes from the three adaptation approaches follow the move-ment of the plume, and each of these approaches captures the local behavior of theadvection-diffusion-adsorption process.

A careful investigation of both the size and the center of the plume revealsthat the HA approach has the least numerical diffusion among the three, and itis most effective in characterizing the plume structure. Moreover, this approachyields better element aspect ratios, which appear to be physics-driven, especiallyduring later times. It is interesting to observe that the mesh generated by HA ismore or less isotropic during early times but becomes more anisotropic later. Thisis because the contaminant plume is circular during early times but transformsinto an anisotropic shape later. Even though the mesh structures from the LI andHI approaches are different, their resolution of the physics appears to be similar.Since the hierarchic error indicator is more expensive to compute, one would preferapplying the L2(L2) error indicator for isotropic mesh refinements.

4.2. A Boundary Layer Case. This test case represents the coupling of a dif-fusion process with an advection-dominated process over two adjacent domains,where a thin concentration boundary layer is formed around the interface betweenthese two processes. The governing equations over the entire domain Ω = (0, 1)2


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Figure 4.6. DG with the LI adaptation applied to the boundarylayer case (left column: the concentration; right column: the meshstructure; top row: t=0.01; bottom row: t=10).

are as follows.

∂φc

∂t+ ∇ · (uc −D∇c) = s, (x, t) ∈ Ω × (0, T ],(4.2)

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

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

c(x, 0) = 0, x ∈ Ω.

The domain Ω is divided into two parts, i.e. the lower half Ωl = (0, 1) × (0, 0.5)and the upper half Ωu = (0, 1) × (0.5, 1). The velocity u is zero in Ωl and is (1, 0)uniformly in Ωu. On the other hand, the contaminant source term s is 1 in Ωl andzero in Ωu (shown in Figure 4.5).

We now solve the problem (4.2) using SIPG with the three mesh adaptationapproaches LI, HI and HA. The initial mesh is a 16× 32 uniform rectangular grid.Here, φ, D, α and the iteration numbers are the same as before. The polynomialspace P2 is used for each element. Simulation results for these approaches are shownin Figures 4.6, 4.7 and 4.8, respectively. For each adaptation approach, we run along-term simulation up to t=10 using a uniform time step ∆t=0.1, and a short-term simulation up to t=0.01 using a uniform time step ∆t=0.0001. For all runs,a time slice contains 10 time steps.


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Figure 4.7. DG with the HI adaptation applied to the boundarylayer case (left column: the concentration; right column: the meshstructure; top row: t=0.01; bottom row: t=10).

Results clearly indicate that a concentration boundary layer is formed almostimmediately. The boundary layer is very thin at early times and grows spatiallywith time. The concentration profile changes rapidly in early times but slows astime increases. It ultimately approaches a steady state shortly after t=2. Theplotted long-term simulation results at t=10 are almost at steady state. The localphysics of diffusion and advection is captured in all three adaptation approaches.The top third part of the domain (y ≥ 2/3) has nearly zero concentration duringearly times, and very little diffusion or advection takes place. Correspondingly, thisarea has very coarse grids for all three approaches. The successful application of DGto this example occurs due to the effectiveness of DG in treating both advection-dominated and diffusion-dominated problems. In addition, DG offers flexible andeffective mesh adaptation in capturing boundary layer phenomena.

Both the HI and HA approaches yield meshes that are dense only around bound-ary layers necessary for resolving the physics. On the other hand, the LI approachunnecessarily refines the mesh in the lower part of the domain far away from theboundary layers; this is especially true in early times. It should be noted that theboundary layer expands during early simulation times, and both the HI and the HAlead to meshes well adapted to this time-evolving phenomena. The HA approachdoes not only provide a proper density distribution of the mesh, but it also yields


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Figure 4.8. DG with the HA adaptation applied to the bound-ary layer case (left column: the concentration; right column: themesh structure; top row: t=0.01; bottom row: t=10).

physics-driven element aspect ratios. During early times, most of the elements arenarrow, mimicking the shape of the thin boundary layer. During later times, theelements are still slim in the advection-dominated region, but they are more or lessisotropic in the diffusion region. This is another physics-driven example, in whichadvection is direction-oriented while diffusion is isotropic. It should be observedthat the corner point (0, 0.5), being the starting point of the boundary layer, playsan important role in the formation of the boundary layer. We note that both HIand HA resolve the starting region of the boundary layer adequately.

4.3. The ANDRA-Couplex1 Case. In this benchmark case, we consider a nu-clear waste repository in a heterogeneous porous medium Ω = (0, 25000m) ×(0, 695m) [12]. The repository lies at a depth of 450m inside a clay layer. Aboveand below the clay layer there are layers of limestone, marl and dogger limestone asshown in Figure 4.9(left). Water flows through the entire porous medium, and oncethe radioactive materials leak from the repository they are diffused and advected bythe groundwater flow. Two chemical elements, plutonium-242 and iodine-129, leakfrom the repository, located in the rectangle defined by the corners (18440m, 244m)and (21680m, 250m). These elements escape in small amounts over time periods


Marl

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Figure 4.9. ANDRA-Couplex1 domain (left) and contaminantrelease rates (right).

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Figure 4.10. The initial mesh used for the ANDRA-Couplex1 case.

Figure 4.11. The pressure field alone (left) and together withthe velocity field (right) for the ANDRA-Couplex1 case.


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Figure 4.12. DG with the HA adaptation applied to theANDRA-Couplex1 case (left column: iodine concentration; rightcolumn: the mesh structure; from top row to bottom: t=200k,500k, 2m and 5m years, respectively).


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Figure 4.13. DG with the HA adaptation applied to theANDRA-Couplex1 case (left column: plutonium concentration;right column: the mesh structure; top row: t=2m years; bottomrow: t=5m years).

that are short compared with the total simulation time. The rates of release areshown in Figure 4.9(right).

The modeling equations for this problem are as follows.

• Flow equation

(4.3) −∇ · (K∇p) = ∇ · u = q, (x, t) ∈ Ω × J,

• Transport equation

(4.4)∂φαcα

∂t+ ∇ · (ucα −Dα(u)∇cα) = sα, (x, t) ∈ Ω × J,

• Dispersion-diffusion tensor

(4.5) Dα(u) = dα,mI + |u| dα,lE(u) + dα,t (I −E(u)) ,

where α = 1 and 2 for iodine and plutonium respectively; dα,m = φτDα,m; andE(u) is the tensor that projects onto the u direction, whose (i, j) component is

(E(u))i.j = uiuj/ |u|2.

The following properties apply to the flow and transport of the contaminants inthe subsurface. The permeabilities of the marl, limestone, clay, and dogger layersare 3.1536e−5 m/yr, 6.3072 m/yr, 3.1536e−6 m/yr, and 25.2288 m/yr, respectively


(note that the unit for the pressure here is m-water). The molecular diffusivity ofiodine and plutonium in the marl, limestone, and dogger layers is 5.0e−4 m2/yr,and it is 9.48e−7 m2/yr for iodine and 4.42e−4 m2/yr for plutonium in the claylayer. The porosity is 0.001 for iodine and 0.2 for plutonium in the clay layer and0.1 elsewhere for both. The half-life of plutonium is 3.76e5 years and 1.57e7 yearsfor iodine. The retardation factor is 10e5 for plutonium in the clay layer and 1everywhere else for plutonium. The retardation factor is 1 everywhere for iodine.Mechanical dispersion is applied in the limestone and dogger limestone only, withαt = 1 m and αl = 50 m.

For all simulations and figures, OBB-DG with uniform time steps of 100 yearsis used. Each time slice contains 1000 time steps. Simulations begin at 0 and goto 10 million years. Again, the modification factor is taken to be α = 0.05, andthe number of iterations is chosen to be 5 for the initial time slice and 2 for theremaining slices. The initial grid consists of 1920 quadrilateral elements where thelayer boundaries fall along the edges of the elements (shown in Figure 4.10). Forboth flow and transport the complete quadratic basis functions are used.

The resulting pressure field (pressure unit in m-water) and the velocity field areshown in Figure 4.11. This DG solution of velocity is used to compute the advectionterm in the transport equation. Figure 4.12 (concentration unit in mol/m2 for allpictures in this ANDRA-Couplex1 case) are concentration results for iodine andcorresponding mesh structures at various simulation times. We have only shownthe results using the HA approach as the other two adaptation approaches do notperform so well. Compared with iodine, the concentration profiles of plutonium,as well as the employed meshes, are localized near the repository (shown in Figure4.13). This is due to the high retardation factor, high effective porosity and shorterhalf-life of plutonium, even though the diffusivity of plutonium is larger in the claylayer. Note that the magnitude of the velocity varies greatly in the different layersdue to discontinuities in the permeabilities of the layers. In addition, in the clayand marl layers, where permeability is small, transport is dominated by moleculardiffusion. In the limestone and the dogger limestone layers, where permeability islarge, transport is dominated by advection and dispersion.

It should be noted that the local conservation of the DG solution is crucial to thesimulations of small concentrations. The low numerical diffusion of the DG methodis also important in this benchmark problem because of the long simulation time. Itis observed that DG performs very well for treating the diffusion dominated domain(the clay layer) and for handling the advection dominated domains (the limestoneand dogger limestone layers). Another attractive feature found in numerical exper-iments is that DG can handle the full dispersion tensor easily and accurately. Wehave simulated the ANDRA-Couplex1 case with and without mechanical dispersion,and we have obtained a dramatic difference between the two (figures produced bya nonadaptive DG were shown in [52]), which suggests the importance of treatingdispersion in the ANDRA-Couplex1 case.

The effectiveness of the mesh adaptation is clearly demonstrated in Figure 4.12.The location of the concentration plume is effectively captured by the meshes dy-namically. As a result, the density of the meshes resembles the concentration pro-files during both early and later times. The area around the repository in the claylayer undergoes a diffusion-dominated process, and, correspondingly, the mesh isextensively refined. The advection regions in the limestone and dogger limestone


layers also have a relatively higher mesh density. In addition, the interfaces betweenlayers are captured with a locally refined grid. For instance, the interface betweenthe clay and limestone layers is refined for resolving the physics coupling there. Inthis example, we again see evidences of physics-driven mesh adaptation for bothshort and long time simulations.

5. Discussion and Conclusions

We have established a new a posteriori error indicator for DG based on hierarchicbases, which is applicable to a large class of problems and is easily implementable.Here we address the dynamic mesh adaptation strategies for DG as applied to reac-tive transport problems. In particular, we focus on dynamic and anisotropic meshadaptation using the proposed hierarchic error indicator. Various numerical exam-ples demonstrate the effectiveness of these strategies. In particular, we see that theflexibility of DG, allowing non-matching meshes substantially, simplifies the imple-mentation of the mesh adaptation as the local element refinement is independent ofneighborhood elements. In addition, this flexibility increases the efficiency of adap-tivities because unnecessary areas do not need to be refined in order to maintain theconformity of the mesh. Moreover, DG errors are localized; in other words, thereis less pollution of errors. This leads to a more effective adaptivity for DG thanfor conforming methods. Numerical examples show that local physical phenomenacan be sharply captured by DG with dynamic mesh adaptations. The anisotropicmesh adaptation allows the flexible aspect ratios of individual elements to locallyand dynamically accommodate simulated phenomena, and this accommodation canlead to further computational savings.

We have numerically investigated three dynamic adaptation strategies, namely,HA, LI, and HI. Results indicate that all approaches resolve time-dependent trans-port processes for both long- and short-term simulations and eliminate the need ofslope limiters. The boundary layer case and the ANDRA-Couplex1 case demon-strate that DG can treat both advection-dominated and diffusion-dominated prob-lems, and they show that anisotropic mesh adaptations are flexible and effective incapturing boundary layer phenomena. Our numerical results further show that thenumber of iterations in each time slice may be as small as 1 or 2 while obtaining anaccurate mesh. We emphasize that, due to the discontinuous spaces employed inDG, the projections of concentration during mesh modifications only involve localcomputations and are locally mass conservative. These features ensure both theefficiency and the accuracy of DG during dynamic mesh modifications.

Comparisons of the three dynamic adaptation approaches clearly demonstratethe superior effectiveness of HA, which results in the most efficient physics-drivenmeshes and has the least numerical diffusion. The HI approach performs similarlyto the LI method for standard reactive transport problems, but it performs betterfor transport problems involving sharp boundary layers. However, the L2(L2) normerror indicator can be computed more efficiently than the hierarchic error indicator,which makes the choice between LI and HI problem-dependent.

The hierarchic error indicator has superior numerical performance to guide ef-fective anisotropic and dynamic mesh modification, but it is computationally ex-pensive. Our future direction is to approximate the hierarchic error indicator by


solutions of only local problems on a fine grid. The goal of this approach is to re-duce the computational cost while maintaining its superior numerical performance,which is a topic we are currently pursuing.

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Anisotropic and Dynamic Mesh Adaptation for Discontinuous ...element, and establish dynamic mesh adaptation strategies. Section 4 is devoted to numerical studies. It is demonstrated