Mesh Adaptation Strategies for Discontinuous Galerkin Methods
Applied to Reactive Transport Problems
Shuyu SUN and Mary F. WHEELER
The Center for Subsurface Modeling (CSM)
The Institute for Computational Engineering and Sciences (ICES)
The University of Texas, Austin, Texas 78712, USA
ABSTRACT
Static and dynamic strategies are formulated and studied for
discontinuous Galerkin (DG) methods applied to reactive
transport problems based on a posteriori error estimators. It is
shown that the flexibility of DG allowing non-matching meshes
substantially simplifies the implementation of the mesh
adaptation. Moreover, DG with adaptivity can effectively
capture local physical phenomena due to the localization of DG
errors. Dynamic strategies are efficient in improving accuracy
and in saving computational time, especially for transport
problems with a long period of simulation time. In contrast,
static approaches perform poorly on transport problems unless
the total simulation time is short.
Keywords: Discontinuous Galerkin Methods, A Posteriori
Error Estimators, Adaptivity, Reactive Transport, Transient
Problems.
1. INTRODUCTION
The discontinuous Galerkin (DG) methods [5, 11, 24, 13, 3, 4]
have recently gained popularity for many attractive properties.
First of all, the methods are locally mass conservative while
most classical Galerkin finite element methods are not. In
addition, they have less numerical diffusion than most
conventional algorithms, thus are likely to offer more accurate
solutions for at least advection-dominated transport problems.
They handle rough coefficient problems and capture the
discontinuity in the solution very well by the nature of
discontinuous function spaces. DG can naturally handle
inhomogeneous boundary conditions and curved boundaries.
The average of the trace of the fluxes from a DG solution along
an element edge is continuous and can be extended so that a
continuous flux is defined over the entire domain. Thus DG can
be easily coupled with conforming methods. Furthermore, for
smooth flow and transport problems, DG with varying degree of
approximation can yield nearly exponential convergence rates.
For time-dependent problems in particular, the mass matrices
are block diagonal for DG, but not for conforming methods.
This provides a substantial computational advantage, especially
if explicit time integrations are used.
From a computer science point of view, the DG methods are
easier to implement than most traditional finite element
methods. The trial and test spaces are easier to construct than
conforming methods because they are local. This also makes
the code shorter and more efficient. For instance, DG methods
are simpler to implement than the finite volume method and the
mixed finite element method which are two other types of
element-wise conservative methods. Unlike traditional finite
element methods, the DG algorithms need only the mesh
information about elements and interfaces, but without the mesh
information about edges and vertexes. Such a property of
spatial dimension independence offers a great convenience to
implement, test and debug DG codes. In addition, the simple
communication pattern between elements ensures that DG
methods are well parallelizable, which is a necessity for many
massive problems having excessive memory and CPU time
requirements.
Reactive transport is a fundamental process arising in many
diversified fields such as petroleum engineering, groundwater
hydrology, environmental engineering, soil mechanics, earth
sciences, chemical engineering and biomedical engineering.
Realistic simulations for simultaneous transport and chemical
reaction present significant computational challenges [16, 10,
14, 15, 17, 8, 23, 26, 18, 9]. Traditional algorithms employ
operator-splitting to treat advection, diffusion-dispersion and
chemical reaction sequentially and separately. Godunov [6] and
characteristics [2] are popular methods for the advectiondiffusion subproblem. The DG methods have recently applied
for flow and transport problems in porous media [20, 12, 25]. It
has been found that the non-symmetric DG has optimal
convergence in L2(H1) for both flow and transport problems [12,
13]. The hp-convergence behaviors in L2(L2) and in negative
norms have been analyzed in [19]. Explicit a posteriori error
estimates of DG for reactive transport have been studied in [22,
21]. In this paper, we formulate and study the static and the
dynamic mesh adaptation strategies for DG applied to reactive
transport problems. We note that the work here can be extended
to other transient problems.
2. GOVERNING EQUATIONS AND DISCONTINUOUS
GALERKIN SCHEMES
Reactive Transport Governing Equations
For convenience, we consider single-species reactive transport
for a single flowing phase in porous media. Results can be
directly extended to multiple species systems with kinetic
reactions. It is assumed that the 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 Ω is a
polygonal and bounded domain in a d-dimensional space (d=1,
2 or 3). We denote by Γin the inflow boundary and Γout the
outflow/no-flow boundary. Let T be the final simulation time.
The classical advection-diffusion-reaction equation for a single
flowing phase in a porous medium is given by
∂φc
+ ∇ ⋅ (u c − D(u)∇c) = qc * + r (c), ( x, t ) ∈ Ω × (0, T ] , (1)
∂t
where the unknown variable c is the concentration of a species
(amount per volume). Here φ is the porosity; D(u) is the
dispersion/diffusion tensor; r(c) is the reaction term; qc* is the
source term, where 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 . This equation is closed
with certain boundary and initial conditions [19, 25].
Discontinuous Galerkin (DG) Methods and A Posteriori
Error Estimators
We consider the four primal discontinuous Galerkin schemes
for the space discretization in this paper. They are Symmetric
Interior Penalty Galerkin (SIPG) [19, 24, 25], Oden-BaumannBabuska version of DG (OBB-DG) [11, 19], Nonsymmetric
Interior Penalty Galerkin (NIPG) [12, 13, 19] and Incomplete
Interior Penalty Galerkin (IIPG) [7, 19] methods. Interested
readers can refer to corresponding references for the detailed
DG formulations, which are omitted here due to the lack of
space.
The SIPG method is a widely used primal discontinuous
Galerkin method because SIPG offers optimal convergence in
L2(L2) norm for the primal unknown and results in an algebraic
system with a symmetric pattern for diffusion-reaction
equations, which could be exploited to construct fast linear
solvers. Using duality argument, we have developed explicit
L2(L2) a posteriori error estimates for SIPG applied to time
dependent problems including reactive transport in porous
media [22]. These a posteriori error estimates provide valuable
error information, which can be used to guide adaptive
modifications of the mesh. These error estimates are especially
valuable in the cases where the primal scalar unknown rather
than the flux is of interest. They are easy to implement and
computationally cheap. Numerical experiments shows that
these a posteriori error estimates are sharp in capturing the
areas with large errors and could guide effective mesh
modifications to achieve efficient adaptivities [22].
In
particular, they are sharp in capturing concentration fronts in
reactive transport problems.
Other family members of primal DG methods are of interest in
many situations. For instance, OBB-DG and NIPG methods are
capable to handle problems involving high varying coefficients,
and IIPG method is are useful for coupled flow and transport
problems [7, 19]. To guide the adaptivities for these DG
methods, we established a unified a posteriori error estimate
approach in L2(H1) [21]. The approaches apply to all the four
versions of DG, namely, OBB-DG, SIPG, NIPG and IIPG.
These a posteriori error estimates in L2(H1) are explicit and
residual based and thus is computationally efficient. General
boundary conditions can be easily took into consideration of
these error estimates in L2(H1). In addition, no regularity
assumption for the dual problem is required for these estimates.
Numerical performance of the a posteriori error estimators
under two major categories, namely estimators in L2(L2) and in
L2(H1) have been investigated and compared [19, 21, 22].
Results indicate that L2(H1) type error indicators are more
flexible, but L2(L2) type error indicators are more effective for
many reactive transport cases. It was found that adaptivities are
essential for large reactive transport problems, in particular for
the problems where strong physics/chemistry occurs locally in a
small part of the domain.
3. ADAPTIVE MESH MODIFICATION
Mesh modification is an important ingredient of adaptive
strategies for finite element methods. In practice, the number of
elements during mesh modification either grows or remains
constant. Thus we naturally have two basic mesh modification
approaches, namely mesh enrichment and mesh adjustment. As
we will see in following sections, mesh enrichment plays an
important role in static adaptive approaches, whereas mesh
adjustment is essential to dynamic adaptive strategies.
Mesh Enrichment using Adaptive Refinement
Mesh enrichment is a popular mesh modification approach for
solving steady problems. In this approach, we start with a very
coarse initial mesh. Then at each adaptive iteration step, we
refine the elements with the largest error indicator values
(Figure 1). The number of elements in the mesh grows as the
adaptive process proceeds. This approach can be extended to
transient problems including reactive transport problems. It can
be implemented by one of the two versions defined by
Algorithms 1 and 2 respectively:
Algorithm 1. (Adaptive mesh enrichment I)
Given a mesh Ξ and a fraction γ ∈ (0,1) :
1. Compute the DG approximation of the PDE based on the
mesh Ξ and compute the error indicator η E for each
element E ∈ Ξ ;
2.
Compute
the
η ∞ = max(η E : E ∈ Ξ ) ;
3. Select Ξ r = {E ∈ Ξ :
maximum
error
indicator
ηE
> γ} ;
η∞
4. Refine all elements E ∈ Ξ r .
Algorithm 2. (Adaptive mesh enrichment II)
Given a mesh Ξ and an integer n:
1. Compute the DG approximation of the PDE based on mesh
Ξ and compute the error indicator η E for each element
E∈Ξ ;
2.
Select
Ξr ⊂ Ξ
such
that
# (Ξ r ) = n
and
min{η E : E ∈ Ξ r } ≥ max{η E : E ∈ Ξ \ Ξ r } ;
3. Refine all elements E ∈ Ξ r .
(A)
(B)
Figure 1. An example of the growing mesh enrichment
approach. The three elements with the largest error
indicator values are locally refined.
Mesh Adjustment using Adaptive Refinement & Coarsening
In mesh enrichment, the number of elements in mesh grows as
adaptivity proceeds. Alternatively, we can fix the number of
elements, but refine some elements and coarsen some other
elements adaptively to make the mesh more suitable for the
problem (Figure 2). It is described in the following algorithm:
Algorithm 3. (Adaptive mesh adjustment)
Given a mesh Ξ and an integer n:
1. Compute the DG approximation of the PDE based on mesh
Ξ and compute the error indicator η E for each element
E∈Ξ ;
2.
Select
Ξr ⊂ Ξ
such
that
# (Ξ r ) = n
and
min{η E : E ∈ Ξ r } ≥ max{η E : E ∈ Ξ \ Ξ r } ;
3. Select Ξ c ⊂ Ξ to minimize max{η E : E ∈ Ξ c } subject to
# (Ξ c ) = n and that Ξ c satisfies the coarsening-compatible
condition with regard to Ξ and Ξ r ;
4. Refine all elements E ∈ Ξ r and coarsen all elements E ∈ Ξ c .
We have used a coarsening-compatible condition in Algorithm
3. We note that, because DG allows for an arbitrary degree of
nonconformity, each element can be refined. However, not
every element is available to be coarsened; for instance, the
element without a father cannot be further coarsened. The
coarsening-compatible condition is defined as below:
Definition 1. (Coarsening-compatible condition)
The coarsening element set Ξ c is said to satisfy the coarseningcompatible condition with regard to the mesh Ξ and the
refining element set Ξ r if and only if
1. Each element in Ξ c has a father;
2. Brothers of an element in Ξ c are active, that is, they sit in
Ξ;
3. None of the elements in Ξ c and their brothers are in Ξ r ;
4. Brothers of an element in Ξ c are not in Ξ c .
(A)
5.
Refine
all
elements E ∈ Ξ k ,r
and
coarsen
all
elements E ∈ Ξ k ,c to form a new mesh Ξ k +1 ;
6. If satisfied (e.g. if global error indicator drops below some
tolerance or simply if k> k ∞ ), report the solution and stop;
otherwise, let k=k+1 and go to step 2.
We present a numerical example to illustrate this adaptive
approach. We consider the following problem in a domain Ω =
(0, 10)2:
∂φ e c
( x, t ) ∈ Ω × (0, T ],
+ ∇ ⋅ (uc − D(u )∇c ) = 0,
∂t
(uc − D(u )∇c ) ⋅ n = c B u ⋅ n,
( x, t ) ∈ Γin × (0, T ],
( − D(u )∇c ) ⋅ n = 0,
( x, t ) ∈ Γout × (0, T ],
c( x,0) = c 0 ( x ),
x ∈ Ω.
( 2)
The porosity is a constant 0.1; the diffusion-dispersion D is a
constant, diagonal tensor with Dii = 0.01; and the velocity is
u=(-0.1,0) uniformly across the domain. The domain Ω is
divided into two parts, i.e. the lower half Ω l =(0,10)x(0,5) and
the upper half Ω u =(0,10)x(5,10). Adsorption occurs only in
the lower part of the domain, which results in an effective
porosity φ e =0.2 in Ω l . The effective porosity φ e in Ω u is still
0.1. The initial total concentration is 0.1 inside the square
centered at (5,5) with size of 0.3125x0.3125 and is 0.0
elsewhere (shown in Figure 3(A)). The total concentration here
is defined as the product of the concentration in fluid and the
effective porosity.
(B)
Figure 2. An example of the non-growing mesh
adjustment approach. The three elements with the largest
error indicator values are locally refined while the other
three elements with the smallest error indicator values are
locally coarsened. The total number of elements remains
constant.
(A)
(B)
4. STATIC ADAPTIVE STRATEGIES
Static adaptive strategies are widely used for solving steady
problems. They can be also extended to transient problems,
especially for transient problems involving a short period of
simulation time.
Non-growing Static Adaptive Strategy
In non-growing static adaptive strategy, we keep the number of
elements constant but modify locally the fineness/coarseness of
the mesh according to the associated error indicator. It usually
starts with a fine uniform mesh.
Algorithm 4. (Non-growing static adaptive strategy)
Given an initial mesh Ξ 0 and a modification factor α ∈ (0,1) :
1. Let k=0;
2. Compute the DG approximation of the PDE based on mesh
Ξ k and compute the error indicator η E for each
element E ∈ Ξ k ;
3.
Select
Ξ k ,r ⊂ Ξ k
such
that
# (Ξ k ,r ) = α # (Ξ k ) 
and min{η E : E ∈ Ξ k ,r } ≥ max{η E : E ∈ Ξ k \ Ξ k ,r } ;
4. Select Ξ k ,c ⊂ Ξ k to minimize max{η E : E ∈ Ξ k ,c } subject
to # (Ξ k ,c ) = α # (Ξ k )  and that Ξ k ,c satisfies the coarseningcompatible condition with regard to Ξ k and Ξ k ,r ;
Figure 3. Problem data and classic DG solution for a
transport-adsorption case: (A) Velocity and initial fluid
concentration; (B) DG solution in a uniform rectangular
mesh without mesh adaptation.
SIPG is employed to solve this problem using the non-growing
static strategy with a modification factor α =0.05. The penalty
parameter is chosen according to Ref. [19]. The simulation time
interval is (0,1) and we use the backward Euler's method to
integrate with regard to time with a uniform time step ∆t =
0.01. The complete quadratic basis function is used for each
element. The initial mesh is a 16x16 uniform rectangular grid.
The error indicator in L2(L2) is computed for each element. The
time derivative term in the interior residual is approximated by a
finite difference [21, 22]. The fluid concentration and the mesh
structure during the mesh adaptation progress are shown in
Figures 4(A)-(F).
Due to retardation effects arising from adsorption, the
contaminant transport is slower in the lower part of the domain.
A continuous concentration profile is observed because of
diffusion/dispersion. We see that this adaptive strategy with the
error indicator in L2(L2) captures the local behavior of the
advection-diffusion-adsorption process. The spot around the
small square originally contaminated and its downwind areas up
to the concentration front are heavily and efficiently refined,
while the areas far from the contaminated regions have a
relatively coarser mesh.
For comparison, DG using the same number of unknowns is
applied to the problem without mesh adaptation (shown in
Figure 3(B)). Figures 5(A) and (B) are the zoom-in versions of
3(B) and 4(F) respectively. The simulation results clearly
indicate that the adaptive DG resolves the concentration plume
more sharply than the non-adaptive counterpart.
(A)
(B)
3. Compute the maximum error indicator defined by
η ∞ = max(η E : E ∈ Ξ k ) ;
4. Select Ξ k ,r = {E ∈ Ξ k :
ηE
> γ};
η∞
5. Refine all elements E ∈ Ξ k ,r to form a new mesh Ξ k +1 ;
6. If satisfied (e.g. if global error indicator drops below some
tolerance or simply if k> k ∞ ), report the solution and stop;
otherwise, let k=k+1 and go to step 2.
We now solve the same example problem (2) using SIPG with
the error indicator in L2(L2) norm. All parameters are taken
with the same values as in the previous example except that the
growing static adaptive strategy is used with γ = 0.8 and that
the initial mesh consists of a single element (i.e. the entire
domain). Simulation results are shown in Figures 6(A)-(F).
(C)
(D)
(E)
(F)
Figure 4. Adaptive DG solutions using the non-growing
static approach (A) after 1 iteration, (B) after 2 iterations,
(C) after 4 iterations, (D) after 8 iterations, (E) after 16
iterations, and (F) after 32 iterations.
(A)
(A)
(B)
(C)
(D)
(E)
(F)
(B)
Figure 6. Adaptive DG solutions using the growing static
approach (A) after 2 iterations, (B) after 4 iterations, (C)
after 8 iterations, (D) after 16 iterations, (E) after 32
iterations, and (F) after 64 iterations.
Figure 5. Zoom-in views of DG solutions (A) without
and (B) with mesh adaptation.
Growing Static Adaptive Strategy
The growing static adaptive strategy increases the number of
elements by adaptively refining the mesh locally according to
the error indicator. It usually starts with a coarse mesh.
Algorithm 5. (Growing static adaptive strategy)
Given an initial mesh Ξ 0 and a growth factor γ ∈ (0,1) .
1. Let k=0;
2. Compute the DG approximation of the PDE based on the
mesh Ξ k and compute the error indicator η E for each
element E ∈ Ξ k ;
Comparing the results of adaptive DG solutions using the
growing and the non-growing static approaches, we find that the
growing static approach performs as effective as the nongrowing static does in term of accuracy. However, the growing
static approach costs less than the non-growing static due to
cheap computational efforts associated with coarse meshes in
the early iteration steps. Therefore, the growing approach is
more favorable than the non-growing approach for static
strategies. This explains why the growing static approach is
widely used for steady problems while the non-growing static
approach is not. Interestingly, as we will see in next section, the
opposite is observed for dynamic strategies. That is, the nongrowing approach is more favorable than the growing approach
for dynamic strategies, especially for transient problems with a
long period of time.
5. DYNAMIC ADAPTIVE STRATEGIES
For transient problems involving a long period of simulation
time, the location of strong physics (including biogeochemistry)
usually moves with time. But many error indicators for
transient problems, including the L2(L2) and the L2(H1) error
indicators discussed in this paper, combine contributions over
the entire simulation time. Naturally, we want the error
indicator account for the physics only in the current time, and it
is favorable to compute the error indicator for only a short time
interval involving the current time, and to modify the mesh
dynamically with time. Because it is very expensive to change
the mesh each time step, we divide the entire simulation time
into a collection of time slices, each of which may in turn
contains a certain number of time steps. The main idea of
dynamic adaptive approaches is to apply the static strategy into
each individual time slice.
Non-growing Adaptive Strategy
The non-growing dynamic adaptive strategy performs mesh
adjustment without changing the number of elements according
to the associated error indicator during each adaptive iteration
step for each time slice. It usually starts with a fine uniform
mesh.
Algorithm 6. (Non-growing dynamic adaptive strategy)
Given an initial mesh Ξ 0 , a modification factor α ∈ (0,1) , time
slices {(T0, T1), (T1, T2), …, (TN-1, TN)} and iteration numbers
for each time slices {M1, M2, …, MN}.
1. Let n = 1;
2. Let m = 1;
3. Compute the initial concentration for time slice (Tn-1, Tn)
using either the initial condition (if n = 1) or the concentration at
the end of last time slice (if n>1) by a local projection;
4. Let Ξ m,n = Ξ 0 if n=1 and m=1; or Ξ m,n = Ξ M n −1 +1,n −1 if
n>1 and m=1;
5. Compute the DG approximation of the PDE for the time slice
(Tn-1, Tn) based on the mesh Ξ m,n and compute the error
indicator η E for each element E ∈ Ξ m,n ;
(
6. Select Ξ r ⊂ Ξ m,n such that # (Ξ r ) = round α # (Ξ m,n )
)
and min{η E : E ∈ Ξ r } ≥ max{η E : E ∈ Ξ m,n \ Ξ r } ;
7. Select Ξ c ⊂ Ξ m,n to minimize max{η E : E ∈ Ξ c } subject to
(
# (Ξ c ) = round α # (Ξ m,n )
)
and
that
Ξc
satisfies
the
coarsening-compatible condition with regard to Ξ m,n and Ξ r ;
8. Refine all elements E ∈ Ξ r and coarsen all elements E ∈ Ξ c
to form a new mesh Ξ m+1,n ;
9. Let m=m+1. If m ≤ M n , go to step 3;
10. Let n=n+1. If n ≤ N , go to step 2;
11. Report the solution and stop.
We now solve the same example problem (2) in the time period
(0,2) using SIPG with the error indicator in the L2(L2) norm. All
parameters are chosen with the same values as in the previous
example except that we now use the non-growing dynamic
adaptive strategy with a modification factor α =0.05. The
initial mesh is a 16x16 uniform rectangular grid. We partition
the simulation time interval (0,2) into 20 time slices uniformly.
The iteration number is chosen to be 5 for the initial time slice
and to be 2 for all other time slices. It should be observed that,
compared with static approaches, this adaptive strategy is more
effective in capturing the local behavior of the advectiondiffusion-adsorption process (shown in Figures 7(A)-(F)).
(A)
(B)
(C)
(D)
(E)
(F)
Figure 7. Adaptive DG solutions using the non-growing
dynamic approach at (A) m=2, n=1; (B) m=5, n=1; (C)
m=2, n=5; (D) m=2, n=10; (E) m=2, n=15; and (F) m=2,
n=20.
Growing Adaptive Strategy
Similarly as we did in the growing static adaptive approach, we
can formulate the growing dynamic adaptive strategy, which
has a varying number of elements during the mesh adaptation
process. It should be pointed out, however, that the growing
dynamic adaptive approach has much worse performance than
the non-growing dynamic adaptive approach (results not
shown). First of all, the projection from a fine into a coarse
mesh is used between two time slices, in which the accuracy
could be seriously jeopardized. Moreover, in each time slice, a
large number of iterations is usually required to reach the
desired fine mesh. In contrast, the non-growing dynamic
adaptive approach needs only 1 or 2 iterations in each time slice
because the mesh in the beginning of the time slice is already
very close to the desired mesh. Therefore, the growing dynamic
adaptive approach is not recommended for practice.
Comparison of Static and Dynamic Adaptive Strategies
From the results in the last section, we know the growing static
approach can achieve a similar adaptive effect as the nongrowing static approach does while has less computational cost.
It should be noted that the non-growing dynamic approach are
even cheaper than the growing static approach because it only
needs a very few adaptive iterations (1 or 2) for each time slice.
In Figures 8(A)-(D), we compare the static and the dynamic
strategies tested using the same problem (2) with the same
number of unknowns. For a short period of simulation time
(t=1), the static and the dynamic strategies provide similar
adaptive sharpness. For a long period of simulation time (t=2),
however, the dynamic strategy results in significantly better
adaptive effectivity. Clearly, the dynamic modification of mesh
structure increases the accuracy of the solution while decreases
the computational cost; this is especially true for transient
problems with a long period of time and with a time-varying
physics.
(A)
(B)
(C)
(D)
cases. The growing dynamic approach has a poor performance
in both of the computational time and the numerical accuracy,
because of the large iteration number in each time slice and the
projection from a fine into a coarse mesh. Among the four
strategies we considered, the non-growing dynamic approach is
often the method of choice for reactive transport. In this
approach, the iteration number in each time slice can be as small
as 1 or 2, which makes it computationally cheap in addition to
its superior adaptive effectivity. Finally, we emphasize that,
due to the discontinuous spaces used in DG, the projections of
concentration during mesh modifications involve only local
computations and are locally mass conservative. These two
features ensure the efficiency and the accuracy of DG during
dynamic mesh modifications.
7. REFERENCES
Figure 8. Comparison of the non-growing static and the
non-growing dynamic strategies for adaptive DG
solutions: (A) at t=1 by the static approach; (B) at t=1 by
the dynamic approach; (C) at t=2 by the static approach;
and (D) at t=2 by the dynamic approach.
6. CONCLUDING REMARKS
Adaptive strategies including both the static and the dynamic
approaches are formulated and studied for discontinuous
Galerkin (DG) methods applied to reactive transport problems
based on mathematically proved a posteriori error estimators.
Various numerical examples demonstrate the advantage of
adaptive approaches over the non-adaptive version.
In
particular, we see that the flexibility of DG allowing nonmatching meshes substantially simplifies the implementation of
the mesh adaptation as the local element refinement is
independent of neighborhood elements. In addition, this
flexibility increases the efficiency of adaptivities because the
unnecessary areas do not need to be refined in order to maintain
the conformity of the mesh. Moreover, DG errors are localized;
in other words, there is less pollution of errors. This leads to a
more effective adaptivity for DG than for nonconforming
methods. Because of this, we see that adaptivity of DG sharply
captures local physical phenomena.
Growing adaptive approaches increase the number of elements
by adaptively refining the mesh locally according to the
associated error indicator, while non-growing adaptive
approaches modify locally the fineness/coarseness of the mesh
according to the error indicator without changing the number of
elements. The growing approaches usually start with a coarse
mesh and the non-growing approaches usually start with a fine
uniform mesh. It is found that the growing approaches are
better fitted in static strategies, while the non-growing
approaches are better suited for dynamic strategies.
In a short period of simulation time, both the dynamic strategies
and the static strategies work well. However, as one might
expect, the dynamic strategies are generally better than the static
strategies for transient problems with a long period of
simulation time in the aspects of accuracy and computational
costs. The growing static approach is generally better than the
non-growing static approach due to its cheap computational cost
associated with the coarse mesh in early iteration steps, but the
non-growing static approach might be more robust in some
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