Multiphase multicomponent flow in porous media with general reactions: efficient problem formulations, conservative discretizations, and convergence analysis Mehrphasen-Mehrkomponenten-Fluss in por¨osen Medien mit allgemeinen chemischen Reaktionen: effiziente Problemformulierungen, massenerhaltende Diskretisierungen und Konvergenzordnungsanalyse DerNaturwissenschaftlichenFakult¨at der Friedrich–Alexander–Universit¨ at Erlangen–N¨ urnberg zur Erlangung des Doktorgrades Dr. rer. nat. vorgelegt von Fabian Brunner aus Weiden i.d.OPf.
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Multiphase multicomponent flow inporous media with general reactions:
efficient problem formulations,conservative discretizations, and
convergence analysis
Mehrphasen-Mehrkomponenten-Fluss in porosen Medien
mit allgemeinen chemischen Reaktionen: effiziente
Problemformulierungen, massenerhaltende
Diskretisierungen und Konvergenzordnungsanalyse
Der Naturwissenschaftlichen Fakultatder Friedrich–Alexander–Universitat Erlangen–Nurnberg
zurErlangung des Doktorgrades Dr. rer. nat.
vorgelegt vonFabian Brunner
aus Weiden i.d.OPf.
Als Dissertation genehmigt von der Naturwissenschaftlichen Fakultatder Friedrich–Alexander–Universitat Erlangen–Nurnberg
Tag der mundlichen Prufung: 22. Dezember 2015
Vorsitzender des Promotionsorgans: Prof. Dr. Jorn Wilms
Gutachter: Prof. Dr. Peter Knabner
Prof. Dr. Florin A. Radu
Contents
Danksagung (German) 11
Zusammenfassung (German) 13
I Efficient formulations and numerical approaches formultiphase-multicomponent flow in porous media withgeneral chemical reaction systems 19
1 Introduction 20
1.1 Current state of research . . . . . . . . . . . . . . . . . . . . . . . 22
1.2 Objective of this work . . . . . . . . . . . . . . . . . . . . . . . . 23
1.3 Overview over this work . . . . . . . . . . . . . . . . . . . . . . . 25
and has one unknown less than the global system (3.65)–(3.73). Nevertheless, in
our numerical tests, this formulation required more computation time than the
formulation based on the global system (3.65)–(3.73). One reason for this is the
structure of the local problem, which consists of the equations
∂t(φs`ξs,kin) = φs`As,kinRkin(cnmin) , (3.111)
∂t(φs`ηs) = 0 , (3.112)
φex(c`, sg) = 0 , (3.113)
φmob(c`) = 0 , (3.114)
φsorp(c`, cs,nmin) = 0 , (3.115)
φmin(c`, cs,min) = 0 , (3.116)
ρmol,g − fg(p` + pc(sg)) = 0 , (3.117)
ρmol,` −I∑i=1
ci` = 0 , (3.118)
ρmass,` −I∑i=1
M ici` = 0 , (3.119)
ρmass,g −M1ρmol,g = 0 . (3.120)
72 CHAPTER 3. THE REDUCTION SCHEME
In the local problem, the equations (3.111)–(3.116) consisting of Is + Jex + Jmob
equations are fully coupled by the saturations and concentrations. Consequently,
they have to be solved simultaneously to obtain the saturations and the local
transformed variables. If the logarithms of the concentration variables are used
as unknowns instead of the transformed variables, the system is even larger since
the defining equations of transformed variables must be added to the system, cf.
Section 6.4.3.
On the other hand, if the additional pressure pc is used, the saturation depends
only on pc and can be computed independently from the rest of the system.
Afterwards, the variables ηs are obtained explicitly, and the chemical problem
consisting of (3.77)–(3.79) along with (3.74) can be solved independently, cf. Sec-
tion 6.4.3. Thus, despite of the fact that pc represents an additional variable,
the local problems – having the same size as in the formulation considered here
– can be solved more efficiently if it is included because they decompose into
several smaller subproblems. Another advantage of using pc as a global variable
is that the derivative of sg with respect to the global variables is zero except for
the derivative with respect to pc. Hence, the global Jacobian can be assembled
faster, and it has fewer nonzero entries than the Jacobian that results from the
formulation considered in this section.
3.4.3 Gas and liquid phase pressures as local unknowns
Another possibility to reduce the number of global pressure variables is to use
only the extended capillary pressure pc as global unknown and to treat pg and p`
as local variables. This is accomplished by shifting the equation (3.73) from the
global to the local system. In the resulting formulation, the property that the
local system decomposes into smaller subproblems is preserved. Nevertheless, in
a numerical experiment related to CO2 sequestration, the computations based on
this formulation required significantly more computation time than the compu-
tations based on the formulation (3.65)–(3.73). This is because more steps of the
Newton solver were necessary to solve the nonlinear systems in each time step.
Potential reasons for this behavior are discussed in Section 7.3.1.
3.4. VARIANTS AND SPECIAL CASES 73
3.4.4 Two-phase two-component flow without reactions
Let us finally consider the special case of two-phase two-component flow without
chemical reactions, where I` = 2, Is = 0, Jex = 1 and Jmob = Jsorp = Jmin =
Jkin = 0. This implies
S =
Sg
S`
=
−1
1
0
, B` = S∗` = S` =
1
0
, S⊥` =
0
1
,
and the variable transformation of Section 3.1 yields the transformed variables
η` = (S⊥`TB⊥` )−1S⊥`
Tc` = c2
` ,
ξ` = (BT` S∗`)−1BT
` c` = c1` .
The resulting system of global equations reads (cf. (3.65)–(3.73))
∂t(φs`η1` ) + L`η1
` = 0 , (3.121)
∂t(φs`ξ1`,ex + φsgρmol,g) + L`ξ1
`,ex + Lgρmol,g = 0 , (3.122)
ρmol,` − f`(c`(Xglob)) = 0 , (3.123)
φex(c`(Xglob), pg) = 0 . (3.124)
Here, (3.121) represents conservation of component 1 (water) and (3.122) repre-
sents conservation of component 2 (e. g., hydrogen, CO2). The global and local
variables are given by
Xglob =
pc
pg
η1`
ξ1`,ex
∈ R4 , Xloc =
sg
p`
ρmol,`
ρmass,`
ρmol,g
ρmass,g
∈ R6 ,
74 CHAPTER 3. THE REDUCTION SCHEME
and the resolution function Xglob 7→Xloc is defined by the local equations
pc − pg + p` = 0 , φcap(sg, pc) = 0
and the constraints
ρmol,` −I∑i=1
ci` = 0 , ρmass,` −I∑i=1
M ici` = 0 ,
ρmol,g − fg(pg) = 0 , ρmass,g −M1ρmol,g = 0 .
In this special case, the number of global equations can be reduced further by
shifting two of the global variables, e. g., η1` and pg, to the set of local vari-
ables and using (3.123)–(3.124) as local equations instead of global equations.
The resulting global system consists of two partial differential equations for two
global unknowns, which is the standard number of unknowns in two-phase two-
component flow, see, e. g., [NBI13, MMK13, BJS09]. This formulation is used for
the benchmark computations in Section 7.1.
Chapter 4
Resolution function
One of the key issues of the reduction mechanism is the partitioning of unknowns
into primary/global unknowns and secondary/local unknowns and the use of
algebraic equations to eliminate local unknowns from the fully coupled DAE
system in terms of a resolution function Xloc = Xloc(Xglob). Since the chemistry
equations and the constitutive laws are nonlinear, this elimination has to be done
implicitly, i. e., the evaluation of the resolution function requires the solution of
a nonlinear system of equations.
Clearly, the efficiency of the numerical solver depends strongly on the choice
of primary and secondary variables. On the one hand, one has to ensure that
the secondary variables Xloc can be uniquely determined for a given set of global
variablesXglob. This is only the case if there are sufficiently many global variables.
On the other hand, it is beneficial to have as many local variables as possible
since the computational effort that is required for solving the local problems
is significantly lower than the effort required for solving the global nonlinear
problem.
Typically, in large parts of the computational domain, only very few steps of
the nested Newton iteration are necessary to determine the local variables, e. g.,
in regions where chemical reactions proceed slowly or do not take place at all.
Moreover, the solution of the previous time step provides a potentially good initial
guess for these local iterations. Finally, let us mention that the local problems
can be solved efficiently on parallel machines without the need for communication
between the processors.
75
76 CHAPTER 4. RESOLUTION FUNCTION
In this chapter, the existence of a resolution function Xglob 7→ Xloc is studied
for the choice (3.64) of primary and secondary variables. More precisely, we will
prove existence of a local resolution function for the general reaction system and
existence of a global resolution function for the simplified case that there are no
kinetic reactions. This implies that our primary variables are persistent.
4.1 Local resolution function
In this section, the existence of a local resolution function Xglob 7→Xloc is shown
with the help of an implicit function theorem. Since the minimum function is a
nonsmooth function, the classical version of the implicit function theorem cannot
be applied to the local equations (3.74)–(3.84). Instead, a generalized version
for piecewise smooth functions is used. We begin with the statement of some
relevant definitions and results.
Definition 4.1.1. Let U ⊆ Rn be open and let F : U → Rm be locally Lip-
schitz continuous. Denote by ΩF ⊆ U the set of points at which F fails to be
differentiable. Then, the B-subdifferential of F at x ∈ U is defined by
∂BF (x) = M ∈ R(m,n) : M = limxk→x,xk 6∈ΩF
JF (xk) ,
where JF (·) denotes the classical Jacobian of F . Note that by Rademacher’s
Theorem, ΩF has measure zero. Similarly, if U = V ×W ⊆ Rn−p × Rp, the set
∂By F (x,y) = My ∈ R(m,p) : M = (Mx|My) ∈ ∂BF (x,y)
provides a generalization of the partial derivatives of F with respect to y.
Definition 4.1.2. The convex hull
∂clF (x) = conv(∂BF (x))
is called (Clarke’s) generalized Jacobian of F at x.
It is indeed a generalization of the classical Jacobian, since the following theorem
holds (see [Cla83, Proposition 2.2.4]).
4.1. LOCAL RESOLUTION FUNCTION 77
Theorem 4.1.3. Let f : Rn → Rm be continuously differentiable in a neighbor-
hood of x ∈ Rn. Then ∂clF (x) = JF (x).
Lipschitz continuity and Clarke’s generalized Jacobian provide a framework in
which an implicit function theorem holds, cf. [Cla83, p. 256]. This theorem,
however, is not strong enough for our needs, since it requires that all elements of
∂clF (x) are of maximal rank, which is hard to verify for our problem. Instead,
we use the additional regularity provided by our local equations, which are not
only Lipschitz continuous but also piecewise differentiable.
Definition 4.1.4. A continuous function F : U → Rm on an open set U ⊆ Rn is
PCr (r ≥ 1) if for every x ∈ U there exists a finite family Gii∈I of Cr selection
functions Gi : O → Rm, i ∈ I, where O ⊆ U is an open neighborhood of x, such
that F (z) ∈ Gi(z) : i ∈ I for all z ∈ O. A selection function Gi of a PCr
function F is called essentially active at x if x ∈ intz ∈ O : Gi(z) = F (z).The set of all essentially active indices of F at x is denoted by Ie(F,x).
The following lemma shows how the B-subdifferential can be evaluated for piece-
wise smooth functions.
Lemma 4.1.5 [RS97]. If Gii∈I is a family of selection functions for the PC1
function F at x, then
∂BF (x) = JGi(x) : i ∈ Ie(F,x) .
The decisive concept for a generalization of the implicit function theorem to
piecewise differentiable functions is the following notion of coherent orientation.
Definition 4.1.6. Let V ⊆ Rm and W ⊆ Rn be open sets, F : V ×W → Rn
be a PC1 function, and (x0,y0) ∈ V × W . Denote by Λ(x0,y0) the set of all
matrices M ∈ R(n,n) with the property that there exist matrices M1, . . . ,Mn ∈∂By F (x0,y0) such that the p-th row of M coincides with the p-th row of Mp for
all p ∈ 1, . . . , n. The function F is called completely coherently oriented with
respect to y at (x0,y0) if all matrices M ∈ Λ(x0,y0) have the same nonvanishing
determinantal sign.
Using the notion of completely coherent orientation, we are now ready to state
an implicit function theorem for piecewise smooth functions.
78 CHAPTER 4. RESOLUTION FUNCTION
Theorem 4.1.7 [RS97] (Implicit function theorem for PCr functions). Let
V ⊆ Rm and W ⊆ Rn be open sets and F : V ×W → Rn be a PCr function,
where r ≥ 1. If F is completely coherently oriented with respect to y at a root
(x0,y0) ∈ V ×W of F , then the equation F (x,y) = 0 determines an implicit
PCr function x 7→ y(x) in a neighborhood of (x0,y0).
Corollary 4.1.8 [RS97]. Let V ⊆ Rm and W ⊆ Rn be open and let F : V ×W →Rn denote the componentwise minimum function, given by
F (x,y) = minh1(x,y),h2(x,y) , h(i)(x,y) =
hi1(x,y)
...
hin(x,y)
, i ∈ 1, 2 ,
where h1,h2 : V ×W → Rn are Cr functions, r ≥ 1. Moreover, let (x0,y0) ∈V ×W be a root of F . If all matrices M = (m1, . . . ,mn) ∈ R(n,n) with mp ∈∇yh1
p(x0,y0),∇yh2
p(x0,y0), for each p ∈ 1, . . . , n, have the same nonzero
determinantal sign, then there exists a PCr function x 7→ y(x) on a neighborhood
of x0 such that (x,y(x)) is the locally unique solution of F (x,y) = 0.
Let us now apply the above results to our local equations and show the existence
of a local resolution function.
Theorem 4.1.9. Let Xglob ∈ Rnglob and Xloc ∈ Rnloc be defined as in (3.64) and
is nonempty, where the concentrations cnmin = cnmin(Xglob,Xloc) are consid-
ered as a function of the transformed variables by means of the retransformation
(3.41). Moreover, let the kinetic and equilibrium reactions be governed by the
mass action laws (2.14), (2.16) and (2.19), and let pc ∈ C1([0, 1);R) be a strictly
increasing function of sg. Assume further that the ODEs (3.45) and (3.50) are
discretized with the implicit Euler method, where ∆t > 0 denotes the time step
size and the superscript (·)old refers to the value at the previous time step. Then,
4.1. LOCAL RESOLUTION FUNCTION 79
if ∆t is sufficiently small and if X0 = (X0glob,X
0loc) ∈ P satisfies
φcap(s0g, p
0c) = mins0
g, pc(s0g)− p0
c = 0 , 0 ≤ s0g < 1 , (4.1)
φeq(c`(X0), cs(X
0)) = 0 , (4.2)
φs0`η
0s − φsold
` ηolds = 0 , (4.3)
φs0`ξ
0s,kin − φsold
` ξolds,kin −∆tφs0
`As,kinRkin(cnmin(X0)) = 0 , (4.4)
there exists a PC1 resolution function Xglob 7→ X loc in a neighborhood of X0,
where s0g, s
0` , p
0c, η
0s and ξ0
s,kin represent entries of X0 according to the partitioning
(3.64).
Proof. First, let us note that equation (3.43) defines an explicit global PC1 reso-
lution function pc 7→ sg by means of
sg(pc) =
0 if pc ≤ pc(0) ,
p−1c (pc) if pc > pc(0) .
Note that since s 7→ pc(s) is strictly increasing, it admits an inverse function
p−1c . In the next step, we show that the chemical equilibrium laws (4.2) and the
discretized ODEs (4.3)–(4.4) define a local PC1 resolution function
Xglob 7→ (ξ`,mob, ξs,sorp, ξs,min,ηs, ξs,kin)
in a neighborhood of X0. For that purpose, we define the function φkin : P →RIs−Jsorp−Jmin associated with the discretized ODEs by
φkin(X) =
φ(s`ηs − sold` η
olds )
φ(s`ξs,kin − sold` ξ
olds,kin)−∆tφs`As,kinRkin(cnmin(X))
.
Moreover, for a subset I ⊆ 1, . . . , Jmin with complement A := 1, . . . , Jmin\I,
80 CHAPTER 4. RESOLUTION FUNCTION
let φIeq : P → RJ∗eq be defined by
φIeq(X) =
φmob(c`(X))
φsorp(c`(X), cs,nmin(X))
φImin(c`(X), cs,min(X))
, (4.5)
where φImin represents the selection functions of φmin associated with the indices
j ∈ I,
φImin,j(c`(X), cs,min(X)) =
ϕmin,j(c`(X)) if j ∈ I ,cjs,min if j ∈ A .
Here, ϕmin,j denotes the j-th component of ϕmin, cf. p. 44. Note that φIeq is not
necessarily an essentially active selection function of φeq at X0 because the min-
imum in the j-th component of φmin is not necessarily attained in the first argu-
ment if j ∈ I. If only the derivatives with respect to (ξ`,mob, ξs,sorp, ξs,min,ηs, ξs,kin)
of all essentially active selection functions of φ in X0 were considered, we could
only show that φ is coherently oriented [RS97, Def. 4] in X0, whereas we need it
to be completely coherently oriented to derive the existence of a local resolution
function with the help of Corollary 4.1.8. Let us now show that the determinant
of
M =
∂φIeq∂(ξ`,mob,ξs,sorp,ξs,min)
∂φIeq∂(ηs,ξs,kin)
∂φkin
∂(ξ`,mob,ξs,sorp,ξs,min)∂φkin
∂(ηs,ξs,kin)
=:
M 11 M 12
0 M 22
is positive at X0 ∈ P for an arbitrary choice of the index set I ⊆ 1, . . . , Jmin.After rearranging the mineral reactions and renumbering the variables ξs,min ac-
cordingly (note that this does not change the determinantal sign of M), we may
assume that I = 1, 2, . . . , |I|, A = |I|+ 1, . . . , Jmin and that S`,min provides
a partitioning
S`,min = (S`,min,I S`,min,A) ,
where S`,min,I denotes the submatrix consisting of the first |I| columns of S`,min
4.1. LOCAL RESOLUTION FUNCTION 81
and S`,min,A the submatrix consisting of the last |A| columns. Then, M 11 reads
for all n ≥ N , which implies G(ξnloc) −→ ∞ as n → ∞, since for arbitrary
constants c1, c2, c3, c4 with c2, c3, c4 > 0 it holds that
limx→∞
c1 + ln(c2x− c3)(c2x− c3)− c4x =∞ .
This implies that G(ξnloc) > α0 for n sufficiently large, which is a contradiction
to the assumption (ξnloc)n∈N ⊂ Lα0 . Hence, Lα0 must be bounded. Since it is also
closed, it is compact and G attains a global minimum ξloc ∈ S, which represents
a solution of (MIN1). Finally, it follows from Proposition 4.2.11 that ξloc ∈ Sand that ξloc solves problem (MIN). The uniqueness of ξloc follows from the
strict convexity of G on S, cf. Lemma 4.2.7 and Proposition 4.2.12.
94 CHAPTER 4. RESOLUTION FUNCTION
The proof of Proposition 4.2.13 appears somewhat technical. This is because, in
general, we have no a priori information on the signs of the entries of µ0, which is
defined as a solution of the linear system (4.9). It should be noted, however, that
the proof can be simplified under the additional assumption that there exists a
vector s⊥ with only strictly positive entries which is orthogonal to all columns
of the stoichiometric matrix S. Then, there exists a solution µ0 of (4.9) having
only strictly positive entries, and the estimate
G(c(ξloc)) ≥ C
I`+Is∑i=1
ci(ξloc) = C‖c(ξloc)‖1
holds for all ξloc ∈ S, which immediately implies the boundedness of the level
set Lα0 . The assumption that such a vector s⊥ exists is called the assumption
of the conservation of the number of atoms and is satisfied for many reactive
networks. It is often used in the literature, e. g., in [Kra08] to prove existence
of a multispecies ODE batch problem with kinetic reactions governed by the law
of mass action, or in [Hof10] to prove existence of solutions for a multispecies
transport problem involving kinetic mineral reactions.
Let us now use the above results to prove existence of a global resolution function
Xglob 7→Xloc in the absence of kinetic reactions.
Theorem 4.2.14. Assume that there are no kinetic reactions and let the global
variables Xglob = (pc, pg,η`, ξ`,ex, ξ`,sorp, ξ`,min, ξsorp, ξmin) be given such that
b b bbbb b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b
b b b b b b b b bb b b
bb b
b b
b
b
b
b
b
b b
b b b b b b b b b bb b b b
b b b b b b b b
b
b
b
b
b
b b bbb
b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b bb b b b b b b b b b b b b b b b b b b
b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b bb b b b
b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b bb b b b b b b b b b b b b b b
b b b b b b b b b
b
b
b
b b
b
Figure 7.4: Evolution of the time step size for SIA and GIA for εabs = 10−10
7.2. INFLUENCE OF DIFFERENT RETENTION CURVES 141
GIA SIA
εabs time st. Newton st. CPU time time st. Newton st. CPU time
10−5 54 206 4.3 min 46 1215 6.4 min
10−6 56 211 4.4 min 75 2060 10.9 min
10−7 67 260 5.9 min 118 3397 17.9 min
10−8 74 299 6.2 min 190 5550 29.5 min
10−9 89 351 7.1 min 324 9591 51.7 min
10−10 101 409 8.4 min 536 15973 86.6 min
Table 7.2: Performance of GIA and SIA for the MoMaS benchmark
7.2 Influence of different retention curves
The performance of the global numerical solver is strongly influenced by the soil
parameters of the problem. In the previous section, hydrogen injection into a very
dense and low-permeable material was simulated, which requires different soil pa-
rameters than CO2 injection into deep saline aquifers consisting of sandstone. To
investigate further the effect of different retention curves on the convergence of
the Newton solver, a simple modification of the MoMaS benchmark considered in
the previous section was proposed in [dC15]. The modified benchmark problem
involves two test cases using two different values of the van Genuchten param-
eter α in the definition of the capillary pressure–saturation curve: the value
α1 = 5 · 10−7 Pa was adopted from the MoMaS benchmark and corresponds to a
very low-permeable rock material, and α2 = 5 · 10−4 Pa represents a sandstone
aquifer. The resulting retention curves are shown in Figure 7.5.
Governing equations
The system is governed by the conservation equations (7.1)–(7.2), where diffusion
of component 1 (hydrogen) in the liquid phase is now expressed in terms of the
gradient of its molar concentration,
j1` = −D1
`∇c1` .
142 CHAPTER 7. NUMERICAL RESULTS
0.0 · 108
0.5 · 108
1.0 · 108
1.5 · 108
2.0 · 108
0.0 · 1080.0 0.2 0.4 0.6 0.8 1.00.0
effective liquid phase saturation Sℓ,e [-]
capillary
pressure
p c[Pa]
0.0 · 108
0.5 · 108
1.0 · 108
0.0 · 1080.0 0.2 0.40.0
α1 = 5 · 10−7
α2 = 5 · 10−4
Figure 7.5: Retention curves of the modified MoMaS benchmark problem
The diffusion-dispersion tensor of hydrogen in the liquid phase follows an ap-
proach of Jin and Jury [JJ96] and is given by
D1` = φ
43 s2`D
1diff,`I .
For the diffusive flux of water in the liquid phase, the relation j2` = −j1
` is used.
Initial and boundary conditions
The computational domain is initially liquid-saturated with no hydrogen being
dissolved in the liquid phase. The initial conditions read
p`(·, 0) = p0` , c1
`(·, 0) = 0 .
As in the MoMaS benchmark, hydrogen is injected through the left part of the
computational domain at a constant rate q1in at Γin. At Tinj = 5 · 105 years, the
injection is stopped and the simulation is continued until the final time Tend =
106 years. Accordingly, the the following boundary conditions are imposed on
∂Ω = Γin∪ΓD∪Γimp:
7.2. INFLUENCE OF DIFFERENT RETENTION CURVES 143
1. No-flux on Γimp:
Q1 · n = 0 , Q2 · n = 0 .
2. Hydrogen injection through Γin:
Q2 · n = 0 , Q1 · n =
q1in if t ≤ Tinj ,
0 otherwise .
3. Pure liquid water and a fixed liquid pressure on ΓD:
p` = p`,out , c1` = 0 .
The computational domain Ω = (0, 200) of this benchmark problem is one dimen-
sional, but since our code was written for 2D and 3D simulations, we consider it
as a pseudo 1D problem and run the computations on a thin stripe consisting of
N rectangles, cf. Figure 7.6. On the upper and lower boundary, no flux conditions
are imposed, and on the right boundary, Dirichlet conditions are imposed for c1`
and p`.
(0, 0) (200, 0)
(200, 1)(0, 1)
T1 T2 T2 TN. . .Γin ΓD
Γimp
Γimp
Figure 7.6: Computational mesh for the modified MoMaS benchmark problem
Closure relationships
For the hydrogen interphase mass exchange, Henry’s law is employed, i. e.,
φex(c`) := Hpg − c1` ,
144 CHAPTER 7. NUMERICAL RESULTS
and the compressibility laws are given by the functions
f`(c`) := c2,std` ,
fg(pg) :=pgRT
,
where the constant c2,std` denotes the molar density of water at standard condi-
tions. As a consequence, the liquid phase has constant molar concentration (but
varying mass density) in this example. All parameters associated with this test
problem are listed in Table 7.3. Since the simulation turned out to be much more
challenging using the parameter α2, we refer to this case as the “hard” test case,
whereas the use of the parameter α1 is referred to as the “easy” test case.
variable symbol value SI unit
porosity φ 0.15 [-]
absolute permeability K 5 · 10−20 m2
temperature T 303 K
Van Genuchten parameter s`,res 0.4 [-]
Van Genuchten parameter sg,res 0.0 [-]
Van Genuchten parameter n 1.49 [-]
Van Genuchten parameter α1 5 · 10−7 Pa
Van Genuchten parameter α2 5 · 10−4 Pa
Henry constant (T = 303K) H 7.65 · 10−6 mol m−3 Pa−1
molar mass of hydrogen M1 2.0 · 10−3 kg mol−1
molar mass of water M2 1.8 · 10−2 kg mol−1
initial value p0` 106 Pa
diffusion coefficient D2diff,` 3 · 10−9 m2 s−1
diffusion coefficient D1diff,g 0 m2 s−1
hydrogen injection rate q1in 1.77 · 10−13 kg m−2 s−1
viscosity of the liq. phase µ` 10−3 Pa.s
viscosity of the gas phase µg 9 · 10−6 Pa.s
standard water density c2,std` 55555.56 mol m−3
Table 7.3: Parameters of the modified benchmark problem
7.2. INFLUENCE OF DIFFERENT RETENTION CURVES 145
Grid convergence
A grid convergence study is carried out for both test cases. For this purpose, the
numerical solution of the problem is calculated on different levels of refinement,
where the number of rectangles in x direction is doubled in each refinement step,
resulting in a grid consisting of N = 100 · 2i cells on level i. As in the previous
section, pc and c1` are used as primary variables. Since an analytical solution of
the problem is not known, the solution on level i = 10 (N = 102400) is used as
a reference solution. It is illustrated in Figures 7.7, where the evolution of the
gas phase saturation and the evolution of the pressures at the inflow boundary
are shown. On each level of refinement, the experimental order of convergence is
determined by
EOCi+1 =1
log(2)| log
(eiei+1
)| ,
where ei denotes the L2 error between the solution on level i and the reference
solution. The relative L2 error ereli is defined as the ratio between ei and the L2
norm of the reference solution. All computations are run with a constant time
step size ∆t = 0.5 years, which is chosen sufficiently small to ensure that the
time discretization error is negligible compared to the space discretization error.
The results of the convergence study are listed in Tables 7.4 and 7.5. For the
easy test case, first order convergence is observed for the variables sg, p` and c1` ,
which is optimal for the upwind LFEM–FV scheme. For the hard test case, the
convergence rate is slightly lower for the variable sg. This is in accordance with
the results reported in [dC15].
Performance
In addition to the grid convergence study, this benchmark problem was proposed
to enable a comparison of the performance of different numerical solvers. For
that purpose, we recompute both test cases using the adaptive time stepping
strategy described in Section 6.2 and the control parameters of Table 7.6. In
146 CHAPTER 7. NUMERICAL RESULTS
0.000
0.005
0.010
0.015
0.020
0.0000 2 · 105 4 · 105 6 · 105 8 · 105 1060
time [years]
saturation
[-]
0.000
0.005
0.010
0.015
0.020
0.0000 2 · 105 4 · 1050
sg
0.70 · 106
0.80 · 106
0.90 · 106
1.00 · 106
1.10 · 106
1.20 · 106
1.30 · 106
1.40 · 106
1.50 · 106
0.70 · 1060 2 · 105 4 · 105 6 · 105 8 · 105 1060
time [years]
pressure
[Pa]
0.70 · 106
0.80 · 106
0.90 · 106
1.00 · 106
1.10 · 106
0.70 · 1060 2 · 105 4 · 1050
pgpℓ
0.00
0.02
0.04
0.06
0.08
0.10
0.000 2 · 105 4 · 105 6 · 105 8 · 105 1060
time [years]
saturation
[-]
0.00
0.02
0.04
0.000 2 · 105 4 · 1050
sg
0.70 · 106
0.80 · 106
0.90 · 106
1.00 · 106
1.10 · 106
1.20 · 106
1.30 · 106
1.40 · 106
1.50 · 106
0.70 · 1060 2 · 105 4 · 105 6 · 105 8 · 105 1060
time [years]
pressure
[Pa]
0.70 · 106
0.80 · 106
0.90 · 106
1.00 · 106
1.10 · 106
0.70 · 1060 2 · 105 4 · 1050
pgpℓ
Figure 7.7: Evolution of gas phase saturation and pressures at Γin for the easytest case (top) and the hard test case (bottom)
sg p` c1`
i N ereli EOCi erel
i EOCi ereli EOCi
0 100 1.78e-02 3.26e-04 9.02e-03
1 200 9.21e-03 0.95 1.92e-04 0.77 4.66e-03 0.95
2 400 4.67e-03 0.98 9.18e-05 1.06 2.36e-03 0.98
3 800 2.35e-03 0.99 4.74e-05 0.95 1.18e-03 0.99
4 1600 1.16e-03 1.01 2.31e-05 1.04 5.97e-04 0.99
5 3200 5.70e-04 1.03 1.13e-05 1.03 2.98e-04 1.00
6 6400 2.80e-04 1.03 5.61e-06 1.01 1.41e-04 1.08
7 12800 1.31e-04 1.10 2.62e-06 1.10 6.58e-05 1.10
Table 7.4: Relative L2-errors and experimental orders of convergence (EOC) forthe easy test case (α1 = 5 · 10−7) at t = 105 years.
7.2. INFLUENCE OF DIFFERENT RETENTION CURVES 147
sg p` c1`
i N ereli EOCi erel
i EOCi ereli EOCi
0 100 7.72e-2 1.68e-03 2.99e-02
1 200 4.88e-2 0.66 1.13e-03 0.57 1.71e-02 0.81
2 400 3.16e-2 0.63 5.82e-04 0.95 9.64e-03 0.82
3 800 2.11e-2 0.59 3.33e-04 0.81 5.33e-03 0.85
4 1600 1.38e-2 0.60 1.94e-04 0.78 2.89e-03 0.88
5 3200 8.97e-3 0.63 1.12e-04 0.80 1.52e-03 0.93
6 6400 5.50e-3 0.70 5.37e-05 1.06 7.71e-04 0.98
7 12800 3.15e-3 0.80 2.64e-05 1.03 3.72e-04 1.05
Table 7.5: Relative L2-errors and experimental orders of convergence (EOC) forthe hard test case (α2 = 5 · 10−4) at t = 105 years.
Tables 7.7 and 7.8, we list the number of time steps and Newton iterations (suc-
cessful and failed) that were necessary to compute the numerical solution on
different levels of refinement. For these computations, the linear extrapolation
of the previous time steps described in Section 6.2 was disabled and the value of
the previous time step was used as an initial guess for the global Newton method
at each time step.
Clearly, a significant higher number of Newton steps is required for the hard
test case than for the easy test case. More precisely, the number of Newton
steps almost doubles in each refinement step for the hard test case, whereas it
remains approximately constant for the easy test case. This is in accordance
with the results reported by R. de Cuveland and P. Bastian from the University
of Heidelberg [dC15], cf. Tables 7.9 and 7.10 . They use capillary pressure and
gas pressure as primary unknowns and discretize the equations with an Euler-
implicit cell-centered finite volume scheme. The nonlinear systems of equations
are solved with an inexact Newton method using the same stopping criterion as
in this work.
Although we solved a 2D problem with approximately twice as many unknowns as
for the corresponding 1D problem, our solver needed fewer time steps and Newton
iterations on all refinement levels. One of the reasons for this might be the fact
that we use the analytical Jacobian in our Newton method. Thus, no numerical
differentiation is required, which may lead to inaccurate approximations of the
148 CHAPTER 7. NUMERICAL RESULTS
control parameter value
εrel 10−6
εabs 0.0
max. line search steps 3
max. Newton iterations 30
∆tmin 1 year
∆tmax 106 years
OptSteps 10
Table 7.6: Control parameters for the performance test
Jacobian. Moreover, our time stepping algorithm and line search strategy is
different from the ones used in [dC15], which may have a strong impact on the
global performance. The choice of primary variables, however, seems to have
only little influence on the performance for this test problem. In further numerical
tests, we obtained the identical number of time steps and Newton iterations using
p` instead if pc or using pg instead of c1` = ξ1
`,ex.
We conclude this section by noting that our solver provides an accurate and
efficient numerical scheme which proved to be competitive to other fully implicit
solvers for the two-phase two-component model considered in this and in the
previous section.
N unknowns time steps (failed) Newton steps
100 404 21 (2) 153
200 804 20 (1) 144
400 1604 18 (3) 123
800 3204 29 (9) 246
1600 64042 31 (12) 253
3200 12804 32 (13) 263
Table 7.7: Benchmark performance results for the easy test case (α1 = 5 · 10−4)
7.2. INFLUENCE OF DIFFERENT RETENTION CURVES 149
N unknowns time steps (failed) Newton steps
100 404 93 (11) 994
200 804 167 (25) 1756
400 1604 273 (46) 3060
800 3204 454 (64) 4932
1600 6404 883 (166) 9659
3200 12804 1490 (380) 16626
Table 7.8: Benchmark performance results for the hard test case (α2 = 5 · 10−4)
N time steps (failed) Newton steps (failed)
100 48 (7) 309 (70)
200 62 (11) 422 (110)
400 76 (15) 511 (150)
800 89 (18) 630 (180)
160 92 (19) 643 (190)
3200 89 (18) 629 (180)
Table 7.9: Computational results reported by Uni Heidelberg (R. de Cuvelandand P. Bastian) for the easy test case (α1 = 5 · 10−7)
N time steps (failed) Newton steps (failed)
100 213 (55) 1335 (550)
200 403 (107) 2641 (1070)
400 781 (206) 5535 (2060)
800 1477 (390) 10701 (3990)
1600 2827 (746) 20751 (7440)
3200 5488 (1441) 38942 (13699)
Table 7.10: Computational results reported by Uni Heidelberg (R. de Cuvelandand P. Bastian) for the hard test case (α2 = 5 · 10−4)
150 CHAPTER 7. NUMERICAL RESULTS
7.3 CO2 sequestration
In this section, we consider the injection of CO2 into a deep aquifer covered by
an impermeable layer of rock for the purpose of permanent geological seques-
tration. This is considered as a promising approach to reduce the emission of
greenhouse gases into the atmosphere. The storage concept is based on different
trapping mechanisms that prevent the injected CO2 from being released into the
atmosphere.
Typically, CO2 is injected in its supercritical state where it adopts properties
midway between a gas and a liquid. More precisely, it has the low viscosity of
a gas and the high density of a liquid. Consequently, the ideal gas law used for
hydrogen injection is not applicable for carbon sequestration. Instead, we employ
the EOS of Duan [DMW92] to compute the density of CO2, cf. Figure 7.8.
When CO2 dissolves into the liquid phase, it causes density variations of the liquid
phase that may influence the flow regime. According to experimental data, the
density of aqueous solutions of CO2 can be as much as 2-3% higher than that of
pure water, which triggers downward migration due to gravitational forces. We
use the approach of Garcia [Gar01] to define the function f` that represents the
molar liquid phase density.
Regarding the solubility of CO2 in water, Henry’s law, which was used in the
previous numerical examples, is invalid at the high pressure conditions prevailing
at a deep underground CO2 storage site. We use the EOS of Spycher and Pruess
[SP05] to define the function φex representing the interphase mass exchange of
CO2. Figure 7.9 shows the solubility curves of CO2 in water for different tem-
peratures, neglecting salinity and the presence of other components in the liq-
uid phase. For temperatures below the critical temperature (Tcrit = 304.15K),
the curves exhibit a non-differentiable kink at the transition point between the
gaseous and liquid phase state. For the viscosity of the gas phase, an approach
of Fenghour et al. [FWV98] is used, cf. Figure 7.10.
Since the coefficient functions described above are given only implicitly, their
evaluation may be computationally expensive. Therefore, lookup tables are ini-
tially generated and cubic splines are used to evaluate the coefficient functions
be consistent with the liquid phase compressibility law. Their values are listed
in Table 7.13. Denoting by Q1 = c1`q` + ρmol,gqg + j1
` the total flux of CO2 and
by Qi = ci`q` + ji` the total flux of component i, i ∈ 2, . . . , 7, the following
boundary conditions are imposed on ∂Ω = Γin∪ΓD∪Γimp:
7.3. CO2 SEQUESTRATION 161
1. No-flux conditions on ∂Ω \ (ΓD ∪ Γin) for all components:
Qi` · n = 0 , i ∈ 1, . . . , 7 .
2. CO2 injection through Γin:
Q1 · n = q1in , Qi · n = 0 , i ∈ 2, . . . , 7 .
3. The Dirichlet values prescribed at ΓD for the pressures and the concentra-
tions coincide with the initial values:
p` = p0` , ci` = ci`,0 , i ∈ 1, . . . , 7 .
The boundary and initial values for all other variables are induced by the variable
transformation and the resolution function.
Numerical results
The numerical results obtained on a grid consisting of 40960 hexahedra are shown
in Figures 7.14–7.16. Shortly after the injection of CO2 has started, a gas phase
appears, which continuously moves upwards due to gravity and accumulates be-
low the impermeable upper boundary. The CO2 that has dissolved into the liquid
phase causes the pH to decrease, inducing the dissolution of calcite and Min A.
Thereby, metal ions and bicarbonate are released, which initiates the precipita-
tion of Min B.
The propagation of the concentration fronts, in particular the appearance of the
gas phase and the precipitation and dissolution of minerals are well reproduced
by our solver. The computations were carried out using a maximum time step
size ∆tmax = 5000s and the control parameter OptSteps=6 for the time step
adaption. With these parameters, 2742 time steps (including 667 unsuccessful
ones) were carried out to simulate a time span of 85 days with an average number
of 4.32 Newton steps per time step.
162 CHAPTER 7. NUMERICAL RESULTS
variable symbol value SI unit
porosity φ 0.2 [-]
absolute permeability K 10−12 m2
temperature T 313.15 K
Brooks–Corey parameter s`,res 0.0 [-]
Brooks–Corey parameter sg,res 0.0 [-]
Brooks–Corey parameter λ 2.0 [-]
Brooks–Corey parameter pentry 1000 Pa
molar mass of CO2 M1 4.4 · 10−2 kg mol−1
molar mass of water M2 1.8 · 10−2 kg mol−1
molar mass of bicarbonate M3 6.1 · 10−2 kg mol−1
molar mass of hydrogen M4 10−3 kg mol−1
molar mass of calcium M5 4 · 10−2 kg mol−1
molar mass of metal ions M6 1.5 · 10−2 kg mol−1
molar mass of silicon dioxide M7 6 · 10−2 kg mol−1
initial concentration of CO2 c1`,0 10−2 kg m−3 mol−1
initial concentration of water c2`,0 55333.33 kg m−3 mol−1
initial concentration of bicarbonate c3`,0 10−2 kg m−3 mol−1
initial concentration of hydrogen c4`,0 10−3 kg m−3 mol−1
initial concentration of calcium c5`,0 10−1 kg m−3 mol−1
initial concentration of metal ions c6`,0 10−4 kg m−3 mol−1
initial concentration of silicon dioxide c7`,0 10−2 kg m−3 mol−1
initial concentration of calcite c1s,0 0.1 kg m−3 mol−1
initial concentration of Min A c2s,0 0.2 kg m−3 mol−1
initial concentration of Min B c3s,0 0.0 kg m−3 mol−1
pure water mass density (T = 313.15K) ρstdmass,` 992 kg m−3
diffusion coefficient D` 2 · 10−9 m2 s−1
longitudinal dispersion coefficient αL 0.1 m2 s−1
transversal dispersion coefficient αT 0.01 m2 s−1
CO2 injection rate q1in 2 · 10−2 kg m−2 s−1
viscosity of the liq. phase (T = 313.15K) µ` 6.526 · 10−4 Pa.s
Table 7.13: Parameters for the mineral trapping scenario
7.3. CO2 SEQUESTRATION 163
Figure 7.14: Results after 7 days
164 CHAPTER 7. NUMERICAL RESULTS
Figure 7.15: Results after 20 days
7.3. CO2 SEQUESTRATION 165
Figure 7.16: Results after 85 days
Part II
Analysis of robust mixed hybrid
finite element discretizations for
advection–diffusion problems
166
Chapter 1
Introduction
In the second part of this work, we study mixed hybrid finite element approxi-
mations of the linear parabolic advection–diffusion problem
∂tc−∇ · (D∇c−Qc) = f in QT , (1.1)
c = c0 on Ω× 0 , (1.2)
c = 0 on ∂Ω× J , (1.3)
where Ω is a bounded domain in R2 with sufficiently smooth boundary, 0 <
Tend < ∞ denotes the final time, J = (0, Tend) and QT = Ω × J . This problem
may be regarded as a model problem for a wide range of applications arising,
e. g., in hydrology, civil engineering, petroleum engineering, economics and many
other fields, and the need for accurate, efficient and reliable schemes to solve
them numerically is well recognized. Despite of the simplicity of the problem and
extensive research over the last decades, the design of such methods remains a
challenging task, in particular when advection is strongly dominant. In this case,
numerical instabilities typically occur when standard numerical methods are used
since sharp layers in the solution cannot be resolved properly.
1.1 Current state of research
Generally, the interest of mixed methods lies in the fact that they are locally
mass conservative and provide accurate approximations of a scalar and a flux
167
168 CHAPTER 1. INTRODUCTION
unknown. Moreover, the flux approximations are continuous across interelement
boundaries.
There is a rich literature on mixed finite element methods and their conver-
gence. For a general overview on mixed finite element methods and their ap-
plications, we refer the reader to the books [BBF13, BBD+08, BF91], the re-
view articles [RT91, YAD10], and the references therein. Approximations of
advection–diffusion problems using the Raviart–Thomas element were studied
in [DR82, DR85], upwind-mixed methods are treated in [Daw98, DA99] and in
[RSH+11, Voh07]. Stabilization techniques based on using quadrature formulas
for the mass matrix are employed in [MSS01, SS97].
For approximations of a diffusion equation using the BDM1 element, we refer to
[BDM85]. The phenomenon of suboptimal convergence of the total flux variable
in the presence of an additional advection term is reported in [Dem02], where a
general second order advection–diffusion equation is analyzed.
1.2 Objective of this work
In this part, we study Euler-implicit mixed hybrid finite element methods to ap-
proximate the advection–diffusion problem (1.1)–(1.3) using the RT0 element and
the BDM1 element. While the discretization of the advective term relies on the
cellwise constant approximations of the scalar unknown if standard mixed meth-
ods are employed, cf. [DR82, DR85, Dem02], the schemes considered here use
the Lagrange multipliers arising in the hybrid problem formulation to discretize
the advective fluxes. This is motivated by the fact that the Lagrange multipliers
represent approximations of the scalar unknown on the edges of the triangula-
tion. They can even be used to define a higher order reconstruction in the space
of linear Crouzeix-Raviart elements; see, e. g., [AB85, BDM85].
For the RT0 element, we derive optimal order convergence estimates for a new
class of Euler-implicit methods designed to improve the robustness of the stan-
dard scheme for problems involving high Peclet numbers, including an upwind-
mixed hybrid scheme which is suitable for strongly advection–dominant problems,
cf. [RSH+11]. The definition of this new class involves a reconstruction of the
advective fluxes based on a general weight function, which may depend on the
Lagrange multipliers and the cellwise constant approximations of the scalar un-
1.3. OVERVIEW OVER THIS WORK 169
known. Moreover, the weight function is purely local, i. e., on each element its
definition involves only quantities defined on that element. As a consequence,
the flux unknowns and the scalar unknowns can be eliminated locally, result-
ing in a linear system for the Lagrange multipliers only. This local elimination
process is called static condensation, cf. Section 3.2.2. On the other hand, the
standard upwind-mixed scheme of Dawson [Daw98] relies on non-local upwind
weights requiring information from neighbor cells, such that static condensation
is not applicable. Therefore, our scheme can be implemented more efficiently
than the standard scheme, while the accuracy obtained by both methods in a
numerical experiment was almost identical, cf. Section 3.5.
If the same ideas are employed to discretizations based on theBDM1 element, it is
not only possible to improve the robustness with respect to advection–dominance:
using the Lagrange multipliers in the discretization of the advective term, the
order of convergence for the total flux variable is increased, and optimal second
order convergence in L2(Ω) is restored, which is not provided by the standard
BDM1 scheme if an advection term is present. The results presented in the
following have been published in the articles [BBKR13, BRK14], and the preprint
[BFK15].
1.3 Overview over this work
The second part of this thesis is structured as follows. In the next section, we
introduce some basic notation and state assumptions on the partial differential
equation and the computational mesh. In Chapter 2, a continuous mixed varia-
tional formulation associated with (1.1)–(1.3) is defined, and sufficient conditions
for existence and uniqueness are given. In Chapter 3, we study approximations
of (1.1)–(1.3) based on the RT0 element. More precisely, a new class of methods
is introduced, and optimal order convergence estimates are derived for the flux
and the scalar variable. At the end of Chapter 3, we discuss different schemes
belonging to this class, and we illustrate the theoretical results by numerical ex-
periments. Finally, in Chapter 4, we study approximations of (1.1)–(1.3) based
on the BDM1 element. The numerical results confirm that our modified scheme
approximates the total flux variable with optimal second order accuracy in L2(Ω),
whereas the standard scheme provides only suboptimal first order.
170 CHAPTER 1. INTRODUCTION
1.4 Notations and assumptions
Throughout the following, the common notations of functional analysis are used.
In particular, let (·, ·) denote the inner product in L2(Ω) or (L2(Ω))2, respectively,
and ‖ · ‖0 the corresponding norm. Further, 〈·, ·〉 indicates the inner product in
L2(∂Ω), and ‖ · ‖k stands for the norm in Hk(Ω) = W k,2(Ω). Similarly, if K ⊂ Ω
is measurable, let (·, ·)K denote the inner product in L2(K), ‖ · ‖k,K the norm
in Hk(K), and 〈·, ·〉∂K the inner product in L2(∂K). As usual, H(div,Ω) is
defined as the space of functions in (L2(Ω))2 having the (weak) divergence in
L2(Ω). For the time discretization, let the time elapsed at the n-th time step
be denoted by tn := n · τ , 0 ≤ n ≤ N , where N is an integer and τ := TendN
denotes the (constant) time step size. The time derivative of a function c will be
approximated by the backward difference quotient ∂cn := 1τ(cn−cn−1), where the
superscript n indicates the evaluation of a function at the discrete time t = tn.
Assumptions on the partial differential equation
Throughout this work, we solve the equation (1.1) along with the initial and
boundary conditions (1.2)–(1.3). The following assumptions are made on the
coefficient functions.
(A1) The coefficient matrix D = (Dij)ij is symmetric and Dij ∈ C(QT ) for
i, j ∈ 1, 2. Furthermore, the uniform ellipticity condition
C1|ξ|2 ≤2∑
i,j=1
Dij(x, t)ξiξj ≤ C2|ξ|2
holds for all ξ = (ξ1, ξ2)T ∈ R2, (x, t) ∈ QT , where C1, C2 > 0 are fixed
constants.
(A2) The velocity field Q and the source term f satisfy Q ∈ C(J ; (W 1,∞(Ω))2)
and f ∈ C(J ;L2(Ω)), respectively.
(A3) The initial value has the regularity c0 ∈ H10 (Ω).
In addition to the regularity obtained by Theorem 2.0.1, we shall assume that
the solution (q, c) of the mixed variational problem (2.5)–(2.6) satisfies
1.4. NOTATIONS AND ASSUMPTIONS 171
(A4) (q, c) ∈ C(J ; (H1(Ω))2)×H1(J ;H1(Ω)) ∩H2(J ;L2(Ω)).
Assumptions on the grid
Let Thh>0 denote a family of triangular decompositions of Ω such that
(M1) Ω =⋃K∈Th K and K1 ∩ K2 = ∅ if K1 6= K2, where the elements K ∈ Th
are closed triangles.
(M2) If K1 ∩K2 6= ∅ and K1 6= K2, then K1 ∩K2 is either a vertex or a full edge
of each.
(M3) If K ⊂ Ω, then K has straight edges only.
(M4) If K is a boundary triangle, the boundary edge can be curved.
(M5) hK := diam(K), h := maxK∈Th
hK .
(M6) Thh>0 is shape-regular, i. e., there exists a constant σmax > 0 such that
hK ≤ σmax ρK for all K ∈ Th, where
ρK := supdiam(S) : S is a disc in R2 and S ⊂ K
is the diameter of the inscribed circle of K.
The collection of all edges of an element K is denoted by E(K), whereas Eh =
EIh ∪ EDh represents the set of all edges of Th consisting of the disjoint subsets EIhand EDh of interior and boundary edges, respectively. By L2(E) for some E ∈ Eh,we denote the L2 space with respect to the surface measure on E. The notation
L2(Eh) refers to the L2 space with respect to the surface measure on the union of
all edges.
Chapter 2
Continuous mixed variational
formulation
Mixed finite element methods for problem (1.1)–(1.3) are based on a mixed re-
formulation of the problem. In this reformulation, the total flux q is introduced
as an additional explicit variable, and (1.1)–(1.3) translates into the system
∂tc = −∇ · q + f in QT , (2.1)
q = −D∇c+Qc in QT , (2.2)
c = c0 on Ω× 0 , (2.3)
c = 0 on ∂Ω× J . (2.4)
A mixed finite element method consists in approximating the quantities q and c
simultaneously in appropriate function spaces by discretizing the system (2.1)–
(2.4). The continuous mixed variational problem associated with (2.1)–(2.4) reads
as follows.
Problem 2. Find (q, c) ∈ L2(J ;H(div,Ω))×H1(J ;L2(Ω)) with c|t=0 = c0 such
It is well known that the linear algebraic system resulting when basis functions
of V h and Wh are employed in (3.4)–(3.5) is typically indefinite. Hence, com-
mon iterative solvers requiring a system matrix that is symmetric and positive
definite may fail to converge. To overcome this problem, a hybridization process
178 CHAPTER 3. MHFE SCHEMES BASED ON THE RT0 ELEMENT
can be employed, where the space V h is replaced by the augmented space V h,
i. e., the continuity of the normal fluxes across interelement boundaries is relaxed,
cf. [BF91]. The continuity constraints are then imposed by requiring additional
equations and introducing Lagrange multipliers from the space Λh. The resulting
hybrid formulation is equivalent to the original formulation. Moreover, by apply-
ing a local elimination procedure, one can derive a linear system in terms of the
Lagrange multipliers only, cf. Sec. 3.2.2. Hence, the resulting system has fewer
unknowns than that of the non-hybrid method. The mixed hybrid formulation
associated with (3.4)–(3.5) reads as follows.
Problem 4. Let n ∈ 1, . . . N and cn−1h ∈ Wh be given. Find (qnh, c
nh, λ
nh) ∈
V h ×Wh × Λh such that
((Dn)−1qnh,vh)−∑K∈Th
(∇ · vh, cnh)K − ((Dn)−1Qncnh,vh)
= −∑K∈Th〈λnh,vh · n∂K〉∂K ,
(3.6)
(∂cnh, wh) +∑K∈Th
(∇ · qnh, wh)K = (fn, wh) , (3.7)
∑K∈Th〈µh, qnh · n∂K〉∂K = 0 (3.8)
for all (vh, wh, µh) ∈ V h ×Wh × Λh.
Here, n∂K denotes the outer unit normal vector of ∂K. The Lagrange multipliers
may be regarded as an approximation of the scalar unknown on the element
interfaces. In fact, they carry some extra information about the exact solution
which can be used to reconstruct a higher order nonconforming approximation in
L2(Ω). We refer to [AB85], where this was shown for an elliptic model problem.
The use of the Lagrange multipliers in the discretization of the advective term is
one of the key ideas in the definition of the new class of methods. Moreover, the
exact velocity field is replaced by the approximation
Qnh := ΠhQ
n =∑K∈Th
∑E∈E(K)
QnKEvKE ∈ V h ,
3.2. A NEW CLASS OF EULER-IMPLICIT MHFE SCHEMES 179
which satisfies (cf. [Dur88])
‖Qn −Qnh‖L∞(Ω) ≤ Ch‖Qn‖W 1,∞(Ω) . (3.9)
Note that in many applications, an approximation of the velocity field Qn is
computed in a preprocessing step by solving Richards’ equation. If the RT0
element is used for this calculation, the coefficients QnKE are directly known.
In the schemes studied here, the advective terms are discretized using the re-
construction operator B : Λh × Wh → V h, which—given a piecewise constant
approximation λnh =∑
E∈EIhλnEµE ∈ Λh of cn on the edges of the triangulation
and the cellwise constant approximation cnh =∑
K∈Th cnKχK—reconstructs an ap-
proximation for the advective flux in the space V h. More precisely, the operator
B is defined to act on Λh ×Wh as
B(λnh, cnh) =
∑K∈Th
∑E∈E(K)
αnKE(cnK , λnE)Qn
KEvKE ,
where the weights αnKE are linear functionals on R×R and satisfy the condition
|αnKE(cnK , λnE)− cnK | ≤ C|λnE − cnK | . (3.10)
In contrast to the standard scheme, where the approximation of the advective
flux relies on the cellwise constant approximations cnK only, the discretization of
the advective fluxes in the mixed methods based on the reconstruction operator Bmay additionally depend on the Lagrange multipliers. Different schemes that are
recovered for specific choices of the weights αnKE are discussed in Section 3.4. The
class of mixed methods based on the reconstruction operator B and the weight
functions αnKE reads as follows.
Problem 5. Let n ∈ 1, . . . N and cn−1h ∈ Wh be given. Find (qnh, c
nh, λ
nh) ∈
V h ×Wh × Λh with the representations
qnh =∑K∈Th
∑E∈E(K)
qnKEvKE , cnh =∑K∈Th
cnKχK , λnh =∑E∈EIh
λnEµE (3.11)
180 CHAPTER 3. MHFE SCHEMES BASED ON THE RT0 ELEMENT
such that
((Dn)−1qnh,vh)−∑K∈Th
(∇ · vh, cnh)K − ((Dn)−1B(λnh, cnh),vh)
= −∑K∈Th〈λnh,vh · n∂K〉∂K ,
(3.12)
(∂cnh, wh) +∑K∈Th
(∇ · qnh, wh)K = (fn, wh) , (3.13)
∑K∈Th〈µh, qnh · n∂K〉∂K = 0 (3.14)
for all (vh, wh, µh) ∈ V h ×Wh × Λh.
Note that an extension of the method to three space dimensions using Raviart–
Thomas elements of lowest order on tetrahedral grids is immediate and requires
only minor changes in the proof of convergence below, cf. Theorem 3.3.4.
For later use, let us finally state the standard upwind-mixed method, cf. [Daw98,
Voh07], which approximates the diffusive flux q = −D∇c.Problem 6. Let n ∈ 1, . . . , N and cn−1
h be given. Find (qnh, cnh) ∈ V h ×Wh
satisfying
((Dn)−1qnh,vh)− (∇ · vh, cnh) = 0 ,
(∂cnh, wh) + (∇ · qnh, wh) +∑K∈Th
∑E∈E(K)
QnKEσ
nKEwK = (fn, wh)
for all (vh, wh) ∈ V h ×Wh, where wK = wh|K and
σnKE =
cnK if QnKE ≥ 0 ,
cnK′ otherwise
if E is a common edge of K and K ′, and
σnKE =
cnK if QnKE ≥ 0 ,
0 otherwise
if E is a boundary edge.
3.2. A NEW CLASS OF EULER-IMPLICIT MHFE SCHEMES 181
3.2.2 Static condensation
Let us assume for a moment that there exists a unique solution (qnh, cnh, λ
nh) ∈ V h×
Wh × Λh of Problem 5 having the basis representations (3.11). Employing these
representations in the equations for the fluxes (3.12), using the basis functions as
hold for all (vh, wh, µh) ∈ V h×Wh×Λh. Next, we proceed similarly as in [AB85]
to obtain an estimate for the errors ‖λnh,1 − λnh,2‖0,E on the edges. For K ∈ Thand E ∈ E(K) let τE denote the unique element of V h with supp(τE) ⊆ K and
τE · nE′ =
λnh,1 − λnh,2 if E = E ′ ,
0 otherwise
3.3. ERROR ANALYSIS OF THE FULLY DISCRETE PROBLEM 183
for all edges E ′ ∈ E(K). Then, it follows from a simple scaling argument that
Finally, combining the last estimate with (3.31) yields∑K∈Th
∑E∈E(K)
|E|2(λnE − cnK)2 ≤ C‖cnh − cnh‖20 ≤ C
(h2‖qn − qnh‖2
0 + ‖cn − cnh‖20
+ h2‖Qnhcnh −Qncn‖2
0 + h2∑K∈Th
∑E∈E(K)
|E|2(λnE − cnK)2 + h2‖cn‖21
),
and (3.21) follows by pushing back the last term on the right hand side for
sufficiently small h.
188 CHAPTER 3. MHFE SCHEMES BASED ON THE RT0 ELEMENT
For later use, we state the following discrete integration by parts formula.
Lemma 3.3.3. Let (an)n∈N0 and (bn)n∈N0 be real sequences and let m ∈ N. Then,
it holds that
m∑n=1
(an − an−1)bn = ambm − a0b0 −m∑n=1
an−1(bn − bn−1) .
Theorem 3.3.4. Let the assumptions (A1)–(A3) hold and denote by (q, c) ∈L2(J ;H(div,Ω))×H1(J ;L2(Ω)) the unique solution of Problem 2. Moreover, let
τ and h be sufficiently small such that for every n ∈ 1, . . . , N there exists a
unique solution (qnh, cnh, λ
nh) ∈ V h × Wh × Λh of Problem 5. Then, if (A4) is
satisfied and c0h = Phc0, there exists a constant C > 0, independent of τ and h,
such that
maxm∈1,...,N
‖c(·, tm)− cmh ‖20 + τ
N∑n=1
‖q(·, tn)− qnh‖20 ≤ C(τ 2 + h2) . (3.32)
Proof. We take t = tn in (2.5)–(2.6), subtract (3.12)–(3.13) from the resulting
equations and use the properties (3.1a)–(3.1b) to obtain the error equations
((Dn)−1(qn − qnh),vh)−∑K∈Th
(∇ · vh, Phcn − cnh)K − ((Dn)−1Qncn,vh)
+∑K∈Th
∑E∈E(K)
QnKEα
nKE((Dn)−1vKE,vh) =
∑K∈Th〈λnh,vh · n∂K〉∂K ,
(3.33)
(∂tcn − ∂cnh, wh) +
∑K∈Th
(∇ · (Πhqn − qnh), wh)K = 0 , (3.34)
∑K∈Th〈(Πhq
n − qnh) · n∂K , µh〉∂K = 0 (3.35)
for all (vh, wh, µh) ∈ V h × Wh × Λh. Note that our regularity assumptions
allow the evaluation of c and q and all coefficient functions at t = tn. Taking
vh := Πhqn−qnh, wh := Phc
n−cnh and µh := −λnh as test functions in (3.33)–(3.35)
3.3. ERROR ANALYSIS OF THE FULLY DISCRETE PROBLEM 189
and adding the resulting equations yields
((Dn)−1(qn − qnh),Πhqn − qnh)− ((Dn)−1Qncn,Πhq
n − qnh)
+ (∂tcn − ∂cnh, Phcn − cnh) +
∑K∈Th
∑E∈E(K)
QnKEα
nKE((Dn)−1vKE,Πhq
n − qnh) = 0 .
The last identity can be rewritten equivalently as
((Dn)−1(qn − qnh),Πhqn − qnh) + ((Dn)−1(Qn
hcnh −Qncn),Πhq
n − qnh)
+∑K∈Th
∑E∈E(K)
QnKE(αnKE − cnK)((Dn)−1vKE,Πhq
n − qnh)
+ (∂tcn − ∂cnh, Phcn − cnh) = 0 .
(3.36)
Further, for any m ∈ N with 1 ≤ m ≤ N , by summing (3.36) from n = 1, . . . ,m,
multiplying by 2τ and expanding, we obtain
T1 + . . .+ T8 := 2τm∑n=1
((Dn)−1(qn − qnh), qn − qnh)
+ 2τm∑n=1
((Dn)−1(qn − qnh),Πhqn − qn)
+ 2τm∑n=1
((Dn)−1(Qnhcnh −Qncn),Πhq
n − qnh)
+ 2τm∑n=1
(∂tcn − ∂cn, cn − cnh)
+ 2τm∑n=1
(∂tcn − ∂cn, Phcn − cn)
+ 2τm∑n=1
(∂(cn − cnh), cn − cnh)
+ 2τm∑n=1
(∂(cn − cnh), Phcn − cn)
+ 2τm∑n=1
∑K∈Th
∑E∈E(K)
QnKE(αnKE − cnK)((Dn)−1vKE,Πhq
n − qnh)
= 0 .
190 CHAPTER 3. MHFE SCHEMES BASED ON THE RT0 ELEMENT
We now proceed to estimate separately each of the terms T1, . . . T8. First, the
terms T1 and T6 are bounded from below. Note that sinceD is uniformly positive
definite, D−1 is also uniformly positive definite. Thus, we have
T1 = 2τm∑n=1
((Dn)−1(qn − qnh), qn − qnh) ≥ Cτm∑n=1
‖qn − qnh‖20 . (3.37)
By the identity 2a(a− b) = a2 − b2 + (a− b)2, we find
T6 = 2m∑n=1
((cn − cnh)− (cn−1 − cn−1h ), cn − cnh)
=m∑n=1
‖cn − cnh‖20 −
m∑n=1
‖cn−1 − cn−1h ‖2
0 +m∑n=1
‖cn − cnh − cn−1 + cn−1h ‖2
0
≥ ‖cm − cmh ‖20 − ‖c0 − c0
h‖20 .
(3.38)
All other terms are passed to the right hand side and bounded from above. Using
a weighted Young’s inequality, we get for any δ2 > 0
|T2| ≤ Cδ2τm∑n=1
‖qn − qnh‖20 +
Cτ
δ2
m∑n=1
‖Πhqn − qn‖2
0 . (3.39)
The term T3 is rewritten as
T3 = 2τm∑n=1
((Dn)−1(Qnhcnh −Qncn), qn − qnh)
+ 2τm∑n=1
((Dn)−1(Qnhcnh −Qncn,Πhq
n − qn) =: T31 + T32 .
Then, for h sufficiently small, it follows from (3.9) and the regularity of Q that
‖Qnhcnh −Qncn‖0 ≤ ‖Qn
h(cn − cnh)‖0 + ‖(Qn −Qnh)cn‖0
≤ ‖Qnh‖L∞(Ω)‖cn − cnh‖0 + ‖Qn −Qn
h‖L∞(Ω)‖cn‖0
≤ C(‖cn − cnh‖0 + h‖cn‖0) .
(3.40)
Moreover, from the uniform boundedness of D−1 and a weighted Young’s in-
3.3. ERROR ANALYSIS OF THE FULLY DISCRETE PROBLEM 191
equality, we obtain that for any δ3 > 0
|T31| ≤Cτ
δ3
m∑n=1
‖Qnhcnh −Qncn‖2
0 + Cδ3τm∑n=1
‖qn − qnh‖20
≤ Cτ
δ3
m∑n=1
‖cn − cnh‖20 +
Ch2τ
δ3
m∑n=1
‖cn‖20 + Cδ3τ
m∑n=1
‖qn − qnh‖20 .
(3.41)
Similarly, we derive the estimate
|T32| ≤ Cτ
m∑n=1
‖cn − cnh‖20 + Ch2τ
m∑n=1
‖cn‖20 + Cτ
m∑n=1
‖Πhqn − qn‖2
0 . (3.42)
Next, we use Bochner’s inequality, the Cauchy-Schwarz inequality and the regu-
larity of c to find
|T4| ≤1
τ
m∑n=1
‖τ∂tcn − cn + cn−1‖20 + τ
m∑n=1
‖cn − cnh‖20
=1
τ
m∑n=1
∥∥∥∥∫ tn
tn−1
∫ tn
s
∂ttc(η) dη ds
∥∥∥∥2
0
+ τm∑n=1
‖cn − cnh‖20
≤ τ 2
m∑n=1
∫ tn
tn−1
‖∂ttc(η)‖20 dη + τ
m∑n=1
‖cn − cnh‖20
≤ τ 2‖∂ttc‖2L2(J ;L2(Ω)) + τ
m∑n=1
‖cn − cnh‖20
≤ Cτ 2 + τm∑n=1
‖cn − cnh‖20 .
(3.43)
Analogously,
|T5| ≤ τ 2‖∂ttc‖2L2(J ;L2(Ω)) + τ
m∑n=1
‖Phcn − cn‖20
≤ Cτ 2 + τ
m∑n=1
‖Phcn − cn‖20 .
(3.44)
192 CHAPTER 3. MHFE SCHEMES BASED ON THE RT0 ELEMENT
For T7, we employ Lemma 3.3.3 and the definition c0h = Phc0 to get
|T7| ≤ δ7‖cm − cmh ‖20 +
1
δ7
‖Phcm − cm‖20 + (2 + τ)‖Phc0 − c0‖2
0
+ τ
m−1∑n=1
‖cn − cnh‖20 +
1
τ
m∑n=1
‖(Ph − I)(cn − cn−1)‖20 ,
(3.45)
where I denotes the identity operator. Denoting the last term on the right hand
side of (3.45) by T71, we obtain from Bochner’s inequality, the Cauchy-Schwarz
inequality and the regularity of c that
T71 ≤Ch2
τ
m∑n=1
‖cn − cn−1‖21 =
Ch2
τ
m∑n=1
∥∥∥∥∫ tn
tn−1
∂tc(η) dη
∥∥∥∥2
1
≤ Ch2
∫ tm
0
‖∂tc(η)‖21 dη ≤ Ch2‖∂tc‖2
L2(J ;H1(Ω))
≤ Ch2 .
Consequently,
|T7| ≤ δ7‖cm − cmh ‖20 +
1
δ7
‖Phcm − cm‖20 + (2 + τ)‖Phc0 − c0‖2
0
+ τm∑n=1
‖cn − cnh‖20 + Ch2 .
(3.46)
For the remaining term T8, we combine (3.26) and (3.3) to find
|T8| ≤Cτ
δ8
m∑n=1
∑K∈Th
∑E∈E(K)
|E|2(λnE − cnK)2 + Cδ8τm∑n=1
‖Πhqn − qn‖2
0
+ Cδ8τm∑n=1
‖qn − qnh‖20 ,
3.4. CHOICE OF THE WEIGHT FUNCTION 193
and it follows from Lemma 3.3.2 and (3.40) that
|T8| ≤C
δ8
(h2τ
m∑n=1
‖cn‖21 + τ
m∑n=1
‖cn − cnh‖20 + h2τ
m∑n=1
‖qn − qnh‖20
+ h2τ
m∑n=1
‖cn − cnh‖20 + h4τ
m∑n=1
‖cn‖20
)+ Cτδ8
m∑n=1
‖Πhqn − qn‖2
0
+ Cτδ8
m∑n=1
‖qn − qnh‖20 .
(3.47)
Finally, by collecting the estimates (3.37)–(3.47), choosing δ2, δ3, δ7, δ8, τ and h
sufficiently small, and pushing back terms to the left hand side, we obtain
‖cm − cmh ‖20 + τ
m∑n=1
‖qn − qnh‖20 ≤ Cτ 2 + Ch2 + Cτ
m∑n=1
‖Πhqn − qn‖2
0
+ Cτm∑n=1
‖Phcn − cn‖20 + C‖Phc0 − c0‖2
0 + C‖Phcm − cm‖20
+ Ch2τm∑n=1
‖cn‖20 + Cτ
m∑n=1
‖cn − cnh‖20 + Ch2τ
m∑n=1
‖cn‖21 ,
and (3.32) follows from the properties of the projectors, the regularity of q and
c, and by applying the discrete Gronwall lemma.
Note that since we applied the discrete Gronwall lemma, the constant in the error
estimate (3.32) is potentially large. This is in accordance with the convergence
analysis of the upwind-mixed methods in [Daw91] and [Daw93]. In [DA99], the
use of the Gronwall lemma was avoided, but only suboptimal convergence was
obtained for the semidiscrete problem.
3.4 Choice of the weight function
The proof of convergence given in the previous section applies to all schemes that
are obtained if the weight functions αnKE satisfy (3.10). This condition holds for
194 CHAPTER 3. MHFE SCHEMES BASED ON THE RT0 ELEMENT
the upwind-mixed method resulting from the choice
αn,upKE (cnK , λ
nE) =
cnK if QnKE ≥ 0 ,
2λnE − cnK otherwise ,
which was studied numerically in the paper [RSH+11] in the context of reactive
transport simulation. This upwind weighting formula is motivated by the fact
that the Lagrange multipliers represent an approximation of the scalar unknown
on the edges. Hence, the expression 2λnE − cnK may be regarded as an extrapo-
lation of the concentration on the adjacent cell. The advantage of the resulting
upwind-mixed hybrid method is that it is fully local, i. e., it does not incorporate
information of adjacent cells. Consequently, static condensation can be employed
to eliminate the flux and the scalar unknowns from the system, cf. Section 3.2.2.
This implies that the method can be implemented more efficiently than the stan-
dard upwind-mixed method stated in Problem 6, where the definition of the
upwind weights explicitly involves concentration values of adjacent grid cells. In
the next section, the upwind-mixed hybrid method resulting from the choice αn,upKE
is compared with the standard upwind-mixed scheme in terms of accuracy and
efficiency.
It is well-known that full upwind methods may introduce a significant amount of
numerical diffusion, which leads to an artificial smearing of the numerical solution.
We refer to the numerical experiments in [RSH+11], where the amount of artificial
diffusion of the upwind-mixed hybrid method was quantified and compared with
other discretization schemes. One possibility to reduce the amount of artificial
diffusion is to use the partial upwind scheme obtained for the choice
αn,partKE (cnK , λ
nE) =
cnK if QnKE ≥ 0 ,
(1− νE)cnK + νE(2λnE − cnK) otherwise ,
which obviously satisfies condition (3.10). Here, νE ∈ [0, 1] denotes a coefficient
describing the amount of upstream weighting. It may depend on the local grid
Peclet number to ensure that upwinding is only employed in those parts of the
domain where the flow is advection–dominant. Obviously, for νE = 1, the full
upwind scheme is recovered.
3.4. CHOICE OF THE WEIGHT FUNCTION 195
Qn
cnK1b cnK2
bλnEb
E
Figure 3.1: Scalar unknowns and Lagrange multiplier associated with the com-mon edge of two adjacent triangles K1 and K2
Further admissible choices for the weight function are given by
αn,lagKE (cnK , λ
nE) = λnE ,
which was tested in [RSH+11] and proved to be robust for moderately advection–
dominant problems, and
αn,stdKE (cnK , λnE) = cnK ,
which represents the standard scheme (3.6)–(3.8) with the only difference that
Qn was approximated by Qnh. Another full upwind scheme is recovered for the
choice
αn,up,2KE (cnK , λ
nE) =
cnK if QnKE ≥ 0 ,
λnE otherwise .
This scheme was tested numerically in the paper [BFK14] and performed favor-
ably compared to an upwind-weighted finite volume scheme in terms of robust-
ness, monotonicity properties, and the amount of artificial numerical diffusion
introduced by both methods.
196 CHAPTER 3. MHFE SCHEMES BASED ON THE RT0 ELEMENT
3.5 Numerical results
In this section, we present a numerical example for the upwind-mixed hybrid
method obtained for the weight αn,upKE defined in the previous section. More
precisely, we provide computational evidence that our error estimates are sharp,
and we compare the upwind-mixed hybrid method with the standard upwind-
mixed method of Dawson [Daw98]. Further numerical examples from the context
of contaminant biodegradation with advection–dominated flow are included in
[Bru10]. Moreover, in [RSH+11], the numerical diffusion of the upwind-mixed
hybrid method was quantified and compared with other discretization methods.
In our computations, we solve problem (1.1)–(1.3) on the unit square Ω = (0, 1)2
and the time interval J = (0, 1). The (scalar) diffusion coefficient and the velocity
field are given by D = 1 and Q = (0,−1)T , respectively. The source term f is
chosen so that c(x, y, t) = x(1− x)y(1− y)e−t represents the analytical solution
of the problem, and c0 := c(·, 0) is imposed as the initial value. A simple uniform
triangular mesh with h =√
2 is used as a coarse grid, and τ = 0.5 is chosen as
the initial time step size. In each level of refinement, h and τ are halved, and the
errors
Eτ,h =
‖c(·, tN)− cNh ‖2
0 + τN∑n=1
‖q(·, tn)− qnh‖20
1/2
are computed. For the upwind scheme of Dawson, we use qnh = qnh + Qcnh ∈RT0(Ω, Th) as an approximation of the total flux, where qh denotes the flux vari-
able of the scheme, which approximates the diffusive flux q = −D∇c. Both
methods were implemented in the software package M++ [Wie, Wie05]. To han-
dle the saddle point problems arising from the standard upwind-mixed scheme,
the linear systems are solved with a direct solver based on the SuperLU library
[Li05]. The results of the computations are summarized in Tables 3.1 and 3.2.
They clearly show optimal first order convergence in h and τ for our upwind-
mixed hybrid scheme. Hence, the error bound of Theorem 3.3.4 is sharp. More-
over, the magnitude of the errors almost coincides for both methods, while the
computation time was 50% lower on the finest mesh for the hybrid scheme.
On the one hand, this is due to a lower number of global unknowns resulting
from hybridization and employing static condensation, cf. Sec. 3.2.2. On the
3.5. NUMERICAL RESULTS 197
Ref. level # unknowns Eτ,h reduction CPU time [s]
1 16 3.685e-2 0.01
2 56 2.297e-2 1.60 0.03
3 208 1.236e-2 1.85 0.09
4 800 6.345e-3 1.95 0.47
5 3136 3.205e-3 1.98 3.88
6 12416 1.610e-3 1.99 33.11
7 49408 8.067e-4 2.00 305.90
Table 3.1: Numerical results for the upwind-mixed hybrid method
Ref. level # unknowns Eτ,h reduction CPU time [s]
1 24 3.507e-2 0.01
2 88 2.264e-2 1.54 0.03
3 226 1.234e-2 1.83 0.10
4 1312 6.349e-3 1.94 0.66
5 5184 3.208e-3 1.98 5.79
6 20608 1.611e-3 1.99 57.52
7 82176 8.068e-4 2.00 709.00
Table 3.2: Numerical results for the upwind-mixed method [Daw98]
other hand, the linear system associated with our method is sparser than that
corresponding to the non-hybrid method. Hence, the linear solver is faster and
less memory is required during the computation.
Chapter 4
Mixed hybrid finite element
discretizations based on the
BDM1 element
In the previous chapter, a new class of mixed hybrid finite element methods
based on the RT0 element was studied, where the advective fluxes are discretized
using the Lagrange multipliers introduced during hybridization. Depending on
the particular choice of the weight functions, different schemes are recovered from
this general class, e. g., the standard scheme or an upwind-mixed hybrid scheme
designed for strongly advection–dominated problems.
In this chapter, similar ideas are applied to approximations based on the BDM1
element. More precisely, we will show that suboptimal convergence for the flux
variable, which is known to arise when the standard BDM1 scheme is employed to
advection–diffusion problems, can be overcome by using the Lagrange multipliers
for the discretization of the advective fluxes. This observation was made and
confirmed numerically in the paper [BRBK12]. The error analysis is provided in
the article [BFK15].
198
4.1. SUBOPTIMAL CONVERGENCE OF THE STANDARD SCHEME 199
4.1 Suboptimal convergence of the standard non-
hybrid scheme
Using the Raviart–Thomas spaces for the discretization of (1.1)–(1.3), the flux
and the scalar unknown are approximated with the same order of convergence.
For applications where the main interest lies in the flux variable, the BDM
spaces were introduced [BDM85], which are able to approximate the flux to one
order higher than the scalar variable. Indeed, if the BDM1 mixed finite element
method is used to discretize a pure second order elliptic diffusion equation, the
flux is approximated with second order accuracy in the L2 norm. Error estimates
in L2(Ω) and L∞(Ω) for general second order elliptic problems with nonvanish-
ing advection using the BDM family of methods were derived in [Dem02] and
[Dem04], respectively. More precisely, it was shown that the total flux variable
consisting of diffusive and advective transport is approximated with first order
accuracy in L2(Ω), which is suboptimal since the same order of convergence is
obtained by the computationally less expensive RT0 element. This phenomenon
of suboptimal convergence may occur whenever the mixed finite element spaces
employed use polynomials of higher order for the approximation of the flux vari-
able than for the approximation of the scalar variable. The order of convergence
is then limited by the first order accuracy of the approximation for the conserved
quantity itself.
In the following, we will present a modified mixed hybrid BDM1 scheme which
restores optimal second order convergence of the flux variable in the presence of
an advective transport term. The modification consists in replacing the cellwise
constant approximation of the scalar unknown in the definition of the discrete
advective flux by a reconstruction based on the interelement Lagrange multipliers.
This is motivated by the well-known fact that the Lagrange multipliers carry some
higher order information about the scalar unknown.
200 CHAPTER 4. MHFE SCHEMES BASED ON THE BDM1 ELEMENT
b
b b
b
b b
K
Figure 4.1: Degrees of freedom of the local BDM1 space
4.2 The modified hybrid scheme
Mixed approximation spaces and projections
In the following, we assume that the domain Ω ⊂ R2 is polygonally bounded
and that Thh>0 denotes a family of shape-regular triangulations of Ω where
all triangles have straight edges only, cf. (M1)–(M3), (M5)–(M6). For the
discretization with the BDM1 mixed finite element method, let
V h = BDM1(Ω, Th) = v ∈ H(div,Ω) : v|K ∈ BDM1(K) for all K ∈ Th ,
Wh = w ∈ L2(Ω) : w|K ∈ P0(K) for all K ∈ Th ,
where BDM1(K) = (P1(K))2. The degrees of freedom for BDM1(K) are illus-
trated in Figure 4.1.
The standard BDM1 mixed finite element method for approximating the total
flux q = −D∇c+Qc reads as follows, cf. [Dem02].
Problem 7. Let n ∈ 1, . . . N and cn−1h ∈ Wh be given. Find (qnh, c
Here, n∂K denotes the outer unit normal vector of ∂K. The basis functions of
Wh are given by characteristic functions χK for each K ∈ Th. With the help of
the basis functions, we expand the unknowns in terms of the basis functions,
qnh =∑K∈Th
∑E∈E(K)
2∑i=1
qnKEiviKE ,
λnh =∑E∈EIh
2∑i=1
λnEiµiE ,
cnh =∑K∈Th
cnKχK .
Expanding the velocity field Qn as
Qn = Π1hQ
n +Qnr =
∑K∈Th
∑E∈E(K)
2∑i=1
QnKEivKEi +Qn
r =: Qnh +Qn
r ,
where Π1h denotes the usual BDM1 projection operator [BDM85], and using the
abbreviations
BnK,EiE′j
:= ((Dn)−1vKEi ,vKE′j) , F nK := (fn, χK) ,
we can approximate the standard mixed hybrid finite element scheme by the
4.2. THE MODIFIED HYBRID SCHEME 203
linear system (the approximation consisting in using Qnh instead of Qn)
|K|cnK − cn−1
K
τ+
∑E∈E(K)
2∑i=1
qnKEi = F nK ∀K ∈ Th , (4.7)
∑E∈E(K)
2∑i=1
BnK,EiE′j
qnKEi − cnK −∑
E∈E(K)
2∑i=1
QnKEi
cnKBnKEiE′j
= −λnE′j
∀K ∈ Th , E ′ ∈ E(K) , j = 1, 2 ,
(4.8)∑K∈Th,E∈E(K)
qnKEi = 0 ∀E ∈ EIh , i = 1, 2 . (4.9)
Obviously, the discretization of the advective term in (4.8) relies on the cellwise
constant approximation cnh ∈ Wh of the scalar unknown cn. As in the previous
chapter, we define a modified scheme by replacing the cellwise constant value
cnK with a reconstruction based on the Lagrange multiplier λnh (evaluated in the
points dividing an edge into three segments of equal length). The resulting linear
system reads
|K|cnK − cn−1
K
τ+
∑E∈E(K)
2∑i=1
qnKEi = F nK ∀K ∈ Th , (4.10)
∑E∈E(K)
2∑i=1
BnK,EiE′j
qnKEi − cnK −∑
E∈E(K)
2∑i=1
QnKEi
λnE1+ λnE2
+ λnEi3
BnKEiE′j
= −λnE′j
∀K ∈ Th , E ′ ∈ E(K) , j = 1, 2 ,
(4.11)∑K∈Th,E∈E(K)
qnKEi = 0 ∀E ∈ EIh , i = 1, 2 . (4.12)
In order to find a representation of the modified scheme on the finite element
level, we define an operator B : Λh → V h, which reconstructs an approximation
for the advective flux in the space V h. It is defined to act on Λh as
B(λnh) :=∑K∈Th
∑E∈E(K)
2∑i=1
λnEi + λnE1+ λnE2
3QnKEivKEi ,
204 CHAPTER 4. MHFE SCHEMES BASED ON THE BDM1 ELEMENT
v1KE1
(x, y) = (−2x, 6x+ 4y − 4)T v2KE1
(x, y) = (4x,−6x− 2y + 2)T
v1KE2
(x, y) = (4x,−2y)T v2KE2
(x, y) = (−2x, 4y)T
v1KE3
(x, y) = (−2x− 6y + 2, 4y)T v2KE3
(x, y) = (4x+ 6y − 4,−2y)T
Table 4.1: BDM1 basis functions on the reference triangle K
i. e., the normal component B(λnh) · n is prescribed at the two points of an edge
dividing the edge into three segments of equal size to match the product of the
normal component of Qnh and the Lagrange multiplier λnh at these points. This
property of the reconstruction operator is crucial for its higher order approxima-
tion property. With the help of the operator B, the modified linear system (4.10)–
(4.12) can be rewritten as the following mixed hybrid finite element method.
Problem 9. Let n ∈ 1, . . . N and cn−1h ∈ Wh be given. Find (qnh, c
nh, λ
nh) ∈
V h ×Wh × Λh such that
((Dn)−1qnh,vh)−∑K∈Th
(∇ · vh, cnh)K − ((Dn)−1B(λnh),vh)
= −∑K∈Th〈λnh,vh · n〉∂K , (4.13)
(∂cnh, wh) +∑K∈Th
(∇ · qnh, wh)K = (fn, wh) , (4.14)
∑K∈Th〈µh, qnh · n〉∂K = 0 (4.15)
for all (vh, wh, µh) ∈ V h ×Wh × Λh.
Note that in three space dimensions, a similar modification of the standard scheme
is possible by defining an appropriate reconstruction operator. The theoretical
results continue to hold for this case, cf. [BFK15].
4.3. OPTIMAL CONVERGENCE OF THE MODIFIED SCHEME 205
4.3 Optimal second-order convergence of the mod-
ified hybrid scheme
The convergence of the modified scheme is provided by the following theorem,
which was shown in [BFK15].
Theorem 4.3.1. Let D ∈ L∞(Ω × [0, T ];R2×2) be uniformly elliptic with el-
lipticity constant λ > 0 and upper bound Λ > 0 and let f ∈ L2(Ω × [0, T ]),
Q ∈ L∞([0, T ];W 2,∞(Ω)). Consider a weak solution u ∈ L2([0, T ];H10 (Ω)) to the
problem (1.1)–(1.3) and let q be defined by (2.2). For every n ∈ 1, . . . , N, let
(qnh, cnh, λ
nh) ∈ V h ×Wh × Λh denote a solution of the numerical scheme (4.13)–
(4.15). Then, there exists some hmax = hmax(Ω, σmax, λ,Λ, ||Q||L∞([0,T ],W 1,∞(Ω)))
and some maximal time step size τmax = τmax(Ω, σmax, λ,Λ, ||Q||L∞(Ω×[0,T ])) such
that the following holds: Provided that the smallness conditions τ < τmax and
h < hmax are satisfied, the a priori error estimate
maxm∈1,...,N
||P 0hc(·, tm)− cmh ||20 +
N∑n=1
τ ||q(·, tn)− qnh||20
≤ C(||P 0
hc0 − c0h||20 + ||∂ttc||2L∞(Ω×[0,T ])τ
2 + ||q||2L∞([0,T ];H2(Ω))h4
+ ||c||2L∞([0,T ];H2(Ω))h4)
holds, where C = C(Tend, σmax, λ,Λ,Ω, ||Q||L∞([0,T ];W 2,∞(Ω))) is independent of the
discretization parameters.
The main step in proving this theorem is the derivation of error estimates for the
Lagrange multipliers by employing techniques from the a posteriori error analysis
presented in [BDM85]. Indeed, it can be shown that the Lagrange multiplier λnhrepresents an approximation of cn on the edges. These estimates are then used
to establish approximation properties of the reconstruction operator B. Given
the Lagrange multiplier λnh (defined only on the edges), the reconstruction B(λnh)
provides an approximation of the advective flux Qncn in the space V h. The
precise definition of B implies that the normal component of B(λnh) coincides
with the normal component of Π1hQ
n · λnh in the two points dividing an edge into
three segments of equal length. The approximation properties of the Lagrange
multipliers and the reconstruction operator B entail the above theorem.
206 CHAPTER 4. MHFE SCHEMES BASED ON THE BDM1 ELEMENT
4.4 Numerical results
In this section, we illustrate our theoretical results and compare the classical and
the modified BDM1 scheme with the help of a computational experiment. Our
numerical results show that the error bounds stated above are sharp. Further
numerical tests, in which the modified BDM1 scheme was applied to nonlinear
reactive transport problems, are presented in [BRBK12, BBKR13].
To be more precise, problem (1.1)–(1.3) is solved on the unit square Ω = (0, 1)2 ⊂R2 on the time interval J = (0, 1). For the (scalar) diffusion coefficient and the
velocity field, we choose D = 1 and Q = (0,−1)T , respectively. Moreover,
homogeneous Dirichlet conditions are imposed on the boundary ∂Ω. The source
term f is prescribed so that the analytical solution of the problem is given by
u(x, y, t) = x(1− x)y(1− y)e−t .
In each refinement step, the mesh (which initially consists of two triangles) is
uniformly refined, and the errors
E(1)h,τ = τ 1/2
(N∑n=1
‖q(·, tn)− qnh‖20
)1/2
,
E(2)h,τ = max
1≤n≤N‖c(·, tn)− cnh‖0 ,
E(3)h,τ = max
1≤n≤N‖P 0
hc(·, tn)− cnh‖0
are computed. For all computations, a fixed time step size τ = 0.001 is used,
which is sufficiently small to ensure that the time discretization error is neg-
ligible compared to the space discretization error. The experimental orders of
convergence are determined by means of
EOC(i)h,τ = log2
(E
(i)2h,τ
E(i)h,τ
), i ∈ 1, 2, 3 .
The results of the computations are listed in Tables 4.2 and 4.3. As expected,
we obtain optimal second order convergence for the flux variable if the advective
fluxes are approximated using the Lagrange multipliers, whereas only suboptimal
first order convergence is observed when the classical scheme is used. Moreover,
4.4. NUMERICAL RESULTS 207
the scalar unknowns are approximated with first order accuracy for both schemes,
the magnitude of the errors being of almost equal size. For the approximation of
the projection P 0hc of the scalar variable into the space of piecewise constants by
our numerical solution cnh, we observe the usual (second order) superconvergence
result both in the case of the classical scheme and in the case of the modified
scheme.
triangles E(1)h,τ EOC E
(2)h,τ EOC E
(3)h,τ EOC
2 · 40 4.36e-02 2.91e-02 1.80e-02
2 · 41 2.01e-02 1.12 1.60e-02 0.86 6.37e-03 1.50
2 · 42 5.94e-03 1.76 8.57e-03 0.90 1.88e-03 1.76
2 · 43 1.56e-03 1.93 4.36e-03 0.97 4.94e-04 1.93
2 · 44 3.96e-04 1.98 2.19e-03 0.99 1.24e-04 1.99
2 · 45 9.88e-05 2.00 1.10e-03 1.00 3.07e-05 2.02
2 · 46 2.39e-05 2.04 5.48e-04 1.00 7.18e-06 2.10
Table 4.2: L2-errors and experimental orders of convergence (EOC) for the mod-ified BDM1 scheme
triangles E(1)h,τ EOC E
(2)h,τ EOC E
(3)h,τ EOC
2 · 40 4.13e-02 2.91e-02 1.79e-02
2 · 41 2.05e-02 1.01 1.60e-02 0.86 6.34e-03 1.50
2 · 42 6.91e-03 1.57 8.57e-03 0.90 1.87e-03 1.76
2 · 43 2.51e-03 1.46 4.36e-03 0.97 4.91e-04 1.93
2 · 44 1.09e-03 1.21 2.19e-03 0.99 1.23e-04 1.99
2 · 45 5.19e-04 1.07 1.10e-03 1.00 3.05e-05 2.02
2 · 46 2.56e-04 1.02 5.48e-04 1.00 7.12e-06 2.10
Table 4.3: L2-errors and experimental orders of convergence (EOC) for the clas-sical BDM1 scheme
Conclusion
In the first part of this work, we developed a numerical framework for efficiently
simulating partially miscible multiphase multicomponent flow in porous media,
modeled by a system of coupled and strongly nonlinear partial differential equa-
tions, ordinary differential equations, and algebraic equations.
For this purpose, the problem was transformed with the help of a reduction
scheme, which preserves the model and allows the unknown equilibrium reac-
tion rates to be eliminated from the system. More precisely, a variant of the
reduction scheme including additional variables was used, which were shown to
be mandatory to obtain a well-conditioned global problem, cf. [HKK10, Hof10].
In contrast to the work mentioned above, we need an additional transformed
variable representing the interphase mass exchange between the liquid and the
gas phase, and there are no partial differential equations decoupling from the
rest of the system after using the reduction scheme. This is due to the nonlinear
coupling of flow and transport in our multiphase model, which implies that each
partial differential equation depends on the liquid pressure variable. If, however,
the primary variables are chosen appropriately, it remains possible to reduce the
size of the global problem by shifting at least one block of ODEs to the local
level.
In the transformed system, the algebraic equations resulting from the chemical
equilibria are eliminated with the help of a nonlinear, implicitly defined resolution
function. Hereby, the equilibrium conditions consisting of equations and inequal-
ity constraints are formulated as a complementarity problem and rewritten as
algebraic equations using the minimum function. The existence of a resolution
function is established by using the connection of the nonlinear system of alge-
braic equations to a constrained minimization problem based on a modified Gibbs
functional, and numerical strategies to evaluate it numerically are presented. This
208
CONCLUSION 209
requires a formulation based on the logarithms of the concentrations. Moreover,
we present a method to ensure that the local nonlinear solver does not produce
nonphysical values while solving the chemical subproblem. As a nonlinear solver,
the Semismooth Newton method is used, which converges locally quadratically.
The use of the resolution function corresponds to an elimination on the level of
the nonlinear solver, which can be efficiently carried out on multiple processors.
The time discretization of the system obtained after the variable transformation
and the use of the resolution function is done using the implicit Euler method,
and the spatial discretization is based on a linear finite element method, enhanced
by an upwind finite volume stabilization for advection–dominated problems.
Using a global implicit approach, the nonlinear systems that remain to be solved
in each time step are treated with Newton’s method. With the help of the
resolution function, we are ready to use a persistent set of primary variables that
are valid in either phase state and for an arbitrary mineral assemblage. Thus,
the discontinuous switch of variables during the simulation as a result of the
appearance or disappearance of the gas phase or the dissolution and precipitation
of minerals can be avoided.
Regarding the definition of primary variables, our numerical tests indicate that
the use of two extended pressure variables leads to better convergence properties
of the nonlinear solver than the use of only one pressure variable in a smaller
global system. This is because the nonlinear coefficient functions depend directly
on one primary variable if two phase pressures are used as primary variables.
Moreover, there are fewer nonlinearities in the transport operators which involve
the gradients of both phase pressures.
Different numerical tests related to hydrogen migration in deep geological repos-
itories of radioactive waste and to CO2 sequestration show that our solver, which
was designed for 2D and 3D computations and implemented in a parallel finite
element toolbox, provides accurate numerical results, and that it is capable of
handling the strongly nonlinear coupling of flow, transport, chemical reactions,
and mass transfer across the phases. If a strict stopping criterion is used for the
nonlinear solver, the global implicit approach required significantly less compu-
tation time than the sequential iterative approach.
210 CONCLUSION
The second part of this work deals with the analysis of robust mixed hybrid finite
element methods of lowest order for advection–diffusion problems. When advec-
tion strongly dominates diffusion, the standard mixed method [DR85] typically
fails to resolve steep gradients in the analytical solution and produces approx-
imations that are polluted by spurious oscillations. One of many methods to
overcome this is to employ upwinding to the method. In the approach studied
here, this is accomplished by extending the classical scheme with the help of the
Lagrange multipliers that arise from the hybrid problem formulation. It relies
on the fact that the Lagrange multipliers represent approximations of the scalar
unknown on the interelement boundaries and uses them in the approximation of
the advective fluxes. Using techniques from the a posteriori error analysis, we are
able to establish optimal order convergence for a new class of methods including
the standard method, a partial upwind method, and a full upwind method. The
advantage of the specific choice of the upwind weights is the fact that the method
remains local such that static condensation can still be employed, whereas the
standard upwind-mixed method requires information from neighbor cells such
that static condensation is no longer applicable.
Concerning approximations with the BDM1 mixed finite element, we show that
the use of Lagrange multipliers in the discretization of the advective term pro-
vides optimal second order convergence for the total flux variable consisting of
advective and diffusive flux in the L2 norm, whereas the standard mixed method
is known to be of suboptimal first order accuracy only. Thus, the modification,
which was intended to enhance the robustness of the standard method against
advection–dominance for moderately advection dominated problems, leads to a
higher accuracy of the method without increasing the computational costs.
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