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Preprint submitted on 21 Jul 2020
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Convergence of nonlinear finite volume schemes for
two-phase porous media flow on general meshes
Léo Agélas, Martin Schneider, Guillaume Enchéry, Bernd Flemisch
To cite this version:
(will be inserted by the editor)
Convergence of nonlinear finite volume schemes for two-phase porous
media flow on general meshes
L´eo Ag´elas · Martin Schneider · Guillaume Ench´ery · Bernd Flemisch
the date of receipt and acceptance should be inserted later
Abstract In this work, we present an abstract finite volume discretization framework for incompress-ible immiscincompress-ible two-phase flow through porous media. A-priori error estimates are derived that allow to prove the existence of discrete solutions and to establish the proof of convergence for schemes belonging to this framework. In contrast to existing publications, the proof is not restricted to a specific scheme and it does neither assume symmetry nor linearity of the flux approximations. Two nonlinear schemes, namely a nonlinear two-point flux approximation (NLTPFA) and a nonlinear multi-point flux approximation (NLMPFA) are presented and some properties of these schemes, e.g. saturation bounds, are proven. Furthermore, the numerical behavior of these schemes (e.g. accuracy, coercivity, efficiency or saturation bounds), is investigated for different test cases.
Keywords two-phase flow · porous medium · monotone schemes · finite volume methods ·
convergence analysis
1 Introduction
1
Flow through porous media occurs in a variety of technical engineering applications such as petroleum
2
exploration and production, geological storage of carbon dioxide, hydrogeology, or geothermal energy.
3
Many challenging problems arise in the numerical simulation of complex fluid processes in reservoir
4
simulation, subsurface contaminant transport and remediation, gas migration through engineered
5
and geological barriers of deep radioactive waste repositories, sequestration of CO2, and other
ap-6
plications. The design of suitable discretization schemes for solving such applications is therefore
7
essential. There is a large variety of discretization schemes that have been used for simulating
multi-8
phase flow in porous media, whereby mainly locally mass conservative schemes are used, which is
9
L´eo Ag´elas
IFP Energies nouvelles, 1 & 4 avenue du Bois-Pr´eau, 92852 Rueil-Malmaison Cedex, France, E-mail: leo.agelas@ifpen.fr
Martin Schneider
Institute for Modelling Hydraulic and Environmental Systems, University of Stuttgart, Pfaffenwaldring 61, 70569 Stuttgart, Germany,
E-mail: martin.schneider@iws.uni-stuttgart.de Guillaume Ench´ery
IFP Energies nouvelles, 1 & 4 avenue du Bois-Pr´eau, 92852 Rueil-Malmaison Cedex, France, E-mail: guillaume.enchery@ifpen.fr
Bernd Flemisch
Institute for Modelling Hydraulic and Environmental Systems, University of Stuttgart, Pfaffenwaldring 61, 70569 Stuttgart, Germany,
essential when solving fluid dynamical processes. This is why finite volume schemes are the most
10
commonly used methods for solving flow through porous media. A comparison and an overview of
11
different schemes can be found in [14, 33, 38]. For subsurface simulations, often corner-point grids are
12
used to account for the different petrophysical properties that are associated to the control volumes
13
(cells) of the grids. Solving partial differential equations on such corner-point grids with highly
het-14
erogeneous and anisotropic properties poses challenges on the discretization scheme. In our previous
15
work, it has been demonstrated that so-called nonlinear finite volume schemes can be used for such
16
grids and for complex applications [37, 36], where the convergence of these nonlinear schemes has
17
been proven for elliptic problems in [34]. This work is an extension of [34] to incompressible
im-18
miscible two-phase porous media flow problems on general meshes. Besides a general discretization
19
framework, including nonlinear flux discretization schemes, a-priori estimates are presented, which
20
are used to show the existence of discrete solutions and to prove the convergence of schemes belonging
21
to the presented discretization framework.
22
Previous publications include the convergence proof of a phase-based fully-upwind scheme with
23
a two-point flux approximation, which was first analyzed for a one-dimensional setup in [7, 32] and
24
then extended in [24] to general higher dimensional grids. In this work, we establish the proof of
25
convergence for the so-called fractional-flow formulation (global pressure-saturation formulation),
26
which was theoretically analyzed (e.g. showing the existence of weak solutions) in [26, 10, 11]. The
27
fractional-flow approach treats the two-phase flow problem as a total fluid flow of a single mixed fluid,
28
and then describes the individual phases as fractions of the total flow. This approach leads to two
29
coupled equations: the global pressure equation and the saturation equation. For the mathematical
30
analysis of different discretization schemes for this fractional-flow formulation we refer to [10, 20, 40,
31
29, 8]. The proof of convergence for schemes belonging to the gradient discretization framework (e.g
32
[18]) has been presented in [23]. The gradient discretization method (GDM) is a recent framework
33
for the numerical discretization and analysis of elliptic and parabolic PDEs. The usual GDM defines
34
reconstruction operators (e.g. discrete gradient operators) on discrete solution spaces and discretizes
35
the PDEs by replacing the continuous operators in the weak formulation by the corresponding
36
discrete ones. The convergence of gradient schemes obtained in [7, 23] is in fact based upon a
weak-37
star convergence (weak-strong convergence) argument which states that if fn * f is weak-star
38
convergent in the dual spaceX∗ of a Banach spaceX andxn → x converges strongly in X, then
39
[fn, xn] → [f, x] as n → ∞. The use of this argument in the case of gradient schemes is possible
40
because only one discrete gradient reconstruction operator∇Dis used in the discrete problem, which
41
allows to get both weak and strong convergence in the duality bracket thanks to the limit-conformity
42
and consistency properties required for these methods. Thus, by using an argument of weak-strong
43
convergence, the proof of convergence follows by establishing some compactness results (see Section
44
3.3 of [8] and Theorem 3.7 in [23]).
45
However, despite its flexibility, the usual GDM does not seem to cover some important families of
46
numerical methods, in particular some finite volume schemes such as the two-point flux
approxima-47
tion (TPFA), the multi-point flux approximation MPFA-L/G schemes, MPFA-O schemes on general
48
meshes except some particular meshes for which they become symmetric (simplex, parallelogram),
49
or nonlinear schemes. These non-symmetric schemes do not belong to the family of gradient schemes,
50
because two different gradient reconstruction operators∇D and e∇D are needed in the discrete
for-51
mulation, where one of the operators is strong (in the sense of the consistency) and the other one
52
is weak (in the sense of the limit-conformity). Due to these two different gradient reconstruction
53
operators, which appear in the duality bracket terms of the weak formulation, the weak-strong
con-54
vergence argument cannot be used to get the proof of convergence. This is the main reason why
55
the proof of convergence for non-symmetric schemes is quite different from the one used for schemes
56
encompassed by the gradient discretisation framework. The proof of convergence for non-symmetric
57
schemes requires to establish a-priori error estimates (see the proof of Theorem 1 in [1], Lemma 5.7
58
and Theorem 5.1 in [2], Theorem 1 in [34] and the asymmetric gradient discretization framework in
59
[16]) depending on duality bracket terms, which involve two discrete gradient reconstruction
oper-60
ators and thus allow the use of weak-strong convergence and compactness arguments. This is done
61
in this article, where we give, after establishing a-priori error estimates, the proof of convergence for
62
the two-phase flow problem of cell-centered finite volume schemes which are possibly unsymmetric
and nonlinear. The proof is based on a-priori error estimates combined with compactness arguments,
64
where our assumptions are compatible with field applications (discontinuous data, fully nonlinear
65
models, etc.). These are, at least to our knowledge, novel results for general parabolic PDEs and
66
differ from recent proofs which are essentially based on weak-strong convergence arguments (see [8,
67
23]). Thus our proof appears to be technically quite difficult because of the a-priori error estimates
68
that have to be additionally established.
69
Furthermore, most of the existing literature either neglect capillary pressure or buoyancy terms
70
in their mathematical analysis and only consider linear flux approximations. This is not done in this
71
work, where all terms are considered and the fluxes are allowed to be nonlinear. Such nonlinear flux
72
approximations have the advantage that they are consistent and satisfy saturation bounds.
73
This work is organized as follows: In Section 2, the mathematical formulation of the two-phase
74
flow problem using the fractional-flow formulation is presented. In Sections 3 and 4, a general finite
75
volume discretization framework is introduced and the proof of convergence is given. This general
76
framework also includes nonlinear schemes. Two representatives of such schemes are presented in
77
Section 5, where also some fundamental properties of these schemes are proven. Finally, these schemes
78
are numerically investigated in Section 6 for a quasi one-dimensional setup and a two-dimensional
79
test case including gravity and capillary pressure effects.
80
2 Mathematical formulation of a two-phase flow problem
81
2.1 Continuous form
82
Let Ω ⊂ Rd, d ∈ N∗, be an open bounded connected polygonal domain with boundary ∂Ω and
83
d-dimensional measure |Ω|. OnΩ and for allt∈ (0, T) (T >0), we define the following two-phase
84
porous-media flow problem, where the phases are assumed to be incompressible and immiscible with
85
a rigid porous matrix,
86
φut− ∇·(λ1(u)Λ(∇p1− %1g)) =f1(c)s+− f1(u)s−, (1a)
φ(1− u)t− ∇·(λ2(u)Λ(∇p2− %2g)) =f2(c)s+− f2(u)s−. (1b) Here, udenotes the saturation of the wetting phase; p1, p2 the wetting and non-wetting pressures
87
linked together through the capillary pressurepc=p2−p1;φthe porosity;Λa symmetric permeability
88
tensor; %1, %2 the phase densities;g= (0,0,−g)T the gravity vector (g >0);s+, s− the source and
89
sink terms;cthe inflow wetting saturation;λ1andλ2 the wetting and non-wetting phase mobilities;
90
andf1, f2the fractional-flow functions, which are given as
91 f1= λ1 λT , f2= λ2 λT , (2)
whereλT=λ1+λ2 is the total mobility.
92
Using these quantities, problem (1) can be rewritten in the fractional-flow form
93
φut+∇· f1vT− Λ∇ψ(u) + (%1− %2)f1λ2Λg =f1(c)s+− f1(u)s−, (3a)
∇·vT=s+− s−, (3b)
where we have introduced the total velocity
vT=−λT(u)Λ ∇p − %fg
,
with the average fluid density
94
%f =%1f1+%2f2, (4)
the global pressure
95
p=p1− Z 1
u
and the following function 96 ψ(u) =− Z u 0 λ1(v)f2(v)p0c(v) dv. (6)
Initial conditions for Problem (3) are given for the wetting saturation
97
u(.,0) =uinit inΩ. (7)
Additionally, for simplicity, we assume homogeneous zero Dirichlet boundary conditions
98
u(x, t) = 0, p(x, t) = 0, on ∂Ω×(0, T). (8) In the following, we consider problem (3) with the two unknowns (u, p) and make the following
99
assumptions. For simplicity we do not introduce residual saturations such that the effective saturation
100
corresponds tou.
101
Hypotheses 1 We assume that:
102
(A1) φ∈ L∞(Ω)with φ∈[φ, φ]almost everywhere (a.e.) in Ω (without loss of generality, we assume φ= 1
103
in the mathematical analysis of the finite volume scheme),
104
(A2) Λ is symmetric and there exist0< α0< β0<+∞ so that the spectrum of Λ is contained in[α0, β0]
105
a.e. in Ω,
106
(A3) uinit∈ L∞(Ω), with uinit∈[0,1]a.e.,
107
(A4) c∈ L∞(Ω×(0, T)), with c∈[0,1] a.e.,
108
(A5) s+, s−∈ L2(Ω×(0, T)), s+≥0and s−≥0a.e.,
109
(A6) λ1: R7→[0, λ1]is a nondecreasing Lipschitz continuous function such that (s.t.)
110
λ1(x) = 0, ∀x ∈(−∞,0], λ1(x) =λ1>0, ∀x ∈[1,∞),
(A7) λ2: R7→[0, λ2]is a nonincreasing Lipschitz continuous function s.t.
111
λ2(x) =λ2>0, ∀x ∈(−∞,0], λ2(x) = 0, ∀x ∈[1,∞),
(A8) ψ∈ C([0,1])with ψ(0) = 0, is a strictly increasing Lipschitz-continuous function. The function ψ is
112
linear outside [0,1]that is
113
ψ(u) =
Ξψ(1)(u−1) +ψ(1) if u >1,
Ξu if u <0, (9)
with Ξ > 0. We denote by Lψ the Lipschitz constant of ψ over R. At last, there exist C1,ψ ≥ 0,
114
C2,ψ ≥0 so that, for all u∈R,
115
|ψ(u)| ≥ C1,ψ|u| − C2,ψ. (10)
Using the assumptions (A6) and (A7), we set λ = minx∈RλT(x) and λ= maxx∈RλT(x). We also
116
introduce the function
117
Ψ(s) = Z s
0
ψ(x) dx, ∀ s ∈R. (11)
Thanks to the Lipschitz continuity of ψ, the fact that ψ is nondecreasing and that ψ(0) = 0, the
118
functionΨ satisfies the following inequality (see proof of Lemma 13 in section Appendix):
119 0≤ ψ(s) 2 2Lψ ≤ Ψ (s) = Z s 0 (ψ(x)− ψ(0)) dx≤ Lψ s2 2. (12)
Furthermore, under assumption (A8), we deduce that 2(ψ(s)2+C2
2,ψ)≥ C12,ψ|s|2. Hence, using (12), 120 we obtain 121 Ψ(s)≥C 2 1,ψ|s|2−2C22,ψ 4Lψ . (13)
The monotonicity ofψimplies thatΨ is a convex function such that for alls1, s2∈R
122
Fig. 1 An example of admissible mesh for d = 2.
2.2 Weak form
123
Under Hypotheses 1, (¯p,¯u) is a weak solution of (3) if
124 – p¯∈ L2(0, T;H01(Ω)), 125 – u¯∈ L2(Ω×(0, T)), 126 – ψ(¯u)∈ L2(0 , T;H01(Ω)), 127
and if, for allϕ∈ L2(0, T;H01(Ω)) s.t.ϕt∈ L2(Ω×(0, T)) andϕ(., T) = 0 a.e., we have
128 Z T 0 Z Ω h − φ¯uϕt− f1vT−Λ∇ψ(¯u) + (%1− %2)f1λ2Λg) ∇ϕidxdt = Z Ω φuinit(x)ϕ(x,0) dx+ Z T 0 Z Ω (f1(c)s+− f1(¯u)s−)ϕdxdt, (15a) − Z T 0 Z Ω vT· ∇ϕdxdt= Z T 0 Z Ω (s+− s−)ϕdxdt. (15b) 129
3 Finite volume discretization
130
Before giving a finite volume discretization of (15), we introduce a few notations and definitions.
131
3.1 Discretization of the space and time domains and their regularity
132
We first define the spatial discretization which includes general polygonal meshes (see Figure 1).
133
Definition 1 (Spatial discretization)A spatial discretizationDis a tripletD= (T , E, P), where
134
(i) T (the cells or control volumes) is a finite family of non-empty connected open disjoint subsets of
135
Ωs.t.Ω=∪K∈TK. For all cellsK∈ T,|K| >0 denotes itsd-dimensional measure (the volume)
136
and∂Kdef= K\Kits boundary. The size of the discretization is defined byhD
def
= supK∈T diam(K).
137
The number of cells is indicated bynT.
138
(ii) E (the faces) is a finite family of subsets ofΩs.t., for allσ∈ E,σis a non-empty closed subset of a
139
hyperplane of Rdwith (d
−1)-dimensional measure|σ| >0 (the area), and the intersection of two
140
different faces has zero (d−1)-dimensional measure. For allK∈ T, we assume that there exists a
141
subsetEK ofE s.t.∂K=∪σ∈EKσ. For anyσ∈ E, eitherTσ
def
= {K ∈ T | σ ∈ EK}has exactly one
142
element (if σ⊂ ∂Ω) orTσ has exactly two elements (inner face); the sets of inner and boundary
143
faces are denoted byEintandEext, respectively. The face evaluation points (interpolation points)
144
are denoted byxσ(not required to be the barycenters). For allK∈ T andσ∈ EK, we denote by
145
nK,σ the unit vector that is normal toσand outward toK.
(iii) P = {xK}K∈T (the cell centers, not required to be the barycenters) is a family of points of Ω
147
s.t. xK ∈ K. We assume that there is α > 0 such that for all K ∈ T, K is star-shaped with
148
respect to all the points in a ball of radiusαdiam(K) and, in particular, toxK. For allK∈ T and
149
for all σ∈ EK, dK,σ denotes the Euclidean distance between xK and the hyperplane including
150
σ, and ∆K,σ denotes the convex hull of xK and σ. In addition, we denote, for all σ ∈ E, by
151
∆σ= Interior SK∈T
σ∆K,σ and by∆={∆σ, σ∈ E}. 152
Let us remark with the notations of Definition 3.1, that, since |σ|dK,σ
d is the measure of the convex
153
hull∆K,σ ofxK andσ, we have
154
∀K ∈ T , X
σ∈EK
|σ|dK,σ=d|K|. (16)
In the following of this work, a few regularity assumptions are made on the spatial discretization
155
for the convergence analysis of the scheme. Therefore, we now introduce the notion of an admissible
156
spatial discretization.
157
Definition 2 (Admissible spatial discretization) Let D be a spatial discretization of Ω in the
158
sense of Definition 1. This discretization is admissible if there exist 0< ζ1, ζ2, ζ3, ζ4, ζ5<+∞s.t.
159 |EK| ≤ ζ1, min K∈T , σ∈EK |σ| diam(K)d−1 ≥ ζ2, (17) 160 min K∈T , σ∈EK dK,σ
diam(K)≥ ζ3, σ∈Eint, Tminσ={K,L}
min(dK,σ, dL,σ) max(dK,σ, dL,σ)≥ ζ 4, min K∈T diam(K) hD ≥ ζ5 . (18)
The next two definitions allow us to precise the concept of admissible discretization for the whole
161
space-time domain and to introduce the notion of admissible family for these discretizations that
162
will be used for the convergence study. For this, we use the definitionJ0, NKdef= {0,· · · , N}.
163
Definition 3 (Admissible space-time discretization) The pair D = (D, Dt) is a space-time
dis-164
cretization ofΩ×(0, T) if:
165
– Dis a spatial discretization in the sense of Definition 1,
166
– Dt = [
n∈J0,NK
In with In = [t(n), t(n+1)[, {t(n)}n=0,··· ,N such that t(0) = 0, t(N+1) = T, and
167
δt(n+12)=t(n+1)− t(n)>0, for alln∈
J0, NK.
168
The maximum time step size of a space-time discretization is denoted by
169
|δt|= max
n=0,··· ,Nδt
(n+1
2). (19)
It is said to be admissible ifDis admissible according to Definition 2.
170
Definition 4 (Admissible family of space-time discretizations)A family of space-time
discretiza-171
tions{Dm}m∈N is admissible ifhDm→0,|δtm| →0 asm→ ∞, and for eachm,Dmis admissible in 172
the sense of Definition 3 where the parametersζ1, ζ2, ζ3, ζ4, ζ5do not depend onm.
173
In what follows, when referring to a generic elementDmof an admissible family of discretizations
174
{Dm}m∈N, the subscriptmwill be dropped for the ease of reading in cases where no ambiguity arises.
175
3.2 Further notations and discrete tools
176
In this section, we introduce further notations and some discrete tools which are needed for the
177
analysis of the discrete scheme.
3.2.1 Notations
179
In the sequel, we use the following notations
180
– for anyV ⊂ ΩandΦ∈ L1(V),hΦiV
def = |V |−1R
V Φdxwhich is meant component-wise for functions
181
with vector or tensor values,
182
– L(E;F) represents the vector space of bounded linear operators fromE toF.
183
3.2.2 Discrete spaces
184
First, we define discrete spaces on Ω. The space of piecewise constant functions onQ ∈ {T , ∆} is
185 defined as 186 HQ(Ω) def = {v ∈ L2( Ω)| vK def= v|K ∈P0(K), ∀K ∈ Q}.
With this, for any v ∈ L2(Ω), we denote by vQ the element of HQ(Ω) such that for all K ∈ Q,
187
(vQ)K =hviK. Forv∈ H∆(Ω) we often use the abbreviationvσ instead ofv∆σ. 188
Next, we define discrete spaces onΩ×(0, T). Here, the space of piecewise constant functions on
189 Q ∈ {T , E }, withT def = T × DtandE def = ∆× Dt, is given as 190 HQ(Ω×(0, T)) def = {v ∈ L2(Ω×(0, T))| vK(n)def= v|K ×In∈P0(K × In), ∀(K , In)∈ Q}.
In the same way, for any v∈ L2(Ω×(0, T)), we denote by vQ the element of HQ(Ω×(0, T)) such
191
that for all (K , In)∈ Q, (vQ)(Kn) =hviK ×In. With this, for eachv∈ HQ(Ω×(0, T)) (Q= ˜Q × D
t
192
and ˜Q ∈ {T , ∆}) and for eacht∈ In, we definev(n)=v(t)∈ HQ˜(Ω) s.t. (v(t))K =hv(·, t)iK for all
193
K ∈Q˜.
194
3.2.3 Discrete operators and norms
195
We now introduce a general trace reconstruction operator, which allows to define discrete gradients
196
andH1-norms on the spacesHQ.
197
Definition 5 (Trace reconstruction operator)A trace reconstruction operator is a set of bounded
198
linear operators I, such thatI = {Iσ}σ∈E, Iσ ∈ L(HT(Ω); P0(σ)), andIσv= 0 for allv∈ HT and
199
σ∈ Eext.
200
Among these operators, we will consider the ones, denoted byΥ def= {Υσ}σ∈E, for which there exist,
201
for allσ∈ Eint withTσ={K, L}, two non-negative values,θK,σ andθL,σ, such thatθK,σ+θL,σ= 1
202
and which are given by
203
Υσv=
θL,σvK+θK,σvL ifσ∈ Eint,
0 ifσ∈ Eext. (20)
We denote byRE the set of operators satisfying (20). Of special interest is the trace reconstruction
204
operatorγ={γσ}σ∈E that is defined, for allv∈ HT(Ω), by
205 γσv= dL,σvK+dK,σvL dK,σ+dL,σ ifσ∈ Eint, 0 ifσ∈ Eext. (21)
Then, for any trace reconstruction operator I = {Iσ}σ∈E matching Definition 5 and for any v ∈
206
HT(Ω), we define
207
– a discrete gradient with values in (HT(Ω))d:
– a discrete norm: 209 kvkT ,I def= X K∈T X σ∈EK |σ| dK,σ|I σv− vK|2 1 2 , (23)
where forI=γ we use the simplified notation||v||T
def
= kvkT ,γ,
210
– a discrete dual semi-norm for all u∈ L2(Ω):
211 ||u||−1,T def = sup Z Ω u(x)w(x)dx:w∈ HT(Ω),||w||T = 1 (24)
– the extensions of both previous norms to the spaceHT(Ω×(0, T)):
||v||T def= Z T 0 ||v(t)||2 Tdt !1/2 and||v||−1,T def = Z T 0 ||v(t)||2 −1,T dt !1/2 .
Finally, for allv∈ HT(Ω×(0, T)) andn= 0,· · · , N −1, we also define the discrete time derivative
212 ofv, 213 (δtv)(Kn)def= v(Kn+1)− vK(n) δt(n+12) (25) and, for allK∈ T andσ∈ EK, we denote by
214
FK,σ:HT(Ω)× HT(Ω)7→P0(σ) (26)
a numerical flux designed to approximate the flow induced by the normal component of a gradient
215
term with respect to nK,σ. In this work, we assume that the fluxes are locally mass conservative,
216
meaning that for anyσ∈ Eint withTσ={K, L}
217
FK,σ(u, v) +FL,σ(u, v) = 0. (27)
3.3 Definition of the scheme
218
Using the notations introduced in Sections 3.1–3.2, a finite volume discretization of problem (15),
219
along with an implicit Euler scheme for the time discretization, consists in computing a pair (u, p)∈
220 [HT(Ω×(0, T))]2s.t., for alln ∈ {0,· · · , N −1}andK∈ T: 221 |K|u (n+1) K − u (n) K δt(n+12) + X σ∈EK f1(u(1n,σ+1))v (n+1) K,σ +f1(u (n+1) 2,σ )λ2(u3(n,σ+1))(%1− %2)GK,σ − X σ∈EK FK,σ(ψ(u(n+1)), ψ(u(n+1))) =|K| f1(cK)s+K− f1(u( n+1) K )s − K , (28a) 222 X σ∈EK v(K,σn+1)=|K|(s+K− s−K). (28b)
Note that (28b) also holds for n = −1. In the previous discrete system (28), we have used the
withδK,σ= dL,στK,σdK,σ+τdL,σK,στL,σ,δL,σ= dL,στdK,σL,σ+τdK,σK,στL,σ,τK,σ=nK,σ·ΛKnK,σ, τL,σ=nL,σ·ΛLnL,σ.
226
The downstream upstream inner-face saturations{u(i,σn+1)}i=1,...,3 are defined according to the sign
227
of the quantitites{Mi}i=1,...,3 in the following way
228 u(i,σn+1)= ( u(Kn+1) ifMi≥0, u(Ln+1) otherwise, (31) withM1=%f(u( n+1) 0,σ )GK,σ− FK,σ(p(n+1), p(n+1)),M2= (%1− %2)GK,σ,M3=−M2. Forσ∈ Eext, we 229
setu(Ln+1)= 0. For the case thatσ∈ Eext, we use the same formulas as introduced above by setting
230
unL+1= 0,δK,σ= 1, andδL,σ= 0.
231
In view of analyzing this discrete scheme, we introduce, for allχ: R→R,α∈ H∆(Ω), (u, v, w)∈
232 [HT(Ω)]3, the form 233 aT ,χ,α(u, v, w) =− X K∈T X σ∈EK χ(ασ)FK,σ(u, v)wK. (32)
4 Analysis of the finite volume discretization
234
The aim of this section is to carry out an analysis of the discrete problem (28) by making the
235
following assumptions.
236
Hypotheses 2 Let {D}m∈N be a family of space-time discretizations matching Definition 4. LetDbe a
237
dense subspace of H01(Ω)s.t.D⊂ C0(Ω), where C0(Ω)denotes the space of continuous functions which
238
vanish on ∂Ω. We suppose that:
239
(P1) for any u∈ HTm(Ω), for all K∈ Tm and for all σ∈ EK, u7→ FK,σ(u,·)is a linear form; 240
(P2) for any bounded function χ and αm∈ H∆m(Ω), aTm,χ,αm is continuous, i.e., there is0< Cχ<+∞ 241
independent of m s.t. for all (u, v, w)∈[HTm(Ω)]
3
242
|aTm,χ,αm(u, v, w)| ≤ CχkvkTmkwkTm; (33)
(P3) the finite volume scheme is coercive, i.e., there is0<Cˆ1<+∞ independent of m s.t. for χ=λTand
243
χ= 1, for all(v, w)∈[HTm(Ω)]
2
and for any αm∈ H∆m(Ω) 244
aTm,χ,α(v, w, w)≥Cˆ1kwk
2
Tm; (34)
(P4) For χ= 1, χ=λTor χ=λ1, aTm,χ,·is weakly consistent on L
2(0, T; D), i.e., for all ϕ
∈ L2(0, T; D), 245 Dm(ϕ)→0as m→ ∞, where, 246 Dm(ϕ)def= max (u,v,w)∈PmΥminf∈REm 1 kwkTm Z T 0 aTm,χ,v(t)(u(t), ϕTm(t), w(t))dt− Z T 0 Z Ω χ(v)Λ∇ϕ ·∇eDm,Υmw dx dt , (35) wherePmdef= {(u, v, w)|(u, w)∈[HTm(Ω×(0, T))]2, w 6 = 0, v∈ HEm(Ω×(0, T))}. 247
By using the fact that D is a dense subspace ofH01(Ω), we extend in Proposition 4 property (P4) to
248
the spaceL2(0, T;H01(Ω)). This result is stated and proved in Section 8.1.
249
4.1 A priori estimates
250
In this section, we establish several estimates that will be used in Sections 4.2 and 4.3 to prove the
251
existence of discrete solutions and the convergence of the scheme.
Lemma 1 (Discrete estimates) Let D be an admissible space-time discretization matching Definition
253
3. Assume that Hypotheses 1 and the continuity and coercivity properties (P2) and (P3) hold. Then, there
254
exist C1, C2, C3, C4>0, depending on Ω, T , ζ3, ζ4, β0, %1, %2, g, λiwith i= 1,2, λ, λ, Lψ, C1,ψ, C2,ψ, s+
255
, s−, uinit,Cˆ1and Cχwith χ=f1λTand χ= 1such that any discrete solution(p, u)∈[HT(Ω×(0, T))]2
256 of problem (28)satisfies 257 sup t∈[0,T[kp (t)kT ≤ C1, (36) kψ(u)kT ≤ C2, (37) sup t∈[0,T[kΨ (u(t))kL1(Ω)≤ C3, (38) sup t∈[0,T[ ku(t)kL2(Ω)≤ C4. (39)
Proof Let n ∈ J0, N−1K. Multiplying equation (28b) by p
(n+1)
K and summing it over K ∈ T give
258 Tp,1=Tp,2+Tp,3 with 259 Tp,1=− X K∈T X σ∈EK λT(u (n+1) 1,σ )FK,σ(p(n+1), p(n+1))p (n+1) K , Tp,2=− X K∈T X σ∈EK λT(u (n+1) 1,σ )%f(u (n+1) 0,σ )GK,σp (n+1) K , Tp,3= X K∈T |K|(s+K− s−K)p(Kn+1).
Thanks to the coercivity assumption (34), we obtain
Tp,1≥Cˆ1kp(n+1)k2T.
By using Lemma 12 withM(x, y) =λT(x)%f(y), we deduce there exists a constantC5>0 depending
260
onλ,λ,λ1 λ2,ρ1,ρ2,g,β0 andΩ such that
261
Tp,2≤ C5kp(n+1)kT.
Using the Cauchy-Schwarz inequality and Proposition 4 of [34] yield
262
Tp,3≤(ks+kL2(Ω)+ks−kL2(Ω))kp(n+1)kL2(Ω)
≤(ks+kL2(Ω)+ks−kL2(Ω))C6kp(n+1)kT,
whereC6 depends onΩ,ζ3andζ4. The previous inequalities thus lead to
kp(n+1)kT ≤ ˆ1 C1 C5+C6(ks+kL2(Ω)+ks−kL2(Ω)) .
Since this estimate is also valid forn=−1, we therefore have
263 sup t∈[0,T[kp (t)kT ≤ ˆ1 C1 C5+C6(ks+kL2(Ω)+ks−kL2(Ω)) , which gives (36). 264
Multiplying equation (28a) by ψ(u(Kn+1)) and summing it up over K ∈ T results in Tψ = Tψ,1+
Tψ,2+Tψ,3 with 266 Tψ= X K∈T |K|u (n+1) K − u (n) K δt(n+12) ψ(u(Kn+1))− X K∈T X σ∈EK FK,σ(ψ(u(n+1)), ψ(u(n+1)))ψ(u( n+1) K ), (40) Tψ,1=− X K∈T X σ∈EK f1(u( n+1) 1,σ )v (n+1) K,σ ψ(u (n+1) K ), (41) Tψ,2= X K∈T X σ∈EK f1(u(2n,σ+1))λ2(u3(n,σ+1))(%2− %1)GK,σψ(u(Kn+1)), (42) Tψ,3= X K∈T |K|f1(cK)s+K− f1(u (n+1) K )s − K ψ(u(Kn+1)). (43)
Using inequality (14) together with the coercivity property (34) withχ= 1, we obtain
267 Tψ≥ X K∈T |K|Ψ(u (n+1) K )− Ψ(u (n) K ) δt(n+12) + ˆC1kψ(u(n+1))k2T. (44)
Let us consider the term Tψ,1. Using the continuity property (33) with χ= f1λT and Lemma 12 withM(x, y) = (f1λT)(x)%f(y), we get
Tψ,1≤ Cχkp(n+1)kTkψ(u(n+1))kT +C7kψ(u(n+1))kT.
Again, using Lemma 12 withM(x, y) =f1(x)λ2(y)(%2− %1) gives
268
Tψ,2≤ C8kψ(u(n+1))kT.
For the termTψ,3we proceed in the same way as previously for the pressure estimate and use Young’s
269
inequality, which leads to
270 Tψ,3≤(ks+kL2(Ω)+ks−kL2(Ω))kψ(u(n+1))kL2(Ω) ≤ Cˆ41kψ(u(n+1))k2T + 1 ˆ C1 C6(ks+kL2(Ω)+ks−kL2(Ω)) 2 .
By combining these estimates we deduce that
271 X K∈T |K|Ψ(u (n+1) K )− Ψ(u (n) K ) δt(n+12) + ˆC1kψ(u(n+1))k2T ≤Cχkp(n+1)kTkψ(u(n+1))kT + (C7+C8)kψ(u(n+1))kT +Cˆ1 4 kψ(u (n+1)) k2T + ˆ1 C1 C6(ks+kL2(Ω)+ks−kL2(Ω)) 2 .
Using again Young inequality and estimate (36), we deduce that there is a constant ˜C1 such that X K∈T |K|Ψ(u (n+1) K )− Ψ(u (n) K ) δt(n+12) + Cˆ1 4 kψ(u (n+1) )k2T ≤C˜1.
Multiplying this inequality by δt(n+12), summing over n = 0,· · · , ` −1 with `∈
J1, NK, and using 272 inequalities (12) to obtain 273 kΨ(u(`))kL1(Ω)+ ˆ C1 4 Z t(`) 0 kψ(u(s))k2Tds≤C˜1T+ Lψ 2 ku (0) k2L2(Ω). (45)
Since (45) is valid for all`∈J1, NK, we deduce, for`=N,
which gives (37).
275
Ift∈[t(0), t(1)[ then u(t) =u(0)and hence (12) gives
276 kΨ(u(t))kL1(Ω)=kΨ(u(0))kL1(Ω)≤ Lψ 2 ku (0) k2L2(Ω).
Ift∈[t(1), T[ then there exists`∈J1, NKsuch thatt∈[t(`), t(`+1)[ and henceu(t) =u(`) and thanks
277 to (45) we get 278 kΨ(u(t))kL1(Ω)≤C˜1T+ Lψ 2 ku (0) k2L2(Ω).
From the two previous inequalities we deduce that
279 sup t∈[0,T[kΨ (u(t))kL1(Ω)≤C˜1T+ Lψ 2 ku (0) k2L2(Ω), (47) which gives (38). 280
Finally, thanks to (13) and (47), we deduce (39).
281
Lemma 2 (Discrete H−1-estimate) Let D be a space-time discretization matching Definition 3.
As-282
sume that Hypotheses 1 and the continuity and coercivity properties, (P2) and (P3), hold. Then, there
283
exists C9 >0, depending on Ω, T , ζ3, ζ4, β0, %1, %2, g, λi with i= 1,2, λ, λ, Lψ, s+ , s−, uinit, Cˆ1,
284
and Cχ with χ=f1λT and χ= 1such that any discrete solution u∈ HT(Ω×(0, T)) of problem (28)
285
satisfies
286
kδtuk−1,T ≤ C9. (48)
Proof For anyw∈ HT(Ω), we deduce, from (28a), that
287 X K∈T |K|u (n+1) K − u (n) K δt(n+1 2) wK ≤ X K∈T X σ∈EK f1(u(1n,σ+1))v (n+1) K,σ wK + X K∈T X σ∈EK f1(u(2n,σ+1))λ2(u3(n,σ+1))(%1− %2)GK,σwK + X K∈T X σ∈EK FK,σ(ψ(u(n+1)), ψ(u(n+1)))wK + X K∈T f1(cK)s+K− f1(u(Kn+1))s−K wK . (49)
The four terms on the right hand side of (49) can be bounded using the same techniques as in the
288
proof of Lemma 1. This provides the existence of a constantC10>0 such that, for allw∈ HT(Ω),
289 Z Ω (δtu)(n)(x)w(x)dx ≤ C10 kp(n+1)kT +kψ(u(n+1))kT +ks+kL2(Ω)+ks−kL2(Ω)+ 1 kwkT,
from which we deduce that
290
||(δtu)(n)||−1,T ≤ C10(kp(n+1)kT +kψ(u(n+1))kT +ks+kL2(Ω)+ks−kL2(Ω)+ 1).
Squaring both sides of the inequality above, multiplying it byδt(n+12), and summing up over n= 291
0, ..., N−1, results in
292
||(δtu)||2−1,T ≤5C102(kpk2T +kψ(u)k2T +T(ks+k2L2(Ω)+ks−k2L2(Ω)+ 1)).
Then, thanks to Lemma 1, we deduce that there existsC9 >0, depending onΩ, T,ζ3, ζ4,β0, %1,
293
%2,g,λi withi= 1,2,λ,λ,Lψ,s+ ,s−,uinit, andCχ withχ=f1λT andχ= 1 such that
294
Lemma 3 (Estimate on the time translates) LetD be a space-time discretization matching Definition
295
3. Assume that Hypotheses 1 and the continuity and coercivity properties, (P2) and (P3), hold. Then, there
296
exists C11 >0depending on Ω, T , ζ3, ζ4, β0, %1, %2, g, λi with i = 1,2, λ, λ, Lψ, s+ , s−, uinit, Cˆ1
297
and Cχ with χ=f1λT and χ= 1, such that any discrete solution u∈ HT(Ω×(0, T)) of problem (28)
298
satisfies
299
kψ(u)(·, ·+τ)− ψ(u)kL2(Ω×(0,T −τ))≤ C11√τ , ∀τ ∈]0, T[. (50)
Proof Thanks to Lemma 2 and to the estimate (37) of Lemma 1, (50) can be obtained by following
300
the proof of Lemma 3.11 in [8].
301
Lemma 4 (H−1-estimate) Let D be a space-time discretization matching Definition 3. Assume that
302
Hypotheses 1 and the continuity and coercivity properties, (P2) and (P3), hold. Then, there exists C12>0,
303
depending on Ω, T , α, ζ1, ζ3, ζ4, β0, %1, %2, g, λi with i = 1,2, λ, λ, Lψ, s+ , s−, uinit, Cˆ1 and Cχ
304
with χ=f1λT and χ= 1, such that any discrete solution u∈ HT(Ω×(0, T)) of problem (28)satisfies
305
kδtukL2(0,T;H−1
0 (Ω))≤ C12
. (51)
Proof For anyw∈ H1
0(Ω) andn∈J0, NK, using Lemma 10, we have
306 Z Ω (δtu)(n)wdx = Z Ω (δtu)(n)wT dx ≤ k(δtu)(n)k−1,TkwTkT ≤ C16k(δtu)(n)k−1,TkwkH1 0(Ω).
We thus deduce that
307
kδtukL2(0,T;H−1
0 (Ω))≤ C16k(δtu)k−1,T
and conclude the proof thanks to Lemma 2.
308
Lemma 5 (L2-estimate on the dual mesh ∆) LetD be a space-time discretization matching Definition
309
3. Assume that Hypotheses 1 and the continuity and coercivity properties, (P2) and (P3), hold. Then, there
310
exists C13 >0, depending on Ω, T , ζ3, ζ4, β0, %1, %2, g, λi with i= 1,2, λ, λ, Lψ, s+ , s−, uinit, Cˆ1
311
and Cχ with χ=f1λT and χ= 1, such that any discrete solution u∈ HT(Ω×(0, T)) of problem (28)
312
satisfies
313
kψ(u)− ψ(v)kL2(Ω×(0,T))≤ C13hD, (52)
for all v∈ HE(Ω×(0, T)) such that for all n∈J0, NK, σ ∈ E , v(σn)∈[min{uK(n), u(Ln)},max{uK(n), u(Ln)}]
314
ifTσ={K, L} otherwise if Tσ={K}, vσ(n)∈[min{uK(n),0},max{u(Kn),0}].
315 Proof We have 316 kψ(u)− ψ(v)k2L2(Ω×(0,T))= N X n=0 δtn+ 1 2kψ(u(n))− ψ(v(n))k2 L2(Ω) = N X n=0 δtn+12 X K∈T X σ∈EK |∆K,σ|(ψ(u(Kn))− ψ(vσ(n)))2.
Sinceψis a monotone function, we notice that for allK∈ T andσ∈ EK:
317
– ifσ∈ Eint withTσ={K, L}then
– ifσ∈ Eext then|ψ(u(Kn))− ψ(v(σn))| ≤ |ψ(u(Kn))|.
319
By using the mesh regularity, we then deduce
320 kψ(u)− ψ(v)k2L2(Ω×(0,T))≤ 2 ζ4 N X n=0 δtn+12 X K∈T X σ∈EK |∆K,σ|(ψ(u (n) K )− γσ(ψ(u (n) )))2 = 2 ζ4 N X n=0 δtn+ 1 2 X K∈T X σ∈EK |σ|dK,σ d (ψ(u (n) K )− γσ(ψ(u (n))))2 ≤ 2h 2 D ζ4d N X n=0 δtn+12 X K∈T X σ∈EK |σ| dK,σ (ψ(u(Kn))− γσ(ψ(u(n))))2 = 2h 2 D ζ4dkψ (u)k2 T.
Finally, using the estimate (37) of Lemma 1, we obtain that there exists a constant C13 >0 such
321
thatkψ(u)− ψ(v)kL2(Ω×(0,T))≤ C13hD. 322
4.2 Existence of discrete solutions
323
Proposition 1 (Existence of discrete solutions) LetD be a space-time discretization matching
Defi-324
nition 3. Assume that Hypotheses 1 and the continuity and coercivity properties, (P2) and (P3), hold. For
325
all n∈J0, N−1K, there exists at least one solution to the equations(28).
326
Proof Let us taken∈J0, N−1Kand let us introduce the following open bounded subset of R
nT× RnT 327 ω=(p(Tn+1), uT(n+1))∈RnT ×RnT kp (n+1) kT < C1+ 1 andku(n+1)kL2(Ω)< C4+ 1 and the applicationην defined overωand for allν∈[0,1], by
328
ην:
RnT ×RnT →RnT ×RnT (p(Tn+1), uT(n+1))7→(ην,1, ην,2) where, for allK∈ T,
329 (ην,1)K =−ν X σ∈EK λT(u(1n,σ+1))(FK,σ(p(n+1), p(n+1))− %f(u(0n,σ+1))GK,σ) +|K|(s+K− s − K) +(1− ν)|K|p(Kn+1), (ην,2)K =ν |K| u(Kn+1)− u(Kn) δt(n+12) + X σ∈EK f1(u(1n,σ+1))v (n+1) K,σ +f1(u(2n,σ+1))λ2(u(3n,σ+1))(%1− %2)GK,σ −FK,σ(ψ(u(n+1)), ψ(u(n+1))) −ν|K|f1(cK)sK+ − f1(u(Kn+1))s − K +(1− ν)|K|u(Kn+1).
ην is continuous with respect toν,p(Tn+1)andu(Tn+1). Thanks to (14) used withs1= 0 and (12), we
330
get that for alls∈R,
331
sψ(s)≥0. (53)
Then proceeding in the same way as in the proof of Lemma 1 and thanks to (53), we deduce that
332
The topological degree d(ην, ω, 0R2nT) is therefore well defined. Forν = 0, the associated system, 333
ην = 0R2nT, admits one solution. Indeed, we have pT(n+1) = 0RnT and u (n+1)
T = 0RnT and both
334
solutions belong toω. Since the degree is homotopy invariant, we have
335
∀ν ∈[0,1], d(ην, ω, 0R2nT) =d(η0, ω, 0
R2nT)6= 0.
As, for ν = 1, the system ην = 0R2nT corresponds to (28), the previous relation guarantees the 336
existence of a solution inω.
337
4.3 Convergence proof
338
Before proving the convergence of the discrete scheme in this section, we present a compactness
339
result and establish some further estimates.
340
Lemma 6 (Compactness of approximate solution) Let Dmbe a sequence of space-time
discretiza-341
tions matching Definition 4 and(pm, um)∈[HT(Ω×(0, T))]2be a discrete solution of problem (28). Let
342
us assume that Hypotheses 1 and the continuity and coercivity properties, (P2) and (P3), hold, then, up
343
to a subsequence (still denoted with the subindex m), we have the following results:
344
(i) ψ(um)converges in L2(Ω×]0, T[) to some ψ(¯u)∈ L2(0, T;H01(Ω)), (and therefore um converges in
345
L2(Ω×]0, T[)tou),¯
346
(ii) for any Υm∈ REm,∇eDm,Υmψ(um)weakly converges in L
2(Ω
×]0, T[)d to
∇ψ(¯u),
347
(iii) pm weakly converges in L2(Ω×]0, T[)to somep¯∈ L2(0, T;H01(Ω)),
348
(iv) for any Υm∈ REm,∇eDm,Υmpmweakly converges in L
2(Ω ×]0, T[)d to∇p,¯ 349 (v) (δtu)mweak-? converges in L2(0, T;H0−1(Ω)) to ∂t¯u, 350
(vi) um weakly converges in L2(Ω)uniformly on[0, T]tou,¯
351 (vii) for Tm→ T , 352 lim inf m→∞ Z Ω Ψ(um(Tm))≥ Z Ω Ψ(¯u(T)).
(viii) ψ((um)i), i∈ {1,2,3}, strongly converge to ψ(¯u)∈ L2(0, T;H01(Ω))in L2(Ω×]0, T[), (and therefore
353
(um)i converge in L2(Ω×]0, T[)to¯u).
354
Proof We first define ˜ψm by ˜ψm = ψ(um) a.e. on Ω×]0, T[ and ˜ψm = 0 a.e. on Rd+1\ Ω×]0, T[.
355
Following the proof of Lemma 3.3 in [21], using the fact that, for allv∈ HT(Ω)
356 (vK− vL)2 dK,σ+dL,σ ≤ (γσv− vK)2 dK,σ +(γσv− vL) 2 dL,σ ,
and (37), we can prove that there exists a constantC14>0 only depending onΩs.t.
357
kψ˜m(·+η,·)−ψ˜mk2L2(Rd+1)≤ C22kηk(kηk+C14hDm), ∀η ∈R
d
. (54)
From (50), (39), and the Lipschitz property ofψ, we also deduce that
358
kψ˜m(·, ·+τ)−ψ˜mk2L2(Rd+1)≤ |τ|(|τ|C112 + 2L2ψC42), ∀τ ∈R. (55) Thus, with (54) and (55), an application of the Kolmogorov theorem gives that ψ(um) strongly
359
converges to someΦinL2(Ω×]0, T[). Therefore,ψ(um) strongly converges toψ(¯u) with ¯u=ψ−1(Φ).
360
Then, thanks to the assumptions made onψ(see Hypotheses 1) and by using Lemma 7.1 in [4] with
361
g=ψ−1, we deduce thatum also strongly converges to ¯u.
362
For anyΥm∈ REm, using (37), the equivalence of the norms|| · ||T and|| · ||T ,Υm (see Lemma 1 in 363
[34]) and a straightforward adaptation of Lemma 4.3 in [22] to time dependent problems, we deduce
364
that there existsGsuch that e∇Dm,Υmψ(um) weakly converges inL
2(Ω
×]0, T[)d toG=
∇Φ.
365
From (36) and Proposition 4 in [34], we have that pm is also bounded in L2(Ω×]0, T[) and thus
366
|| · ||T and|| · ||T ,Υm and Lemma 4.3 in [22], we deduce that ¯p∈ L 2(0 , T;H01(Ω)) and that e∇Dm,Υmpm 368 weakly converges inL2(Ω×]0, T[)d to∇p¯. 369
From (51), we deduce that (δtu)mweak-?converges inL2(0, T;H0−1(Ω)) to someU. In the same way
370
as in the proof of Theorem 4.18 in [17] and by using the strong convergence ofumone can show that
371
U =∂tu¯.
372
Statement (vi) is obtained by using (39), (51) and Theorem 4.19 of [17], whereas (vii) is a consequence
373
of (vi), the convexity ofΨ, and Lemma D.11 of [17]. Finally, from (52) and (i), we obtain (viii).
374
Lemma 7 LetD be a space discretization matching Definition 2, m: R×R7→Ra bounded function and
375
Υ the trace reconstruction operator obtained by taking θK,σ=δK,σ (see Definition 5 and equation (30)).
376
For all(u, v)∈[HT(Ω)]2and(u1, u2)∈[H∆(Ω)]2 there exists a constantC¯1m>0 such that
377 − X K∈T X σ∈EK m(u1,σ, u2,σ)GK,σvK− Z Ω m(u, u)Λg·∇eD,Υvdx ≤C¯1mkm(u1, u2)− m(u, u)kL2(Ω)kvkT. Proof Let I = −P K∈T P
σ∈EKm(u1,σ, u2,σ)GK,σvK. Inserting the definition ofGK,σ and by rear-378
range of terms, we deduce that
379 I= X K∈T X σ∈EK |σ|m(u1,σ, u2,σ)ΛKnK,σ· g(Υσv− vK).
We thus getI=I1+I2 with
380 I1= X K∈T X σ∈EK |σ|(m(u1,σ, u2,σ)− m(uK, uK))ΛKnK,σ· g(Υσv− vK), I2= X K∈T X σ∈EK |σ|m(uK, uK)ΛKnK,σ· g(Υσv− vK) = Z Ω m(u, u)Λg·∇eD,Υvdx.
Thanks to Cauchy-Schwarz inequality and the the equivalence of the discrete norms (see Lemma 1
381 in [34]) we infer 382 I1≤ dβ0|g| X K∈T X σ∈EK |∆K,σ|(m(u1,σ, u2,σ)− m(uK, uK))2 1 2 kvkT ,Υ ≤ d2β0 ζ4 |g|km (u1, u2)− m(u, u)kL2(Ω)kvkT.
Lemma 8 (Error estimate for the strong convergence) LetD be a space-time discretization
match-383
ing Definition 3. Let us assume that Hypotheses 1, the linearity, continuity and coercivity properties,
384
(P1), (P2) and (P3), hold. Let (p, u)∈ [HT(Ω×(0, T))]2 be a discrete solution of problem (28) and
385
(q, v)∈[HT(Ω×(0, T))]2 two test functions. Then there existC¯
with m0(x, y) =λT(x)%f(y), andC¯4,C¯5,C¯6,C¯7>0 such that, 387 kψ(u)− vk2 T ≤C¯4 −(kΨ(uN)kL1(Ω)− kΨ(u0)kL1(Ω)) + Z T 0 Z Ω δtu v + ¯C5km1(u1, u0)− m1(u, u)k2L2(Ω×(0,T)) + ¯C6km2(u2, u3)− m2(u, u)k2L2(Ω×(0,T)) + ¯C7kp − qk2T − Z T 0 aT ,λTf1,u1(t)(p(t), q(t), ψ(u(t))− v(t)) + Z T 0 Z Ω (f1λT%f+f1λ2(%1− %2))(u)Λg·∇eD,Υ(ψ(u)− v) + Z T 0 Z Ω (f1(c)s+− f1(u)s−)(ψ(u)− v) − Z T 0 aT ,1,1(ψ(u(t)), v(t), ψ(u(t))− v(t)) , (57) with m1(x, y) = (f1λT)(x)%f(y)and m2(x, y) =f1(x)λ2(y)(ρ2− ρ1). 388
Proof Let n∈J0, N−1K. We set
389 Tp,1=− X K∈T X σ∈EK λT(u(1n,σ+1))FK,σ(p(n+1), p(n+1)− q(n+1))(p(Kn+1)− q (n+1) K ).
Thanks to assumption (34), withχ=λTandα=u (n+1)
1 , we first have
Tp,1≥Cˆ1kp(n+1)− q(n+1)k2T.
Multiplying equation (28b) byp(Kn+1)−q(Kn+1)and summing it overK∈ T results in ˜Tp,1= ˜Tp,2+ ˜Tp,3
390 with 391 ˜ Tp,1=− X K∈T X σ∈EK λT(u1(n,σ+1))FK,σ(p(n+1), p(n+1))(p(Kn+1)− q(Kn+1)), ˜ Tp,2=− X K∈T X σ∈EK λT(u(1n,σ+1))%f(u (n+1) 0,σ )GK,σ(pK(n+1)− q(Kn+1)), ˜ Tp,3= X K∈T |K|(s+K− s−K)(pK(n+1)− q(Kn+1)) = Z Ω (s+− s−)(p(n+1)− q(n+1)). (58) After setting 392 ˜ Tp,4=− X K∈T X σ∈EK λT(u (n+1) 1,σ )FK,σ(p(n+1), q(n+1))(p (n+1) K − q (n+1) K ),
we observe thatTp,1= ˜Tp,2+ ˜Tp,3−T˜p,4. Then, by using Lemma 7 withm0(x, y) =λT(x)%f(y), we
393
get that there exists a constant ¯C1m0>0 such that 394 ˜ Tp,2≤C¯1m0km0(u (n+1) 1 , u (n+1) 0 )− m0(u(n+1)u(n+1))kL2(Ω)kp(n+1)− q(n+1)kT + Z Ω (λT%f)(u(n+1))Λg·∇eD,Υ(p(n+1)− q(n+1)). (59)
By gathering the results and using the fact that ˜Tp,4=aT ,λ
First using Young’s inequality, then multiplying the obtained inequality byδt(n+12)and summing it 397 up overn= 0, ..., N−1, yield 398 kp − qk2 T ≤3 ˆC41 C¯12 m0 ˆ C1 km0 (u1, u0)− m0(u, u)k2L2(Ω×(0,T)) + Z T 0 Z Ω (λT%f)(u)Λg·∇eD,Υ(p− q) + Z T 0 Z Ω (s+− s−)(p− q) − N −1 X n=0 δt(n+12)a T ,λT,u(n+1)1 ( p(n+1), q(n+1), p(n+1)− q(n+1)). (60)
Let us now consider the discrete saturation equation. We set
399 Tu,1=− X K∈T X σ∈EK FK,σ(ψ(u(n+1)), ψ(u(n+1))− v(n+1))(ψ(u( n+1) K )− v (n+1) K ).
Thanks to the coercivity property (34),
Tu,1≥Cˆ1kψ(u(n+1))− v(n+1)k2T.
Multiplying equation (28a) by ψ(u(Kn+1))− vK(n+1) and summing it over K ∈ T results in ˜Tu,1 =
400 ˜ Tu,2+ ˜Tu,3+ ˜Tu,4+ ˜Tu,5, with 401 ˜ Tu,1=− X K∈T X σ∈EK FK,σ(ψ(u(n+1)), ψ(u(n+1)))(ψ(u (n+1) K )− v (n+1) K ), ˜ Tu,2=− X K∈T |K|u (n+1) K − u (n) K δt(n+12) (ψ(u(Kn+1))− v(Kn+1)), ˜ Tu,3=− X K∈T X σ∈EK f1(u(1n,σ+1))v (n+1) K,σ (ψ(u (n+1) K )− v (n+1) K ), ˜ Tu,4= X K∈T X σ∈EK f1(u(2n,σ+1))λ2(u(3n,σ+1))(%2− %1)GK,σ(ψ(uK(n+1))− v(Kn+1)), ˜ Tu,5= X K∈T |K|f1(cK)s+K− f1(u( n+1) K )s − K (ψ(u(Kn+1))− vK(n+1)). After setting 402 ˜ Tu,6=− X K∈T X σ∈EK FK,σ(ψ(u(n+1)), v(n+1))(ψ(u( n+1) K )− v (n+1) K ) =aT ,1,1(ψ(u(n+1)), v(n+1), ψ(u(n+1))− v(n+1)),
we observe thatTu,1= ˜Tu,2+ ˜Tu,3+ ˜Tu,4+ ˜Tu,5−T˜u,6. From inequality (14), we deduce that
403 ˜ Tu,2≤ − X K∈T |K|Ψ(u (n+1) K )− Ψ(u (n) K ) δt(n+12) + Z Ω (δtu)(n)v(n+1).
By using the expression ofv(K,σn+1), we observe that
On one hand, thanks to Lemma 7, we infer 405 − X K∈T X σ∈EK (f1λT)(u( n+1) 1,σ )%f(u( n+1) 0,σ )GK,σ(ψ(u( n+1) K )− v (n+1) K ) ≤C¯1m1km1(u (n+1) 1 , u (n+1) 0 )− m1(u(n+1), u(n+1))kL2(Ω)kψ(u(n+1))− v(n+1)kT + Z Ω (f1λT%f)(u(n+1))Λg·∇eD,Υ(ψ(u(n+1))− v(n+1)).
On the other hand, we have
406 −aT ,λTf1,u(n+1)1 ( p(n+1), p(n+1), ψ(u(n+1))− v(n+1)) =−aT ,λTf1,u(n+1)1 ( p(n+1), p(n+1)− q(n+1), ψ(u(n+1))− v(n+1)) − aT ,λTf1,u(n+1)1 (p(n+1), q(n+1), ψ(u(n+1))− v(n+1)) ≤ Cχkp(n+1)− q(n+1)kTkψ(u(n+1))− v(n+1)kT − aT ,λTf1,u(n+1)1 ( p(n+1), q(n+1), ψ(u(n+1))− v(n+1)),
where we have used the continuity property (P2) of the formaT ,λ
Tf1,u(n+1)1 . Then, we obtain 407 ˜ Tu,3≤C¯1m1km1(u (n+1) 1 , u (n+1) 0 )− m1(u(n+1), u(n+1))kL2(Ω)kψ(u(n+1))− v(n+1)kT + Z Ω (f1λT%f)(u(n+1))Λg·∇eD,Υ(ψ(u(n+1))− v(n+1)) +Cχkp(n+1)− q(n+1)kTkψ(u(n+1))− v(n+1)kT −aT ,λTf1,u(n+1)1 ( p(n+1), q(n+1), ψ(u(n+1))− v(n+1)). (61)
Thanks again to Lemma 7, we have
408 ˜ Tu,4≤C¯1m2km2(u (n+1) 2 , u (n+1) 3 )− m2(u(n+1), u(n+1))kL2(Ω)kψ(u(n+1))− v(n+1)kT + Z Ω (f1λ2)(u(n+1))(%1− %2)Λg·∇eD,Υ(ψ(u(n+1))− v(n+1)). (62)
By gathering all the results, we obtain
Using Young inequality, multiplying this new inequality byδt(n+12)and summing it overn= 0, ..., N− 410 1 give 411 Z T 0 kψ (u)− vk2 T ≤ ˆ4 C1 −(kΨ(u(N))kL1(Ω)− kΨ(u0)kL1(Ω)) + Z T 0 Z Ω δtu v +C¯1 2 m1 ˆ C1 km1(u1, u0)− m1(u, u)k2L2(Ω)×(0,T) +C¯1 2 m2 ˆ C1 km2(u2, u3)− m2(u, u)k2L2(Ω×(0,T)) +C 2 χ ˆ C1 Z T 0 kp − qk 2 T − N −1 X n=0 δt(n+12)a T ,λTf1,u(n+1)1 (p(n+1), q(n+1), ψ(u(n+1))− v(n+1)) + Z T 0 Z Ω (f1λT%f+f1λ2(%1− %2))(u)Λg·∇eD,Υ(ψ(u)− v) + Z T 0 Z Ω (f1(c)s+− f1(u)s−)(ψ(u)− v) − N −1 X n=0 δt(n+12)a T ,1,1(ψ(u(n+1)), v(n+1), ψ(u(n+1))− v(n+1)) . (63)
Lemma 9 (Strong convergence) Let Dm be a sequence of space-time discretizations matching
Defi-412
nition 4 and (pm, um)∈ [HT(Ω×(0, T))]2 be a discrete solution of problem (28). Let us assume that
413
Hypotheses 1 and 2 hold. Then, there exist ψ(¯u),p¯∈ L2(0, T;H01(Ω))such that
414 lim m→∞kpm−p¯TmkTm= 0, lim m→∞kψ(um)− ψ(¯u)TmkTm = 0 (64)
where, for any m∈Nand v∈ L2(Ω×(0, T)), vTm is defined in Section 3.2.2.
415
Proof Letψ(¯u),p¯∈ L2(0, T;H01(Ω)) be the functions given from Lemma 6 and letϕ, χ∈ L2(0, T; D).
416
For anym∈N, we have
417
kpm−p¯TmkTm ≤ kpm− ϕTmkTm+kϕTm−p¯TmkTm,
kψ(um)− ψ(¯u)TmkTm≤ kψ(um)− χTmkTm+kχTm− ψ(¯u)TmkTm.
Thanks to Lemma 11, there exists a constantC17>0 such that
418
kϕTm−p¯TmkTm≤ C17kϕ −p¯kL2(0,T;H1 0(Ω)),
Thanks to Lemma 8, (i)–(viii) of Lemma 6, (35) and the properties of the functionsλT,λ1,λ2, we 419 infer that 420 lim supm→∞kpm−p¯TmkTm ≤C¯2 Z T 0 Z Ω λT%f(¯u)Λg· ∇(¯p− ϕ) + Z T 0 Z Ω (s+− s−)(¯p− ϕ) − Z T 0 Z Ω λT(¯u)Λ∇ϕ · ∇(¯p− ϕ) +C17kϕ −p¯kL2(0,T;H1 0(Ω)), lim supm→∞kψ(um)− ψ(¯u)TmkTm≤C¯4 −(kΨ(¯u(T))kL1(Ω)− kΨ(¯u(0))kL1(Ω)) + Z T 0 Z Ω ∂tuψ¯ (¯u) + ¯C7lim sup m→∞ kp m−p¯TmkTm − Z T 0 Z Ω (λTf1)(¯u)Λ∇ϕ · ∇(ψ(¯u)− χ) + Z T 0 Z Ω (f1λT%f+f1λ2(%1− %2))(u)Λg· ∇(ψ(¯u)− χ) + Z T 0 Z Ω (f1(c)s+− f1(u)s−)(ψ(¯u)− χ)− Z T 0 Z Ω Λ∇χ · ∇(ψ(¯u)− χ) +C17kχ − ψ(¯u)kL2(0,T;H1 0(Ω))
SinceL2(0, T; D) is dense inL2(0, T;H01(Ω)), by lettingϕ→p¯inL2(0, T;H01(Ω)) andχ→ ψ(¯u), the
421
previous inequality leads to
422 lim sup m→∞ kp m−p¯TmkTm= 0, lim sup m→∞ kψ (um)− ψ(¯u)TmkTm = 0.
The convergence of the discrete solutions to some pair (¯u,p¯) has already been shown in Lemma 6
423
and 9, it remains to show that (¯u,p¯) is indeed a weak solution. This is done in the following theorem.
424 425
Theorem 1 (Convergence of the scheme) LetDmbe a sequence of space-time discretizations
match-426
ing Definition 4 and (pm, um)∈[HT(Ω×(0, T))]2 be a discrete solution of problem (28). Assume that
427
Hypotheses 1 and 2 hold. Then(pm, um)converges, as m→ ∞, to the pair (¯p,u¯)(according to Lemma 6
428
and 9), and this pair is a weak solution of problem (15).
429
Proof Let ϕ∈ L2(0, T; D). For anym∈N, n∈J0, NmK, we multiply equation (28b) by (ϕTm)
(n+1)
K
430
and sum it overK∈ Tmandn∈J0, NmKto obtain
Thanks to the properties (P1), (P2), the Cauchy-Schwarz inequality and Lemma 11, we deduce 433 Z T 0 aTm,λT,(um)1(t)(pm(t), pm(t)−p¯Tm(t), ϕTm(t)) ≤ Cχkpm−p¯TmkTmkϕTmkTm ≤ C17Cχkpm−p¯TmkTmkϕkL2(0,T;H1 0(Ω)). (66)
Similarly, by using the same ideas as for (59) and by means of the Cauchy-Schwarz inequality and
434 Lemma 11, we get 435 Nm X n=0 δt(n+1)T˜p,(n2+1)− Z T 0 Z Ω (λT%f)(um)Λg·∇eDm,ΥmϕTm ≤ C17C¯1m0km0((um)1,(um)0)− m0(um, um)kL2((0,T)×Ω)kϕkL2(0,T;H01(Ω)), (67) withm0(x, y) =λT(x)%f(y). Thanks to (65)–(67), we obtain
Z T 0 aTm,λT,(um)1(t)(pm(t),p¯Tm(t), ϕTm(t))− Z T 0 Z Ω (s+− s−)ϕTm− Z T 0 Z Ω (λT%f)(um)Λg·∇eDm,ΥmϕTm ≤ C17max(Cχ,C¯1m0) kpm−p¯TmkTm+ km0((um)1,(um)2)− m0(um, um)kL2((0,T)×Ω) kϕkL2(0,T;H1 0(Ω)).
By using the Lemma 6 and 9 and Proposition 4, asm→ ∞, we obtain
436 Z T 0 Z Ω λT(¯u)Λ∇p¯· ∇ϕ − Z T 0 Z Ω (s+− s−)ϕ− Z T 0 Z Ω (λT%f)(¯u)Λg· ∇ϕ= 0.
For anym∈N,n∈J0, NmK, we multiply equation (28a) by (ϕTm)(Kn+1) and sum it overK∈ Tmand
437 n∈J0, NmKto obtain 438 Nm X n=0 δt(n+1) ˜Tu,(n1+1)−T˜u,(n2+1)−T˜u,(n3+1)−T˜u,(n4+1)−T˜u,(n5+1) = 0, (68) with 439 ˜ Tu,(n1+1)=− X K∈Tm X σ∈EK FK,σ(ψ(u (n+1) m ), ψ(u(mn+1)))(ϕTm) (n+1) K =aTm,1,1(ψ(u (n+1) m ), ψ(u(mn+1)), ϕ(Tn+1) m ), ˜ Tu,(n2+1)=− X K∈Tm |K|(um) (n+1) K −(um) (n) K δt(n+1 2) (ϕTm)(Kn+1)=− Z Ω (δtum)(n+1)ϕ(Tn+1) m , ˜ Tu,(n3+1)=− X K∈Tm X σ∈EK f1((um)(1n,σ+1))v (n+1) K,σ (ϕTm) (n+1) K , ˜ Tu,(n4+1)= X K∈Tm X σ∈EK f1((um)2(n,σ+1))λ2((um)3(n,σ+1))(%2− %1)GK,σ(ϕTm) (n+1) K , ˜ Tu,(n5+1)= X K∈Tm |K|f1(cK)s+K− f1((um)(Kn+1))s−K (ϕTm)(Kn+1)= Z Ω (f1(c)s+− f1(um(n+1))s−)(ϕTm) (n+1) .
In the same way as for (66) and (67), we can prove that
441 Nm X n=0 δt(n+1)T˜u,(n3+1)+ Z T 0 Z Ω (f1λT%f)(um)Λg·∇eDm,ΥmϕTm+ Z T 0 aTm,λTf1,(um)1(t)(pm(t),p¯Tm(t), ϕTm(t)) ≤ C17max(Cχ,C¯1m1)(km1((um)1,(um)0)− m1(um, um)kL2((0,T)×Ω)+kpm−p¯TmkTm)kϕkL2(0,T;H10(Ω)) (70) withm1(x, y) = (f1λT)(x)%f(y) and 442 Nm X n=0 δt(n+1)T˜u,(n4+1)+ Z T 0 Z Ω (f1λ2)(um)(%2− %1)Λg·∇eDm,ΥmϕTm ≤ C17C¯1m2km2((um)2,(um)3)− m2(um, um)kL2((0,T)×Ω)kϕkL2(0,T;H01(Ω)) (71)
withm2(x, y) =f1(x)λ2(y)(%2− %1). Thanks to (68)–(71), we obtain
443 Z T 0 aTm,1,1(ψ(um(t)), ψ(¯u)Tm(t), ϕTm(t)) + Z T 0 Z Ω δtumϕTm + Z T 0 Z Ω (f1λT%f)(um)Λg·∇eDm,ΥmϕTm+ Z T 0 aTm,λTf1,(um)1(t)(pm(t),¯pTm(t), ϕTm(t)) + Z T 0 Z Ω (f1λ2)(um)(%1− %2)Λg·∇eDm,ΥmϕTm− Z T 0 Z Ω (f1(c)s+− f1(um)s−)ϕTm ≤ C17max(Cχ,C¯1m1,C¯1m2) kψ(um)− ψ(¯u)TmkTm +km1((um)1,(um)0)− m1(um, um)kL2((0,T)×Ω) +kpm−p¯TmkTm +km2((um)2,(um)3)− m2(um, um)kL2((0,T)×Ω) kϕkL 2(0,T;H1 0(Ω)). (72) By using Lemma 6 and 9 and Proposition 4, asm→ ∞, we therefore obtain
Z T 0 Z Ω− ¯ uϕt− Z Ω uinit(x)ϕ(x,0)+ Z T 0 Z Ω (λTf1)(¯u)Λ∇p¯· ∇ϕ − Z T 0 Z Ω (f1λT%f)(¯u)Λg· ∇ϕ− Z T 0 Z Ω (f1λ2)(¯u)(%1− %2)Λg· ∇ϕ+ Z T 0 Z Ω Λ∇ψ(¯u)· ∇ϕ− Z T 0 Z Ω (f1(c)s+− f1(¯u)s−)ϕ= 0. 5 Flux discretizations 444
In this section, we introduce a specific familiy of discrete diffusive fluxes FK,σ, which are used in
445
the equations (28). Hereby, we mainly follow the ideas that have been presented in [34, 36]. Please
446
note that the fluxes do not include the phase mobilities which are evaluated separately such that the
447
fluxes can be constructed analogously.
448
Please note that in the following, we will define the fluxes based on some u, v∈ HT(Ω), where
449
uis not meant to be the saturation. In the discrete formulation (28), the fluxes are then evaluated
450
for u=p(n+1) oru= ψ(u(n+1)). An established idea to obtain monotone or
extremum-principles-451
preserving schemes, as those developed in [30, 41, 12, 27, 19, 28, 35, 31], is to compute for each interior
452
face σ ∈ Eint, with Tσ = {K, L}, two consistent linear flux approximations, ˜FK,σ(u) and ˜FL,σ(u),
453
which depend on the unknown u ∈ HT(Ω), and to define the final flux FK,σ(u, u) as a convex
454
combination (with weightsµK,σ, µL,σ that also depend onu) of these linear fluxes:
455
FK,σ(u, u) =µK,σ(u) ˜FK,σ(u)− µL,σ(u) ˜FL,σ(u),
withµK,σ(u)≥0, µL,σ(u)≥0 andµK,σ(u) +µL,σ(u) = 1,
where the linear fluxes are defined by 456 ˜ FK,σ(u) =|σ| X σ0∈S K,σ αK,σσ0(Iσ0u− uK), (74)
where SK,σ denotes the face stencil and Iσ0 ∈ L(HT(Ω); P0(σ)) a trace reconstruction operator 457
according to Definition 5. The stencil and the coefficients αK,σσ0 are determined by the conormal 458 decomposition: 459 ΛKnK,σ= X σ0∈S K,σ αK,σσ0(xσ0− xK). (75)
This means that the conormalΛKnK,σ is decomposed into a basis (xσ0− xK){σ0∈S
K,σ}with coeffi-460
cients (αK,σσ0){σ0∈SK,σ}which are computed to be non-negative, i.e. αK,σσ0≥0 if possible. By only 461
including neighboring cells into the face stencils, the non-negativity cannot always be guaranteed.
462
In [34, 36], we have thus introduced the idea of formulating the conormal decomposition as an
opti-463
mization problem. The authors of [39] solve this issue by also including non-neighboring cells. For
464
further details the reader is referred to the given references. By using this conormal decomposition,
465
it can be shown that the linear sub-fluxes (74) are strongly consistent if the trace reconstruction
466
operators{Iσ0}σ∈E are of second order accuracy [34]. 467
For any K ∈ T, σ ∈ EK ∩ Eint andL ∈ T such that Tσ = {K, L}, we thus get from (73) the
468
numerical flux functionFK,σ(·, ·), defined for all (u, v)∈[HT(Ω)]2 by
469
FK,σ(u, v) =µK,σ(u) ˜FK,σ(v)− µL,σ(u) ˜FL,σ(v). (76)
These flux functions are constructed to be conservative, such that equation (27) holds.
470
In the following, different choices for the weightsµK,σ, µL,σare presented for the family of schemes
471
(73). Moreover, a general trace reconstruction operatorIσ(see Definition 5), which is needed for the
472
derivation of the different nonlinear schemes, is introduced and given for each faceσ∈ Eintby
473 Iσu= X M ∈Iσ ωM,σuM, X M ∈Iσ ωM,σ= 1, ωM,σ≥0, (77)
where the subsetIσ⊂ T stands for the interpolation index set (see [34] for further details).
474
AvgMPFA scheme
475
The most simple choice of coefficients isµK,σ=µL,σ= 0.5 resulting in a linear finite volume scheme,
476
which is in the following calledAvgMPFA.
477
NLTPFA scheme
478
To derive a nonlinear two-point flux approximation (NLTPFA), the different terms are reordered
479
such that the flux is written as
480 FK,σ(u) =tL,σ(u)uL− tK,σ(u)uK−(µL,σ(u)λL,σ(u)− µK,σ(u)λK,σ(u)) | {z } def =RK,σ(u) , (78)
with the transmissibilities
and 482 λK,σ(v) =|σ| X σ0∈S K,σ X M ∈{Iσ0\{K,L}} αK,σσ0ωM,σ0vM, λL,σ(v) =|σ| X σ0∈S L,σ X M ∈{Iσ0\{K,L}} αL,σσ0ωM,σ0vM. (80)
The idea of the NLTPFA scheme is to choose the weights such thatRK,σ(u) = 0. From a numerical
483
point of view, it is sufficient that|RK,σ(u)| ≤ . Under the assumption thatλK,σλL,σ≥0, this can
484 be ensured, by taking 485 µK,σ(u) = |λ L,σ(u)|+ |λK,σ(u)|+|λL,σ(u)|+ 2 , µL,σ(u) = |λ K,σ(u)|+ |λK,σ(u)|+|λL,σ(u)|+ 2 . (81)
With this, the residual term is given as
486
RK,σ(u) =
λL,σ(u)− λK,σ(u)
|λK,σ(u)|+|λL,σ(u)|+ 2
,
for which it holds that
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|RK,σ(u)| ≤ . (82)
Instead of directly neglecting the termRK,σ(u), we will incorporate it partly intotK,σ, tL,σand then
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only neglect a smaller part RK,σ(u) for which it holds that |RK,σ(u)| ≤ |RK,σ(u)|. Details can be
489
found in [36]. This results in the final fluxes
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FK,σ(u, u) = ˜tL,σ(u)uL−˜tK,σ(u)uK, (83)
with the new transmissibilities ˜tK,σ,˜tL,σ which are greater or equal to the correspondingtK,σ, tL,σ.
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Therefore, the positivity of the new transmissibilities directly follow from the positivity of the old
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ones. Here,is chosen such that 0< ≤ hDmin σ∈E|σ|.
493
NLMPFA scheme
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A nonlinear multi-point flux approximation (NLMPFA) is derived by splitting the linear fluxes into
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a two-point part and a residual flux part as follows
496 ˜ FK,σ(u) =|σ|βσ(uL− uK) + ˜FK,σres(u), ˜ FL,σ(u) =|σ|βσ(uK− uL) + ˜FL,σres(u), (84) withβσ= min(αK,σσωL,σ, αL,σσωK,σ) and ˜FK,σres,F˜L,σres defined by
497 ˜ FK,σres(v) =|σ|(αK,σσωL,σ− βσ)(vL− vK) +|σ|αK,σσ X M ∈{Iσ\{L}} ωM,σ(vM− vK) + X σ0∈{SK,σ\{σ}} |σ|αK,σσ0(Iσ0v− vK), ˜ FL,σres(v) =|σ|(αL,σσωK,σ− βσ)(vK− vL) +|σ|αL,σσ X M ∈{Iσ\{K}} ωM,σ(vM − vL) + X σ0∈{S L,σ\{σ}} |σ|αL,σσ0(Iσ0v− vL). (85)
Here, the weights are chosen as
