A combined finite volume–nonconforming finite element scheme for compressible two phase flow in porous media

Abstract

We propose and analyze a combined finite volume–nonconforming finite element scheme on general meshes to simulate the two compressible phase flow in porous media. The diffusion term, which can be anisotropic and heterogeneous, is discretized by piecewise linear nonconforming triangular finite elements. The other terms are discretized by means of a cell-centered finite volume scheme on a dual mesh, where the dual volumes are constructed around the sides of the original mesh. The relative permeability of each phase is decentred according the sign of the velocity at the dual interface. This technique also ensures the validity of the discrete maximum principle for the saturation under a non restrictive shape regularity of the space mesh and the positiveness of all transmissibilities. Next, a priori estimates on the pressures and a function of the saturation that denote capillary terms are established. These stabilities results lead to some compactness arguments based on the use of the Kolmogorov compactness theorem, and allow us to derive the convergence of a subsequence of the sequence of approximate solutions to a weak solution of the continuous equations, provided the mesh size tends to zero. The proof is given for the complete system when the density of the each phase depends on its own pressure.

1 Introduction

The simultaneous flow of immiscible fluids in porous media occurs in a wide variety of applications. A large variety of methods have been proposed for the discretization of degenerate parabolic systems modeling the displacement of immiscible incompressible two-phase flows in porous media. We refer to [1] and [24] for the finite difference method. The finite volume methods have been proved to be well adapted to discretize conservative equations. The cell-centered finite volume scheme has been studied e.g. by [16, 22] and [4]. Recently, the convergence analysis of a finite volume scheme for a degenerate compressible and immiscible flow in porous media has been studied by Bendahmane et al. [3] when the densities of each phase depend on the global pressure, and by B. Saad and M. Saad [26] for the complete system when the density of the each phase depends on the own pressure. In these works, the medium is considered homogeneous, the permeability tensor is proportional to the matrix identity and the mesh is supposed to be admissible in the sense that two control volumes with a common face, satisfying an orthogonality property e.g. [14]. The cell-centered finite volume method with an upwind discretization of the convection term ensures the stability and is extremely robust and have been used in industry because they are cheap, simple to code and robust. However, standard finite volume schemes do not permit to handle anisotropic diffusion on general meshes see e.g. [14].

Various multi-point schemes where the approximation of the flux through an edge involves several scalar unknowns have been proposed, see e.g. Coudière et al. [10], Eymard et al. [15], Faille [18], etc. However, such schemes require using more points than the classical four points for triangular meshes and five points for quadrangular meshes in space dimension two, making the schemes less robust.

On the other hand finite element method allows a very simple discretization of the diffusion term with a full tensor and does not impose any restrictions on the meshes, they were used a lot for the discretization of a degenerate parabolic problems modeling of contaminant transport in porous media. The mixed finite element method by Dawson [11], the conforming piecewise linear finite element method has been studied e.g. by Barrett and Knabner [2], Chen and Ewing [8], Nochetto et al. [23], and Rulla et al. [25]. However, it is well-known that numerical instabilities may arise in the convection-dominated case.

To avoid these instabilities, the theoretical analysis of the combined finite volume–finite element method has been carried out for the case of a degenerate parabolic problems with a full diffusion tensors. The combined finite volume–conforming finite element method proposed and studied by Debiez et al. [12] or Feistauer et al. [19] for fluid mechanics equations, are indeed quite efficient.

This ideas is extended by [17] for the degenerate parabolic problems, to the combination of the mixed-hybrid finite element and finite volume methods, to inhomogeneous and anisotropic diffusion–dispersion tensors, to space dimension three, and finally to meshes only satisfying the shape regularity condition. In order to solve this class of equations, Eymard et al. [17] discretize the diffusion term by means of piecewise linear nonconforming (Crouzeix–Raviart) finite elements over a triangularization of the space domain, or using the stiffness matrix of the hybridization of the lowest order Raviart–Thomas mixed finite element method. The other terms are discretized by means of a finite volume scheme on a dual mesh, with an upwind discretization of the convection term to ensure the stability, where the dual volumes are constructed around the sides of the original triangularization. The intention of this paper is to extend these ideas to a fully nonlinear degenerate parabolic system modeling immiscible gas-water displacement in porous media without simplified assumptions on the state law of the density of each phase, to the combination of the nonconforming finite element and finite volume methods, to inhomogeneous and anisotropic permeability tensors, to space dimension three, and finally to meshes only satisfying the shape regularity condition.

Following [17], let us now introduce the combined scheme that we analyze in this paper. We consider a triangulation of the space domain consisting of simplices (triangles in space dimension two and tetrahedra in space dimension three). We next construct a dual mesh where the dual volumes are associated with the sides (edges or faces). To construct a dual volume, one connects the barycentres of two neighboring simplices through the vertices of their common side. We finally place the unknowns in the barycentres of the sides. The diffusion term, which can be anisotropic and heterogeneous, is discretized by piecewise linear nonconforming triangular finite elements. The other terms are discretized by means of a cell-centered finite volume scheme on a dual mesh, where the dual volumes are constructed around the sides of the original mesh, hence we obtain the combined scheme. To ensure the stability, the relative permeability of each phase is decentred according to the sign of the velocity at the dual interface. This technique ensures the validity of the discrete maximum principle for the saturation in the case where all transmissibilities are non-negative.

In [26] the authors study, for a compressible two phase flow, a pure finite volume scheme constructed on a mesh satisfying the orthogonal property and for identity permeability tensor. The present paper deals with construction and convergence analysis of a combined finite volume–nonconforming finite element for two compressible and immiscible flow in porous media without simplified assumptions on the state law of the density of each phase and for anisotropic permeability tensor. The proposed scheme is constructed on a general mesh satisfying a non restrictive shape regularity of the space mesh. The choice of $P_{1}$ nonconforming Crouzeix–Raviart finite element allows to define the dual mesh (diamond mesh) and consequently to treat the convective terms in a very simple way. Our scheme is easier to implement with this choice, especially if we consider the 3D case. The mathematical difficulties are the use of nonconforming finite element combined with finite volume approach and the nonstandard nonlinear degenerate system.

The analysis of this model is based on new energy estimates on the velocities of each phase. These estimates are obtained by using the same nonlinear test function as the continuous case in [20, 21] and in the discrete case as [26]. The analysis in [26] is based on the finite volume two points flux reconstruction, but here the reconstruction of the solutions is both finite volume and noncomforming finite element which complicates the analysis. Nevertheless, these estimates are degenerate in the sense that they do not permit the control of gradients of pressure of each phase, especially when a phase is not locally present in the domain. We use the formulation of phase-by-phase upstreaming and we consider a coupled parabolic problem instead of used the “global pressure” formulation. In the convergence proof, the global pressure is used to control the discrete gradients weighted with the relative permeabilities. Individual control of the phase pressure gradients is not possible (for example in the region where the phase is not present), whereas the gradient of the global pressure can be controlled in the whole domain, that allow us, based on the use of the Kolmogorov relative compactness theorem, to derive the convergence of these approximation to a weak solution of the continuous problem in this paper provided the mesh size tends to zero.

The organization of this paper is as follows. In Sect. 2, we introduce the nonlinear parabolic system modeling the two-compressible and immiscible fluids in a porous media and we state the assumptions on the data and present a weak formulation of the continuous problem. In Sect. 3, we describe the combined finite volume–nonconforming finite element scheme and we present the main theorem of convergence. In Sect. 4, we derive three preliminary fundamental lemmas. In fact, we present some properties of this scheme and we will see that we can’t control the discrete gradient of pressure since the mobility of each phase vanishes in the region where the phase is missing. So we are going to use the feature of global pressure. We show that the control of velocities ensures the control of the global pressure and a dissipative term on saturation in the whole domain regardless of the presence or the disappearance of the phases. Section 5 is devoted to a maximum principle on saturation, a priori estimates on the discrete velocities and existence of discrete solutions. In Sect. 6, we derive estimates on difference of time and space translates for the approximate solutions. In Sect. 7 using the Kolmogorov relative compactness theorem, we prove the convergence of a subsequence of the sequence of approximate solutions to a weak solution of the continuous problem.

2 Mathematical formulation of the continuous problem

Consider a porous medium saturated with a fluid composed of two mmiscible phases (liquid and gas). We focus on the water and gas phase, but the considerations below are also valid for a general wetting phase and a non-wetting phase.

Let $T>0$ be the final time fixed, and let be $\Omega$ a bounded open subset of $\mathbb {R}^d\ (d\ge 1)$. We set $Q_T=(0,T)\times \Omega$, $\Sigma _T = (0,T)\times \partial \Omega$. In order to define the physical model, we write the mass conservation of each phase in $Q_T$

$$\begin{aligned}&\phi (x)\partial _{t}( \rho _{\alpha }(p_\alpha )s_\alpha )(t,x) + \mathrm {div}(\rho _{\alpha }(p_\alpha ) \mathbf V _{\alpha })(t,x) +\rho _\alpha (p_\alpha ) s_\alpha f_{P}^{~}(t,x)\nonumber \\&\quad = \rho _\alpha (p_\alpha ) s^I_\alpha f_{I}^{~}(t,x), \end{aligned}$$

(2.1)

where $\phi$, $\rho _\alpha$ and $s_\alpha$ are respectively the porosity of the medium, the density of the $\alpha$ phase and the saturation of the $\alpha$ phase. Here the functions $f_{I}^{~}$ and $f_{P}^{~}$ are respectively the injection and production terms. Note that in Eq. (2.1) the injection term is multiplied by a known saturation $s^I_\alpha$ corresponding to the known injected fluid, whereas the production term is multiplied by the unknown saturation $s_\alpha$ corresponding to the produced fluid.

The velocity of each fluid $\mathbf V _\alpha$ is given by the Darcy law:

$$\begin{aligned} \mathbf V _{\alpha }= - \mathbf{\Lambda } \frac{k_{r_\alpha }(s_\alpha )}{\mu _{\alpha }}\big ( \nabla p_{\alpha }-\rho _\alpha (p_\alpha )\mathbf{g}\big ),\qquad \quad \alpha = l, g. \end{aligned}$$

(2.2)

where $\mathbf{\Lambda }$ is the permeability tensor of the porous medium, $k_{r_\alpha }$ the relative permeability of the $\alpha$ phase, $\mu _\alpha$ the constant $\alpha$-phase’s viscosity, $p_\alpha$ the $\alpha$-phase’s pressure and $\mathbf{g }$ is the gravity term. Assuming that the phases occupy the whole pore space, the phase saturations satisfy

$$\begin{aligned} s_{l}+ s_{g} = 1. \end{aligned}$$

(2.3)

To close the system, we introduce the capillary pressure law which links the jump of pressure of the two phases to the saturation (2.1)–(2.3)

$$\begin{aligned} p_c(s_l(t,x)) = p_{g}(t,x) - p_{l}(t,x), \end{aligned}$$

(2.4)

the application $s_l\mapsto p_c(s_l)$ is non-increasing, $(\frac{\mathrm {d}p_c}{\mathrm {d}s_l}(s_l) < 0, \text{ for } \text{ all } s_l\in [0,1])$, and usually $p_c(s_l=1)=0$ when the wetting fluid is at its maximum saturation.

2.1 Assumptions and main result

Following Chavent et al. [7], in order to obtain a weak formulation (which will be shown to be the limit of the numerical scheme), we introduce some artificial pressures, which are however not actually used in the implementation of the scheme. These artificial pressures are denoted by $\bar{p}$ and $\tilde{p}$ and defined by

$$\begin{aligned} \tilde{p}(s_l)=-\int \limits _{0}^{s_l} \frac{M_{l}(z)}{M(z)} p_c^{'}(z)\mathrm {d}z\quad \text { and }\quad \overline{p}(s_l)=\int \limits _{0}^{s_l} \frac{M_{g}(z)}{M(z)} p_c^{'}(z)\mathrm {d}z, \end{aligned}$$

(2.5)

with the $\alpha$-phase’s mobility $M_\alpha$ and the total mobility are defined by

$$\begin{aligned} M_{\alpha }(s_{\alpha })=k_{r_\alpha }(s_{\alpha })/ \mu _{\alpha }, \quad M(s_l) = M_l(s_l)+M_g(s_g). \end{aligned}$$

Note that,

$$\begin{aligned} p=p_g+\tilde{p}(s_l)=p_l+\bar{p}(s_l), \end{aligned}$$

(2.6)

due to the definition of the capillary pressure, where $p$ denoted the global pressure. This concept of the global pressure, described in [7], leads to

$$\begin{aligned} M(s_l)\nabla p = M_l(s_l) \nabla p_l+ M_g(s_g) \nabla p_g, \end{aligned}$$

We also define the capillary terms by

$$\begin{aligned} \gamma (s_l)=-\frac{M_{l}(s_l)M_{g}(s_g)}{M(s_l)}\frac{\mathrm {d}p_c}{\mathrm {d}s_l}(s_l)\ge 0, \end{aligned}$$

and let us finally define the function $\mathcal {B}$ from $[0,1]$ to $\mathbb {R}$ by:

$$\begin{aligned} \mathcal {B}(s_l)&=\int \limits _{0}^{s_l}\gamma (z) \mathrm {d}z =- \int \limits _{0}^{s_l}\frac{M_l(z)M_g(z)}{M(z)}\frac{\mathrm {d}p_c}{\mathrm {d}s_l}(z) \mathrm {d}z\nonumber \\&=- \int \limits _{0}^{s_g} M_l(z)\frac{\mathrm {d}\bar{p}}{\mathrm {d}s_l}(z) \mathrm {d}z = \int \limits _{0}^{s_l} M_g(z)\frac{\mathrm {d}\tilde{p}}{\mathrm {d}s_l}(z) \mathrm {d}z. \end{aligned}$$

(2.7)

Using these notations, we derive the fundamental relationship between the velocities and the global pressure and the capillary term:

$$\begin{aligned} M_l(s_l)\nabla p_l=M_l(s_l)\nabla p +\nabla \mathcal {B}(s_l),\quad M_g(s_g)\nabla p_g=M_g(s_l)\nabla p -\nabla \mathcal {B}(s_l). \end{aligned}$$

(2.8)

We complete the description of the model (2.1) by introducing boundary conditions and initial conditions. To the system (2.1)–(2.4) we add the following mixed boundary conditions. We consider the boundary $\partial \Omega =\Gamma _l\cup \Gamma _{imp }$, where $\Gamma _l$ denotes the water injection boundary and $\Gamma _{imp }$ the impervious one.

$$\begin{aligned} \left\{ \begin{array}{l} p_l(t,x) = p_g(t,x)=0 \quad \text { on } (0,T)\times \Gamma _l,\\ \rho _l\mathbf V _l\cdot \mathbf n = \rho _g\mathbf V _g\cdot \mathbf n = 0 \quad \text { on } (0,T)\times \Gamma _{imp}, \end{array}\right. \end{aligned}$$

(2.9)

where $\mathbf n$ is the outward normal to $\Gamma _{imp}$.

The initial conditions are defined on pressures

$$\begin{aligned} p_{\alpha }(t=0) =p^{0}_{\alpha } \quad \text { for } \alpha =l,g\text { in } \Omega . \end{aligned}$$

(2.10)

The studied system represents two kinds of degeneracy: the degeneracy for evolution terms $\partial _t(\rho _\alpha s_\alpha )$ and the degeneracy for dissipative terms $\mathrm {div}(\rho _\alpha M_\alpha \nabla p_\alpha )$ when the saturation vanishes. As mentioned above the mobilities vanish and consequently the control of the gradient of the pressure of each phase is not possible. A main point of the paper is to give sense for the term $\nabla p_\alpha$, $\alpha =l,g$. This term is a distribution and it is not enough to give a sense of the velocity. Our approach is based on the control of the velocity of each phase. Thus the gradient of the global pressure and the function $\mathcal {B}$ are bounded which give a rigorous justification for the degenerate problem.

We will see in the Sect. 5 that we can’t control the discrete gradient of pressure since the mobility of each phase vanishes in the region where the phase is missing. So, we are going to use the feature of global pressure to obtain uniform estimates on the discrete gradient of the global pressure and the discrete gradient of the capillary term ${\mathcal B}$ (defined in (2.7)) to treat the degeneracy of this system. The main point is to handle a priori estimates on the approximate solution.

Next we introduce some physically relevant assumptions on the coefficients of the system.

• (A1) There is two positive constants $\phi _{0}$ and $\phi _{1}$ such that $\phi _{0}\le \phi (x) \le \phi _{1}$ almost everywhere $x\in \Omega$.

• (A2) $\mathbf{\Lambda }_{ij} \in L^{\infty }(\varOmega )$, $\left| \mathbf{\Lambda }_{ij}\right| \le \frac{C_\mathbf{\Lambda }}{d} \text { a.e. in } \Omega$, $1 \le i,j\le d$, $C_\mathbf{\Lambda }>0$, $\mathbf{\Lambda }$ is a symmetric and there exist a constant $c_\mathbf{\Lambda }>0$ such that

  $$\begin{aligned} \left<\mathbf{\Lambda }(x) \xi ,\xi \right> \ge c_\mathbf{\Lambda }| \xi |^{2}, \forall \xi \in \mathbb {R}^d. \end{aligned}$$

• (A3) The functions $M_l$ and $M_g$ belongs to ${\mathcal C}^{0}([0,1],\mathbb {R}^{+})$, $M_{\alpha }(s_{\alpha }=0)=0.$ In addition, there is a positive constant $m_{0}>0$ such that for all $s_l\in [0,1]$,

  $$\begin{aligned} M_l(s_l) + M_g(s_g)\ge m_{0}. \end{aligned}$$

• (A4) $(f_{P}^{~},f_{I}^{~})\in (L^2(Q_T))^2$, $f_{P}^{~}(t,x)$, $f_{I}^{~}(t,x) \ge 0$ almost everywhere $(t,x)\in Q_T$.

• (A5) The density $\rho _{\alpha }$ is ${\mathcal C}^{1}(\mathbb {R})$, increasing and there exist two positive constants $\rho _{m}>0$ and $\rho _{M}>0$ such that $0<\rho _{m}\le \rho _{\alpha }(p_{g})\le \rho _{M}.$

• (A6) The capillary pressure fonction $p_c\in \mathcal {C}^{1}([0,1];\mathbb {R}^{+})$, decreasing and there exists $\underline{p_c}>0$ such that $0<\underline{p_c}\le |\frac{\mathrm {d}p_c}{\mathrm {d}s_l}|$.

• (A7) The function $\gamma \in C^{1}\left( [0,1];\mathbb {R}^{+} \right)$ satisfies $\gamma (s_l)>0$ for $0<s_l< 1$ and $\gamma (s_l=1)=\gamma (s_l=0)=0.$ We assume that $\mathcal {B}^{-1}$ (the inverse of $\mathcal {B}(s_l)=\int _{0}^{s_l}\ \gamma (z) \mathrm {d}z$) is a Hölder¹ function of order $\theta$, with $0<\theta \le 1, \text { on } [0,\mathcal {B}(1)]$.

The assumptions (A1)–(A7) are classical for porous media. Note that, due to the boundedness of the capillary pressure function, the functions $\tilde{p}$ and $\bar{p}$ defined in (2.5) are bounded on $[0,1]$.

We now give the definition of a weak solution of the problem (2.1)–(2.4).

Definition 1

$\left( \textit{Weak solutions} \right)$ Under assumptions (A1)–(A7) and suppose ($p^{0}_l,\ p^{0}_g$) belong to $(L^2(\Omega ))^2$ and $0\le s^0_\alpha (x)\le 1$ almost everywhere in $\Omega$, then the pair $\left( p_l, p_g\right)$ is a weak solution of problem (2.1) satisfying:

$$\begin{aligned}&p_\alpha \in L^2(Q_T),~0\le s_\alpha (t,x)\le 1 \text { a.e in } Q_T, \; (\alpha =l,g),\end{aligned}$$

(2.11)

$$\begin{aligned}&p\in L^{2}(0,T;H^{1}(\Omega )),\; \mathcal {B}(s_l)\in L^{2}(0,T;H^{1}_{\Gamma _l}, (\Omega )),\end{aligned}$$

(2.12)

$$\begin{aligned}&M_l(s_l)\nabla p_l:=M_l(s_l)\nabla p +\nabla \mathcal {B}(s_l) \in (L^2(Q_T))^d,\end{aligned}$$

(2.13)

$$\begin{aligned}&M_g(s_g)\nabla p_g:=M_g(s_l)\nabla p -\nabla \mathcal {B}(s_l) \in (L^2(Q_T))^d, \end{aligned}$$

(2.14)

such that for all $\varphi ,\, \psi \in \mathbb {C}^1([0,T];H^1_{\Gamma _l}(\Omega ))\, \text { with } \, \varphi (T,\cdot )=\psi (T,\cdot )=0$,

$$\begin{aligned}&- \int \limits _{Q_T} \phi \rho _l(p_l) s_l\partial _t\varphi \mathrm {d}x \mathrm {d}t- \int \limits _\Omega \phi (x) \rho _l(p_l^0(x)) s_l^0(x)\varphi (0,x)\mathrm {d}x \nonumber \\&+ \int \limits _{Q_T} \rho _l(p_l) M_l(s_l) \mathbf{\Lambda }\nabla p_l\cdot \nabla \varphi \mathrm {d}x \mathrm {d}t-\int \limits _{Q_T}\mathbf{\Lambda }M_l(s_l)\rho _l^2(p_l) \mathbf{g} \cdot \nabla \varphi \mathrm {d}x \mathrm {d}t\\&+\int \limits _{Q_T}\rho _l(p_l)s_lf_{P}^{~}\varphi \mathrm {d}x\mathrm {d}t = \int \limits _{Q_T}\rho _l(p_l) s^I_lf_{I}^{~}\varphi \mathrm {d}x \mathrm {d}t,\nonumber \end{aligned}$$

(2.15)

$$\begin{aligned}&-\int \limits _{Q_T} \phi \rho _g(p_g) s_g\partial _t\psi \mathrm {d}x \mathrm {d}t-\int \limits _\Omega \phi (x) \rho _g(p_g^0(x)) s_g^0(x)\psi (0,x)\,dx \nonumber \\&+\int \limits _{Q_T}M_g(s_g)\rho _g(p_g) \mathbf{\Lambda }\nabla p_g\cdot \nabla \psi \mathrm {d}x \mathrm {d}t -\int \limits _{Q_T} \mathbf{\Lambda }M_g(s_g)\rho _g^2(p_g)\mathbf{g} \cdot \nabla \psi \mathrm {d}x \mathrm {d}t\nonumber \\&+\int _{Q_T}\rho _g(p_g)s_gf_{P}^{~}\psi \mathrm {d}x \mathrm {d}t =\int _{Q_T}\rho _g(p_g) s^I_gf_{I}^{~}\psi \mathrm {d}x \mathrm {d}t. \end{aligned}$$

(2.16)

3 Combined finite volume–nonconforming finite element scheme

We will describe the space and time discretizations, define the approximation spaces, and introduce the combined finite volume–nonconforming finite element scheme in this section.

3.1 Space and time discretizations

In order to discretize the problem (2.1), we perform a triangulation $\mathcal {T}_h$ of the domain $\Omega$, consisting of closed simplices such that $\overline{\Omega }=\cup _{K\in \mathcal {T}_h} K$ and such that if $K, \L \in \mathcal {T}_h$, $K\ne \L$, then $K\cap \L$ is either an empty set or a common face, edge, or vertex of $K$ and $\L$. We denote by $\mathcal {E}_h$ the set of all sides, by $\mathcal {E}_h^{\text {int}}$ the set of all interior sides, by $\mathcal {E}_h^{\text {ext}}$ the set of all exterior sides, and by $\mathcal {E}_K$ the set of all the sides of an element $K\in \mathcal {T}_h$. We define $h:= max\{\text {diam}(K),K\in \mathcal {T}_h \}$ and make the following shape regularity assumption on the family of triangulations $\{ \mathcal {T}_h \}_h$:

There exists a positive constant $\kappa _\mathcal {T}$ such that

$$\begin{aligned} \min _{K\in \mathcal {T}_h} \frac{\left| K\right| }{\text {diam}(K)^d} \ge \kappa _T, \quad \forall h>0. \end{aligned}$$

(3.1)

Assumption (3.1) is equivalent to the more common requirement of the existence of a constant $\theta _\mathcal {T}>0$ such that

$$\begin{aligned} \max _{K\in \mathcal {T}_h} \frac{\text {diam}(K)}{\mathfrak {D}_K} \ge \kappa _T, \quad \forall h>0, \end{aligned}$$

(3.2)

where $\mathfrak {D}_K$ is the diameter of the largest ball inscribed in the simplex $K$.

We also use a dual partition $\mathcal {D}_h$ of $\Omega$ such that $\overline{\Omega } = \cup _{D\in \mathcal {D}_h} D$. There is one dual element $D$ associated with each side $\sigma _D\in \mathcal {E}_h$. We construct it by connecting the barycentres of every $K\in \mathcal {T}_h$ that contains $\sigma _D$ through the vertices of $\sigma _D$. For $\sigma _D\in \mathcal {E}^\text {ext}_h$, the contour of $D$ is completed by the side $\sigma _D$ itself. We refer to Fig. 1 for the two-dimensional case. We denote by $Q_D$ the barycentre of the side $\sigma _D$. As for the primal mesh, we set $\mathcal {F}_h$, $\mathcal {F}_h^\text {int}$, $\mathcal {F}_h^\text {ext}$ and $\mathcal {F}_D$ for the dual mesh sides. We denote by $\mathcal {D}_h^\text {int}$ the set of all interior and by $\mathcal {D}_h^\text {ext}$ the set of all boundary dual volumes. We finally denote by $\mathcal N(D)$ the set of all adjacent volumes to the volume $D$,

$$\begin{aligned} \mathcal N(D) := \{ E\in \mathcal {D}_h; \exists \sigma \in \mathcal {F}_h^\text {int}\text { such that } \sigma = \partial D\cap \partial E\} \end{aligned}$$

and remark that

$$\begin{aligned} \left| K\cap D\right| = \frac{\left| K\right| }{ d+ 1}, \end{aligned}$$

(3.3)

for each $K\in \mathcal {T}_h$ and $D\in \mathcal {D}_h$ such that $\sigma _D\in \mathcal {E}_K$. For $E\in \mathcal N(D)$, we also set $d_{D,E}:= \left| Q_E- Q_D\right|$, $\sigma _{D,E}:= \partial D\cap \partial E$ and $K_{D,E}$ the element of $\mathcal {T}_h$ such that $\sigma _{D,E}\subset K_{D,E}$.

Fig. 1

Triangles $K$, $\L \in \mathcal {T}_h$ and dual volumes $D$, $E\in \mathcal {D}_h$ associated with edges $\sigma _D,\sigma _E\in \mathcal {E}_h$

The problem under consideration is time-dependent, hence we also need to discretize the time interval $(0,T)$. The time discretization of $(0,T)$ is given by an integer value $N$ and by a strictly increasing sequence of real values $(t^n)_{n\in [0,N]}$ with $t^0=0$ and $t^{N} = T$. Without restriction, we consider a uniform step time $\delta t= t^{n} - t^{n-1}$, for $n\in [1,N]$.

We define the following finite-dimensional spaces:

$$\begin{aligned} X_h&:= \{ \varphi _h\in L^2(\Omega ); \varphi _h|_K\text { is linear } \forall K\in \mathcal {T}_h,\\&\qquad \varphi _h\text { is continuous at the points } Q_D, D\in \mathcal {D}_h^\text {int}\},\\ X_h^0&:= \{ \varphi _h\in X_h; \varphi _h(Q_D) = 0 \quad \forall D\in \mathcal {D}_h^\text {ext}\}. \end{aligned}$$

The basis of $X_h$ is spanned by the shape functions $\varphi _D$, $D\in \mathcal {D}_h$, such that $\varphi _\mathcal {D}(Q_E) = \delta _{DE}$, $E\in \mathcal {D}_h$, $\delta$ being the Kronecker delta. We recall that the approximations in these spaces are nonconforming since $X_h\not \subset H^1(\Omega )$. We equip $X_h$ with the seminorm

$$\begin{aligned} \left\| u_h\right\| ^2_{X_h} := \sum _{K\in \mathcal {T}_h} \int \limits _{K} \left| \nabla u_h\right| ^2 \mathrm {d}x, \end{aligned}$$

which becomes a norm on $X_h^0$.

For a given value $u_D, D\in \mathcal {D}_h$ (resp. $u_D^n, D\in \mathcal {D}_h, n\in [0,N]$), we define a constant piecewise function as : $u(x)=u_D$ for $x\in D$ (resp. $u(t,x) = u_D^n$ for $x\in D$, $t\in ]t^{n-1}, t^{n}]$. Next, we define the discret differential operateur:

$$\begin{aligned} \delta _{D|E}(u) = u_E-u_D, \quad \text { and }\quad \delta _{D|E}^{n}(u) = u_E^n-u_D^n. \end{aligned}$$

Definition 2

Let the values $u_D^n$, $D\in \mathcal {D}_h$, $n \in \{0,1, \ldots , N\}$. As the approximate solutions of the problem by means of the combined finite volume–nonconforming finite element scheme, we understand:

• (1) A function $u_{\delta t,h}$ such that

  $$\begin{aligned}&u_{\delta t,h} (x,0) = u_h^0(x) \quad \text { for } x \in \Omega , \nonumber \\&u_{\delta t,h} (x,t) = u_h^n(x) \quad \text { for } x \in \Omega , t \in (t_{n-1},t_n] \qquad n \in \{1, \ldots , N\} , \end{aligned}$$

  (3.4)

  where $u_h^n = \sum _{D\in \mathcal {D}_h} u_D^n \varphi _D$;

• (2) A function $\tilde{u}_{\delta t,h}$ such that

  $$\begin{aligned}&\tilde{u}_{\delta t,h}(x,0) = u_D^0\quad \text { for } x \in \mathring{D}, D\in \mathcal {D}_h, \nonumber \\&\tilde{u}_{\delta t,h}(x,t) = u_D^n \quad \text { for } x \in \mathring{D}, D\in \mathcal {D}_h, t \in (t_{n-1},t_n] \qquad n \in \{1, \ldots , N\}. \end{aligned}$$

  (3.5)

The function $u_{\delta t,h}$ is piecewise linear and continuous in the barycentres of the interior sides in space and piecewise constant in time; we will call it a nonconforming finite element solution. The function $\tilde{u}_{\delta t,h}$ is given by the values of $u_D^n$ in side barycentres and is piecewise constant on the dual volumes in space and piecewise constant in time; we will call it a finite volume solution.

3.2 The combined scheme

For clarity and simplicity, we restrict the theoretical demonstration to a horizontal field, i.e. we neglect the gravity effect. In Remark 1, we indicate how to modify the scheme to include the gravity terms.

Definition 3

(Combined scheme) The fully implicit combined finite volume–nonconforming finite element scheme for the problem (2.1) reads: find the values $p_{\alpha ,D}^n$, $D\in \mathcal {D}_h$, $n \in \{1,\ldots ,N\}$, such that

$$\begin{aligned}&p_{\alpha ,D}^0 = \frac{1}{\left| D\right| } \int \limits _{D} p^0_\alpha (x) \mathrm {d}x,\; s_{\alpha ,D}^0=\frac{1}{\left| D\right| } \int \limits _{D} s^0_\alpha (x) \mathrm {d}x, \text { for all } D\in \mathcal {D}_h^{\text {int}},\end{aligned}$$

(3.6)

$$\begin{aligned}&\quad \left| D\right| \phi _D\frac{\rho _l(p^{n}_{l,D}) s^{n}_{l,D}- \rho _l(p^{n-1}_{l,D}) s^{n-1}_{l,D}}{\delta t}\nonumber \\&\qquad - \sum _{E\in \mathcal N( D)} \rho ^{n}_{l,{D,E}} \; M_{l}(s_{l,{D,E}}^{n}) \; \Lambda _{D,E}\; \delta _{D|E}^{n}(p_l)\nonumber \\&\qquad +\left| D\right| \rho _l(p_{l,D}^{n}) s_{l,D}^{n} f_{P,D}^{n} = \left| D\right| \rho _l(p_{l,D}^{n}) (s^I_{l,D})^{n}f_{I,D}^{n},\end{aligned}$$

(3.7)

$$\begin{aligned}&\quad \left| D\right| \phi _D\frac{\rho _g(p^{n}_{g,D}) s^{n}_{g,D} - \rho _g(p^{n-1}_{g,D}) s^{n-1}_{g,D}}{\delta t}\nonumber \\&\qquad -\sum _{E\in \mathcal N( D)} \rho ^{n}_{g,{D,E}} \; M_{g}(s_{g,{D,E}}^{n}) \; \Lambda _{D,E}\; \delta _{D|E}^{n}(p_g)\nonumber \\&\qquad + \left| D\right| \rho _g(p_{g,D}^{n}) s_{g,D}^{n} f_{P,D}^{n} = \left| D\right| \rho _g(p_{g,D}^{n}) (s^I_{g,D})^{n}f_{I,D}^{n},\end{aligned}$$

(3.8)

$$\begin{aligned}&\quad p_c(s^{n}_{l,D}) = p_{g,D}^{n}-p_{l,D}^{n}. \end{aligned}$$

(3.9)

We refer to the matrix $\mathbf{\Lambda }$ of elements $\mathbf{\Lambda }_{D,E}$, $D, E\in \mathcal {D}_h^{\text {int}}$, as the diffusion matrix. The stiffness matrix of the nonconforming finite element method, is defined as follow

$$\begin{aligned} \Lambda _{D,E}:= - \sum _{K\in \mathcal {T}_h} \left( \mathbf{\Lambda }(x) \nabla \varphi _E, \nabla \varphi _D\right) _{0,K} \quad D, E\in \mathcal {D}_h. \end{aligned}$$

(3.10)

Notice that the source terms are, for $n\in \{1,\ldots ,N\}$

$$\begin{aligned} f^{n}_{P,D}:= \frac{1}{\delta t\left| D\right| } \int \limits _{t^{n-1}}^{t^{n}} \int \limits _Df_P(t,x)\,dx dt,\quad f^{n}_{I,D}:= \frac{1}{\delta t\left| D\right| } \int \limits _{t^{n-1}}^{t^{n}} \int \limits _Df_I(t,x)\,dx dt. \end{aligned}$$

The mean value of the density of each phase on interfaces is not classical since it is given as

$$\begin{aligned} \frac{1}{\rho ^{n}_{\alpha ,{D,E}}}=\left\{ \begin{array}{ll} \frac{1}{p_{\alpha ,E}^{n}-p_{\alpha ,D}^{n}} \int \nolimits _{p_{\alpha ,D}^{n}}^{p_{\alpha ,E}^{n}} \frac{1}{\rho _\alpha (\zeta )}\,d\zeta &{} \text { if } p_{\alpha ,D}^{n} \ne p_{\alpha ,E}^{n},\\ \frac{1}{\rho ^{n}_{\alpha ,D}} &{} \text { otherwise}. \end{array}\right. \end{aligned}$$

(3.11)

This choice is crucial to obtain estimates on discrete pressures.

We denote

$$\begin{aligned} G_\alpha (s^{n}_{\alpha ,D},s^{n}_{\alpha ,E}; \delta _{D|E}^{n}(p_{\alpha })) = -M_{\alpha }(s_{\alpha ,{D,E}}^{n}) \delta _{D|E}^{n}(p_{\alpha }), \end{aligned}$$

(3.12)

the numerical fluxes, where $M_{\alpha }(s_{\alpha ,{D,E}}^{n})$ denote the upwind discretization of $M_\alpha (s_\alpha )$ on the interface $\sigma _{{D,E}}$ and

$$\begin{aligned} s_{\alpha ,{D,E}}^{n}=\left\{ \begin{array}{ll} s_{\alpha ,D}^{n}&{}\quad \text { if } (D,E)\in \mathcal {E}_\alpha ^{n},\\ s_{\alpha ,E}^{n} &{}\quad \text { otherwise, } \end{array}\right. \end{aligned}$$

(3.13)

with the set $\mathcal {E}_\alpha ^{n}$ is subset of $\mathcal {E}_h$ such that

$$\begin{aligned} \mathcal {E}_\alpha ^{n}=\{ (D,E)\in \mathcal {E}_h, \Lambda _{D,E}\; \delta _{D|E}^{n}(p_\alpha ) = \Lambda _{D,E}\left( p_{\alpha ,E}^{n}-p_{\alpha ,D}^{n}\right) \le 0\}. \end{aligned}$$

(3.14)

Remark 1

To take into account the gravity term, it is enough to modify, for example, the numerical fluxes to be

$$\begin{aligned} G_\alpha (s^{n}_{\alpha ,D},s^{n}_{\alpha ,E}; \delta _{D|E}^{n}(p_{\alpha }))&= -M_{\alpha }(s_{\alpha ,{D,E}}^{n}) \delta _{D|E}^{n}(p_{\alpha })\\&+\rho _\alpha (p^{n}_{\alpha ,{D,E}})\Big ( M_{\alpha }(s_{\alpha ,\mathcal {D}}^{n}) g_{D|E}^+ - M_{\alpha }(s_{\alpha ,E}^{n}) g_{D|E}^-\Big ), \end{aligned}$$

with $g_{D|E}^\pm =(g\cdot \eta _{D|E})^\pm$. This numerical flux is consistent, conservative and monotone, then the convergence results remains valid.

In the sequel we shall consider apart the following special case:

All transmissibility’s are non-negative, i.e.

$$\begin{aligned} \Lambda _{D,E}\ge 0 \quad \forall D\in \mathcal {D}_h^\text {int}, E\in \mathcal N(D). \end{aligned}$$

(3.15)

Since

$$\begin{aligned} \nabla \varphi _{D|K} = \frac{\left| \sigma _\mathcal {D}\right| }{\left| K\right| }\mathbf n _{\sigma _\mathcal {D}}, \quad K\in \mathcal {T}_h, \sigma _\mathcal {D}\in \mathcal {E}_K\end{aligned}$$

with $\mathbf n _{\sigma _\mathcal {D}}$ the unit normal vector of the side $\sigma _\mathcal {D}$ outward to $K$, one can immediately see that Assumption (3.15) is satisfied when the diffusion tensor reduces to a scalar function and when the magnitude of the angles between $\mathbf n _{\sigma _\mathcal {D}}$, $\sigma _\mathcal {D}\in \mathcal {E}_K$, for all $K\in \mathcal {T}_h$ is greater or equal to $\pi /2$.

The main result of this paper is the following theorem.

Theorem 1

There exists an approximate solutions $(p_{\alpha ,D}^{n})_{n,D}$ corresponding to the system (3.7)–(3.8), which converges (up to a subsequence) to a weak solution $p_\alpha$ of (2.1) in the sense of the Definition 1.

4 Preliminary fundamental lemmas

In this section, we will first present several technical lemmas that will be used in our latter analysis to obtain a priori estimate of the solution of the discrete problem.

Lemma 1

For all $u_h= \sum _{D\in \mathcal {D}_h} u_D\varphi _D\in X_h$,

$$\begin{aligned} \sum _{D\in \mathcal {D}_h} \sum _{E\in \mathcal N(D)} \Lambda _{D,E}(\delta _{D|E}{u})^2 \ge c_\mathbf{\Lambda }\left\| u_h\right\| _{X_h}^2. \end{aligned}$$

Proof

We have

$$\begin{aligned} \sum _{D\in \mathcal {D}_h} \sum _{E\in \mathcal N(D)} \Lambda _{D,E}(\delta _{D|E}{u})^2&=\sum _{D\in \mathcal {D}_h} \sum _{E\in \mathcal N(D)} \Lambda _{D,E}(u_E- u _D)^2 \\&= - 2\sum _{D\in \mathcal {D}_h} u_D\sum _{E\in \mathcal N(D)} \Lambda _{D,E}(u_E- u _D)\\&= -2\sum _{D\in \mathcal {D}_h} u_D\sum _{E\in \mathcal {D}_h} \Lambda _{D,E}u_E\\&= 2 \sum _{K\in \mathcal {T}_h} ( \mathbf{\Lambda }\nabla u_h, \nabla u_h)_{0,K} \ge c_\mathbf{\Lambda }\left\| u_h\right\| _{X_h}^2, \end{aligned}$$

using (3.10) and assumption (A2). $\square$

We state this lemma without a proof as well (cf. [17, Lemma 3.1]):

Lemma 2

For all $u_h= \sum _{D\in \mathcal {D}_h} u_D\varphi _D\in X_h$, one has

$$\begin{aligned}&\sum _{\sigma _{D,E}\in \mathcal {F}_h^\text {int}} \frac{\left| \sigma _{D,E}\right| }{d_{D,E}} \left( u_E- u_D\right) ^2 \le \frac{d+ 1}{2(d-1)\kappa _\mathcal {T}} \left\| u_h\right\| _{X_h}^2. \end{aligned}$$

(4.1)

In the continuous case, we have the following relationship between the global pressure, capillary pressure and the pressure of each phase

$$\begin{aligned} M_{l} |\nabla p_{l}|^2 + M_{g} |\nabla p_{g}|^2=M |\nabla p |^{2} + \frac{M_{l}M_{g}}{M}|\nabla p_c |^{2}. \end{aligned}$$

(4.2)

This relationship, means that, the control of the velocities ensures the control of the global pressure and the capillary terms $\mathcal {B}$ in the whole domain regardless of the presence or the disappearance of the phases.

In the discrete case, these relationship, are not obtained in a straightforward way. This equality is replaced by three discrete inequalities which we state in the following lemma.

Lemma 3

$\left( \mathrm {Total\, mobility, global\, pressure, Capillary\, term}~ \mathcal {B}~ \mathrm { and~Dissipative\, terms } \right)$ Under the assumptions (A1)–(A7) and the notations (2.6). Then for all $(D,E)\in \mathcal {D}$ and for all $n\in [0,N]$ the following inequalities hold:

$$\begin{aligned}&M_{l,D|E}^{n} + M_{g,D|E}^{n} \ge m_0,\end{aligned}$$

(4.3)

$$\begin{aligned}&\quad m_0\Big (\delta _{D|E}^{n}(p)\Big )^2 \le M_{l,D|E}^{n}\Big (\delta _{D|E}^{n}(p_l)\Big )^2 + M_{g,D|E}^{n}\Big (\delta _{D|E}^{n}(p_g)\Big )^2.\end{aligned}$$

(4.4)

$$\begin{aligned}&\quad (\delta _{D|E}^{n}(\mathcal {B}(s_l)))^2 \le M_{l,D|E}^{n}\Big (\delta _{D|E}^{n}(p_l)\Big )^2 + M_{g,D|E}^{n}\Big (\delta _{D|E}^{n}(p_g)\Big )^2,\end{aligned}$$

(4.5)

$$\begin{aligned}&\quad M_{l,D|E}^{n}(\delta _{D|E}^{n}(\bar{p}(s_l)))^2 \le M_{l,D|E}^{n}\Big (\delta _{D|E}^{n}(p_l)\Big )^2 + M_{g,D|E}^{n}\Big (\delta _{D|E}^{n}(p_g)\Big )^2,\qquad \quad \end{aligned}$$

(4.6)

and

$$\begin{aligned} M_{g,D|E}^{n}(\delta _{D|E}^{n}(\tilde{p}(s_l)))^2 \le M_{l,D|E}^{n}\Big (\delta _{D|E}^{n}(p_l)\Big )^2 + M_{g,D|E}^{n}\Big (\delta _{D|E}^{n}(p_g)\Big )^2. \end{aligned}$$

(4.7)

In [26], the authors prove this lemma on primal mesh satisfying the orthogonal condition. This proof use only two neighbors elements and it is based only on the definition of the global pressure. Thus, we state the above Lemma without proof since the proof made in [26] remains valid on the dual mesh.

5 A priori estimates and existence of the approximate solution

We derive new energy estimates on the discrete velocities $M_\alpha (s_{\alpha ,D|E}^{n}) \delta _{D|E}^{n}(p_\alpha )$. Nevertheless, these estimates are degenerate in the sense that they do not permit the control of $\delta _{D|E}^{n}(p_\alpha )$, especially when a phase is missing.

In this section, firstly we show that the saturation of each phase is admissible. Next, we establish the major estimates on the discrete velocities and their consequences. Then we establish that the finite volume scheme is well defined.

5.1 The maximum principle

Let us show in the following Lemma that the phase by phase upstream choice yields the $L^\infty$ stability of the scheme which is a basis to the analysis that we are going to perform.

Lemma 4

$\left( \mathrm {Maximum principle} \right)$ Under assumptions (A1)–(A7). Let $(s_{\alpha ,D}^{0})_{D\in \mathcal {D}_h}\in [0,1]$ and assume that $(p_{\alpha ,D}^{n})_{D\in \mathcal {D}_h}$ is a solution of the finite volume (3.6)–(3.9). Then, the saturation $(s_{\alpha ,D}^{n})_{D\in \mathcal {D}_h,}$ remains in $[0,1]$ for all $D\in \mathcal {D}_h, \; n \in \{1,\ldots ,N\}$.

Proof

Let us show by induction in $n$ that for all $D\in \mathcal {D}_h, ~ s^n_{\alpha ,D}\ge 0$ where $\alpha =l,g$. For $\alpha =l$, the claim is true for $n=0$ and for all $D\in \mathcal {D}_h$. We argue by induction that for all $D\in \mathcal {D}_h$, the claim is true up to order $n$. We consider the control volume $D$ such that $s^{n}_{l,D}=\min {\{s^{n}_{l,E}\}}_{E\in \mathcal {D}_h}$ and we seek that $s^{n}_{l,D}\ge 0$.

For the above mentioned purpose, multiply the equation in (3.7) by $-(s_{l,D}^{n})^-$, we obtain

$$\begin{aligned}&- \left| D\right| \phi _D\frac{\rho _l(p^{n}_{l,D}) s^{n}_{l,D}-\rho _l(p^{n-1}_{l,D}) s^{n-1}_{l,D}}{\delta t} (s_{l,D}^{n})^-\nonumber \\&\quad - \sum _{E\in \mathcal N(D) } \tau _{D|E}\rho ^{n}_{l,{D,E}} \; \Lambda _{D,E}\; G_l(s^{n}_{l,D},s^{n}_{l,E};\delta _{D|E}^{n}(p_{l})) (s_{l,D}^{n})^-\nonumber \\&\quad - \left| D\right| \rho _l(p_{l,D}^{n}) s_{l,D}^{n} f_{P,D}^{n}(s_{l,D}^{n})^- = -\left| D\right| \rho _l(p_{l,D}^{n}) (s^I_{g,D})^{n}f_{I,D}^{n}(s_{l,D}^{n})^-\le 0.\qquad \quad \end{aligned}$$

(5.1)

The numerical flux $G_l$ is nonincreasing with respect to $s_{l,E}^{n}$, and consistent, hence we get

$$\begin{aligned} G_l(s^{n}_{l,D},s^{n}_{l,E};\delta _{D|E}^{n}(p_{l}))\,(s_{l,D}^{n})^-&\le G_l(s^{n}_{l,D},s^{n}_{l,D} ;\delta _{D|E}^{n}(p_{l}))\,(s_{l,D}^{n})^-\nonumber \\&=-\delta _{D|E}^{n}(p_{l})\,M_l(s^{n}_{l,D}) \,(s_{l,D}^{n})^- = 0. \end{aligned}$$

(5.2)

Using the identity $s_{l,D}^{n}=({s_{l,D}^{n}})^+-(s_{l,D}^{n})^-$, and the mobility $M_l$ extended by zero on $]-\infty , 0]$, then $M_l(s^{n}_{l,D}) (s_{l,D}^{n})^- = 0$ and

$$\begin{aligned} -\left| D\right| \rho _l(p_{l,D}^{n}) s_{l,D}^{n} f_{P,K}^{n}(s_{l,D}^{n})^- = \left| D\right| \rho _l(p_{l,D}^{n}) f_{P,K}^{n}((s_{l,D}^{n})^-)^2 \ge 0. \end{aligned}$$

(5.3)

Then, we deduce from (5.1) that

$$\begin{aligned} \rho _l(p^{n}_{l,D})|(s^{n}_{l,D})^{-}|^2 +\rho _l(p^{n-1}_{l,D})s^{n-1}_{l,D}(s_{l,D}^{n})^-\le 0, \end{aligned}$$

and from the nonnegativity of $s^{n-1}_{l,D}$, we obtain $(s_{l,D}^{n})^-=0$. This implies that $s_{l,D}^{n}\ge 0$ and

$$\begin{aligned} 0\le s^{n}_{l,D} \le s^{n}_{l,E} \quad \text { for all } n \in [0,N-1] \text { and } E\in \mathcal {D}_h. \end{aligned}$$

In the same way, we prove $s_{g,D}^{n}\ge 0$. $\square$

5.2 Estimations on the pressures

We now give a priori estimates satisfied by the solution values $p_D^n$, $D\in \mathcal {D}_h$, $\{1,\ldots ,N\}$.

Proposition 1

Let $(p_{l,D}^n,p_{g,D}^n)$ be a solution of (3.6)–(3.9). Then, there exists a constant $C>0$, which only depends on $M_\alpha$, $\Omega$, $T$, $p^{0}_\alpha$, $s^0_\alpha$, $s^I_\alpha$, $f_P$, $f_I$ and not on $\mathcal {D}_h$, such that the solution of the combined scheme satisfies

$$\begin{aligned} \sum _{n=1}^{N}\delta t\sum _{D\in \mathcal {D}_h} \sum _{E\in \mathcal N(D)}\Lambda _{D,E}M_\alpha (s_{\alpha ,D|E}^{n}) \left| p_{\alpha ,E}^{n}-p_{\alpha ,D}^{n}\right| ^2\le C, \end{aligned}$$

(5.4)

and

$$\begin{aligned} \sum _{n=1}^{N}\delta t\left\| p_h\right\| _{X_h}^2\le C. \end{aligned}$$

(5.5)

Proof

We define the function $\mathcal {A}_{\alpha }(p_{\alpha }) := \rho _{\alpha }(p_{\alpha })g_{\alpha }(p_{\alpha }) - p_{\alpha },$ $\mathcal {P}_c(s_l) := \int _{0}^{s_l}p_c(z)\mathrm {d}z$ and $g_{\alpha }(p_{\alpha }) = \int _{0}^{p_{\alpha }}\frac{1}{\rho _{\alpha }(z)}\mathrm {d}z$. In the following proof, we denote by $C_i$ various real values which independent on $\mathcal {D}$ and $n$. To prove the estimate (5.4), we multiply(3.7) and (3.8) respectively by $g_l(p_{l,D}^{n})$, $g_g(p_{g,D}^{n})$ and adding them, then summing the resulting equation over $D\in \mathcal {D}_h$ and $n \in \{1,\ldots , N\}$. We thus get:

$$\begin{aligned} E_{1}+E_{2}+E_{3}= 0, \end{aligned}$$

(5.6)

where

$$\begin{aligned} E_{1}&= \sum _{n=1}^{N}\sum _{D\in \mathcal {D}_h} \left| D\right| \phi _D\Big ( (\rho _l(p^{n}_{l,D})s^{n}_{l,D} -\rho _l(p^{n-1}_{l,D})s^{n-1}_{l,D})\; g_l(p^{n}_{l,D})\nonumber \\&+ (\rho _g(p^{n}_{g,D})s^{n}_{g,D} -\rho _g(p^{n-1}_{g,D})s^{n-1}_{g,D})\; g_g(p^{n}_{g,D}) \Big ),\nonumber \\ E_{2}&= \sum _{n=1}^{N}\delta t\sum _{D\in \mathcal {D}_h} \sum _{E\in \mathcal N(\mathcal {D})} \Lambda _{D,E}\Big (\rho ^{n}_{l,{D,E}} G_l(s^{n}_{l,D},s^{n}_{l,E};\delta _{D|E}^{n}(p_{l}))\; g_l(p_{l,D}^{n})\nonumber \\&\quad + G_g(s^{n}_{g,D},s^{n}_{g,E};\delta _{D|E}^{n}(p_{g}))\; g_g(p_{g,D}^{n})\Big ),\nonumber \\ E_3&= \sum _{n=1}^{N}\delta t\sum _{D\in \mathcal {D}_h}\left| D\right| \Big (\rho _l(p_{l,D}^{n}) s_{l,D}^{n} f_{P,D}^{n} g_l(p_{l,D}^{n}) - \rho _l(p_{l,D}^{n}) (s^I_{g,D})^{n}f_{I,D}^{n} g_l(p_{l,D}^{n}) \nonumber \\&+ \rho _g(p_{g,D}^{n}) s_{g,D}^{n} f_{P,D}^{n} g_g(p_{g,D}^{n}) - \rho _g(p_{g,D}^{n}) (s^I_{g,D})^{n}f_{I,D}^{n} g_g(p_{g,D}^{n}) \Big ). \end{aligned}$$

To estimate the first term on the left hand side of the equality (5.6), let us recall the following inequality:

$$\begin{aligned}&\bigl (\rho _l(p_l^n)s_l^n-\rho _l(p_{l}^{n-1}) s^{n-1}_{l}\bigr )g_l(p_l^n)+ \bigl (\rho _g(p_g^n)s_g^n-\rho _g(p_g^{n-1}) s^{n-1}_g\bigr )g_g(p_g^n)\nonumber \\&\quad \ge \mathcal {A}_l(p_l^{n})s_l^{n}-\mathcal {A}_l(p_l^{n-1}) s^{n-1}_l+\mathcal {A}_g(p_g^{n})s_g^{n}-\mathcal {A}_g(p_g^{n-1})s^{n-1}_g- \mathcal {P}_c(s_l^{n})\nonumber \\&\qquad +\mathcal {P}_c(s_l^{n-1}), \end{aligned}$$

(5.7)

using the concavity property of $g_\alpha$ and $\mathcal {P}_c$, in [21] the authors prove the above inequality.

So, this yields to

$$\begin{aligned}&E_1 \ge \sum _{D\in \mathcal {D}_h} \phi _D\left| D\right| \Big (s_{l,D}^N \mathcal {A}(p_{l,D}^N) - s_{l,D}^0 \mathcal {A}(p_{l,D}^0) + s_{g,D}^N \mathcal {A}(p_{g,D}^N) - s_{g,D}^0 \mathcal {A}(p_{g,D}^0) \Big )\nonumber \\&\qquad \quad - \sum _{D\in \mathcal {D}_h}\phi _D\left| D\right| \mathcal {P}_c(s_{l,D}^N) + \sum _{D\in \mathcal {D}_h} \phi _D\left| D\right| \mathcal {P}_c(s_{l,D}^0). \end{aligned}$$

(5.8)

Using the fact that the numerical fluxes $G_l$ and $G_g$ are conservative, we obtain by discrete integration by parts

$$\begin{aligned} E_2&= \frac{1}{2}\sum _{n=1}^{N}\delta t\!\sum _{D\in \mathcal {D}_h} \sum _{E\in \mathcal N(D)}\!\!\!\!\! \Lambda _{D,E}\Big ( \rho _{l,{D,E}}^{n} G_l(s^{n}_{l,D},s^{n}_{l,E};\delta _{D|E}^{n}(p_{l})) (g_l(p_{l,D}^{n})\!-\!g_l(p_{l,E}^{n}))\nonumber \\&\quad + \rho _{g,{D,E}}^{n} G_g(s^{n}_{g,D},s^{n}_{g,E};\delta _{D|E}^{n}(p_{g})) (g_g(p_{g,D}^{n})-g_g(p_{g,E}^{n})) \Big ), \end{aligned}$$

and due to the correct choice of the density of the phase $\alpha$ on each interface,

$$\begin{aligned} \rho _{\alpha ,{D,E}}^{n} (g_{\alpha }(p_{\alpha ,D}^{n})-g_{\alpha }(p_{\alpha ,E}^{n}))= p_{\alpha ,D}^{n}-p_{\alpha ,E}^{n}, \end{aligned}$$

(5.9)

we obtain

$$\begin{aligned} E_2&= \frac{1}{2}\sum _{n=1}^{N}\delta t\sum _{D\in \mathcal {D}_h} \sum _{E\in \mathcal N(D)} \Lambda _{D,E}\Big ( G_l(s^{n}_{l,D},s^{n}_{l,E};\delta _{D|E}^{n}(p_{l})) (p_{l,D}^{n}-p_{l,E}^{n})\nonumber \\&\quad + G_g(s^{n}_{g,D},s^{n}_{g,E};\delta _{D|E}^{n}(p_{g})) (p_{g,D}^{n}-p_{g,E}^{n}) \Big ). \end{aligned}$$

The definition of the upwind fluxes in (3.12) implies

$$\begin{aligned}&G_l(s^{n}_{l,D},s^{n}_{l,E};\delta _{D|E}^{n}(p_{l})) (p_{l,D}^{n}-p_{l,E}^{n}) + G_g(s^{n}_{g,D},s^{n}_{g,E};\delta _{D|E}^{n}(p_{g})) (p_{g,D}^{n}-p_{g,E}^{n})\\&\quad = M_{l}(s_{l,{D,E}}^{n})(\delta _{D|E}^{n}(p_{l}))^2 + M_{g}(s_{g,{D,E}}^{n})(\delta _{D|E}^{n}(p_{g}))^2. \end{aligned}$$

Then, we obtain the following equality

$$\begin{aligned} E_2&= \frac{1}{2}\sum _{n=1}^{N}\delta t\sum _{D\in \mathcal {D}_h} \sum _{E\in N(\mathcal {D})}\nonumber \\&\quad \times \Lambda _{D,E}\Big ( M_{l}(s_{l,{D,E}}^{n})(\delta _{D|E}^{n}(p_l))^2 + M_{g}(s_{g,{D,E}}^{n})(\delta _{D|E}^{n}(p_g))^2 \Big ). \end{aligned}$$

(5.10)

In order to estimate $E_3$, using the fact that the densities are bounded and the map $g_\alpha$ is sublinear $(\text {a.e.}|g(p_\alpha )|\le C |p_\alpha |)$, we have

$$\begin{aligned} \left| E_3\right| \le C_1\sum _{n=1}^{N}\delta t\sum _{D\in \mathcal {D}_h} \left| D\right| (f_{P,D}^{n}+f_{I,D}^{n})(|p_{l,D}^{n}|+|p_{g,D}^{n}|), \end{aligned}$$

then

$$\begin{aligned} \left| E_3\right| \le C_1\sum _{n=1}^{N}\delta t\sum _{D\in \mathcal {D}_h} \left| D\right| (f_{P,D}^{n}+f_{I,D}^{n}) (2|p_{D}^{n}|+|\bar{p}_{D}^{n}|+|\tilde{p}_{D}^{n}|). \end{aligned}$$

Hence, by the Hölder inequality, we get that

$$\begin{aligned} \left| E_3\right| \le C_2\left\| f_P+f_I\right\| _{L^2(Q_T)} \left( \sum _{n=1}^{N}\delta t\left\| p^{n}_h\right\| _{L^2({\Omega })}^2 \right) ^{\frac{1}{2}} + C_{3} \end{aligned}$$

and, from the discrete Poincaré–Friedrichs inequality [27], we get

$$\begin{aligned} \left| E_{3}\right| \le C_4 \left( \sum _{n=1}^{N}\delta t\left\| p^{n}_h\right\| _{X_h}^2\right) ^{\frac{1}{2}} + C_5. \end{aligned}$$

(5.11)

The equality (5.6) with the inequalities (5.8), (5.10), (5.11) give (5.4). Then we deduce (5.5) from (4.4) and Lemma 1. $\square$

We now state the following corollary, which is essential for the compactness and limit study.

Corollary 1

From the previous Proposition, we deduce the following estimates:

$$\begin{aligned}&\sum _{n=1}^{N}\delta t\left\| \mathcal {B}(s_{l,h}^n)\right\| _{X_h}^2 \le C,\end{aligned}$$

(5.12)

$$\begin{aligned}&\sum _{n=1}^{N}\delta t\sum _{D\in \mathcal {D}_h} \sum _{E\in \mathcal N(D)}\Lambda _{D,E}M_{l,D|E}^{n}(\delta _{D|E}^{n}(\bar{p}(s_l)))^2 \le C, \end{aligned}$$

(5.13)

and

$$\begin{aligned} \sum _{n=1}^{N}\delta t\sum _{D\in \mathcal {D}_h} \sum _{E\in \mathcal N(D)}\Lambda _{D,E}M_{g,D|E}^{n}(\delta _{D|E}^{n}(\tilde{p}(s_l)))^2 \le C. \end{aligned}$$

(5.14)

Proof

The prove of the estimates (5.12), (5.13) and (5.14) are a direct consequence of the inequality (4.5), (4.6), (4.7), the Lemma 1 and the Proposition 1. $\square$

5.3 Existence of the finite volume scheme

Proposition 2

The problem (3.7)–(3.8) admits at least one solution $(p^{n}_{l,D}, p^{n}_{g,D})_{(D,n) \in \mathcal {D}_h\times \{1,\ldots ,N\}}$.

In [3] and [26] the authors characterize the solution of the nonlinear scheme as the zeros of a vector field. Next, the authors use a fixed point theorem as in [13] to conclude with the existence of a solution. So, it is easy to adopt their proof in our case, thus we omit it.

6 Compactness properties

In this section we derive estimates on differences of space and time translates of the function $\mathbf U _{\alpha ,{\delta t,h}} \,{=}\, \phi \rho _\alpha (p_{\alpha ,{\delta t,h}}) s_{\alpha ,{\delta t,h}}$ which imply that the sequence $\phi \rho _\alpha (p_{\alpha ,{\delta t,h}})s_{\alpha ,{\delta t,h}}$ is relatively compact in $L^1(Q_T)$.

The following important relation between $\mathbf U _{\delta t,h}$ and ${\tilde{\mathbf{U }}}_{\delta t,h}$ (see Definition 2) is valid:

Lemma 5

(Relation between  $\mathbf U _{\alpha , \delta t,h}$  and  ${\tilde{\mathbf{U }}}_{\alpha , \delta t,h}$) There holds

$$\begin{aligned} \left\| \mathbf U _{\alpha ,\delta t,h} - {\tilde{\mathbf{U }}}_{\alpha ,\delta t,h}\right\| _{L^1(Q_T)} \longrightarrow 0 \quad \text { as } h\rightarrow 0. \end{aligned}$$

Proof

$$\begin{aligned}&\left\| \mathbf U _{l,\delta t,h} - {\tilde{\mathbf{U }}}_{l,\delta t,h}\right\| _{L^1(Q_T)} = \int \limits _{Q_T}\left| U_{l,{\delta t,h}}(t,x) - {\tilde{U}}_{l,{\delta t,h}}(t,x)\right| \mathrm {d}x \; \mathrm {d}t\nonumber \\&\quad \le \int \limits _{Q_T} \left| s_{l,{\delta t,h}}(t,x) \Big ( \rho _l(p_{l,{{\delta t,h}}}(t,x)) - \rho _l({\tilde{p}}_{l,{{\delta t,h}}}(t,x)) \Big ) \right| \mathrm {d}x \; \mathrm {d}t \nonumber \\&\qquad + \int \limits _{Q_T} \left| \rho _l({\tilde{p}}_{l,{{\delta t,h}}}(t,x)) \Big ( s_{l,{\delta t,h}}(t,x) - {\tilde{s}}_{l,{\delta t,h}}(t,x) \Big ) \right| \mathrm {d}x\; \mathrm {d}t \nonumber \\&\quad \le \mathtt T_1+ \mathtt T_2, \end{aligned}$$

where $\mathtt T_1$ and $\mathtt T_2$ defined as follows

$$\begin{aligned} \mathtt T _1&= \rho _M \int \limits _{Q_T} \left| s_{l,{\delta t,h}}(t,x) - {\tilde{s}}_{l,{\delta t,h}}(t,x) \right| \mathrm {d}x\; \mathrm {d}t,\end{aligned}$$

(6.1)

$$\begin{aligned} \mathtt T_2&= \int \limits _{Q_T} \left| \rho _l(p_{l,{{\delta t,h}}}(t,x)) - \rho _l({\tilde{p}}_{l,{{\delta t,h}}}(t,x)) \right| \mathrm {d}x \; \mathrm {d}t. \end{aligned}$$

(6.2)

To handle the term on saturation $\mathtt T_1$, we use the fact that $\mathcal {B}^{-1}$ is a Hölder function, then

$$\begin{aligned} \mathtt T_1 \le \rho _M C \int \limits _{Q_T} \left| \mathcal {B}(s_{l,{\delta t,h}}(t,x)) - \mathcal {B}({\tilde{s}}_{l,{\delta t,h}}(t,x)) \right| ^\theta \mathrm {d}x\; \mathrm {d}t, \end{aligned}$$

and by application of the Cauchy-Schwarz inequality, we deduce

$$\begin{aligned} \mathtt T_1 \le C \left( \,\, \int \limits _{Q_T} \left| \mathcal {B}(s_{l,{\delta t,h}}(t,x)) - \mathcal {B}({\tilde{s}}_{l,{\delta t,h}}(t,x)) \right| \mathrm {d}x\; \mathrm {d}t \right) ^\theta \le C \; (\mathtt T_1^\prime )^\theta , \end{aligned}$$

where $\mathtt T_1^\prime$ defined as follows

$$\begin{aligned} T_1^\prime = \int \limits _{Q_T} \left| \mathcal {B}(s_{l,{\delta t,h}}(t,x)) - \mathcal {B}({\tilde{s}}_{l,{\delta t,h}}(t,x)) \right| \mathrm {d}x\; \mathrm {d}t. \end{aligned}$$

We have

$$\begin{aligned} T_1^\prime&= \sum _{n=1}^{N}\delta t\sum _{K \in \mathcal {T}_h}\sum _{\sigma _D\in \mathcal {E}_K} \int \limits _{K\cap D} \left| \mathcal {B}(s_{l,{\delta t,h}}(t,x)) - \mathcal {B}({\tilde{s}}_{l,{\delta t,h}}(t,x))\right| \mathrm {d}x\nonumber \\&= \sum _{n=1}^{N}\delta t\sum _{K \in \mathcal {T}_h}\sum _{\sigma _D\in \mathcal {E}_K} \int \limits _{K\cap D} \left| \mathcal {B}(s_{l,{\delta t,h}}(t,x)) - \mathcal {B}(s_{l,{\delta t,h}}(t,Q_D))\right| \mathrm {d}x\nonumber \\&= \sum _{n=1}^{N}\delta t\sum _{K \in \mathcal {T}_h}\sum _{\sigma _D\in \mathcal {E}_K} \int \limits _{K\cap D} \left| \nabla \mathcal {B}(s_{l,{\delta t,h}}(t,x)) \cdot (x - Q_D)\right| \mathrm {d}x\nonumber \\&\le \sum _{n=1}^{N}\delta t\sum _{K \in \mathcal {T}_h}\sum _{\sigma _D\in \mathcal {E}_K} \Big | \nabla \mathcal {B}(s_{l,{\delta t,h}})|_K\Big |\text {diam}(D) \left| K\cap D\right| \nonumber \\&\le h\sum _{n=1}^{N}\delta t\sum _{K \in \mathcal {T}_h}\Big | \nabla \mathcal {B}(s_{l,{\delta t,h}})|_K\Big | \left| K\right| \nonumber \\&\le h\left( \sum _{n=1}^{N}\delta t\left\| \mathcal {B}(s_{l,{\delta t,h}})\right\| ^2_{X_h} + C \right) \le C h, \end{aligned}$$

(6.3)

where we have used the definitions of $s_{l,{\delta t,h}}$ and ${\tilde{s}}_{l,{\delta t,h}}$, the Cauchy-Schwarz inequality and the estimate (5.12), thus

$$\begin{aligned} \mathtt T_1 \le C h^\theta . \end{aligned}$$

(6.4)

To treat $\mathtt T_2$, we use the fact that the map $\rho _l^\prime$ is bounded and the relationship between the gas pressure and the global pressure, namely : $p_l=p-\bar{p}$ defined in (2.6), then we have

$$\begin{aligned} \mathtt T_2&\le \max _\mathbb {R}|\rho _l^\prime |\int \limits _{Q_T} \left| p_{l,{{\delta t,h}}}(t,x) - {\tilde{p}}_{l,{{\delta t,h}}}(t,x)\right| \mathrm {d}x\mathrm {d}t\nonumber \\&\le \max _\mathbb {R}|\rho _l^\prime | \int \limits _{Q_T} \left| p_{{\delta t,h}}(t,x) - {\tilde{p}}_{{\delta t,h}}(t,x)\right| \mathrm {d}x\mathrm {d}t\nonumber \\&\quad + \max _\mathbb {R}|\rho _l^\prime | \int \limits _{Q_T}\left| \bar{p}(s_{l,{\delta t,h}}(t,x)) -\bar{p}( {\tilde{s}}_{l,{\delta t,h}}(t,x)) \right| \mathrm {d}x\mathrm {d}t, \end{aligned}$$

(6.5)

furthermore one can easily show that $\bar{p}$ is a $C^1([0,1];\mathbb {R})$, it follows, there exists a positive constant $C>0$ such that

$$\begin{aligned} \mathtt T_2 \le C \int \limits _{Q_T} |p_{{\delta t,h}}(t,x) - {\tilde{p}}_{{\delta t,h}}(t,x) | \mathrm {d}x \mathrm {d}t + C \int \limits _{Q_T} |s_{l,{\delta t,h}}(t,x) - {\tilde{s}}_{l,{\delta t,h}}(t,x)| \mathrm {d}x \mathrm {d}t. \end{aligned}$$

The last term in the previous inequality is proportional to $\mathtt T_1$, and consequently it remains to show that the term on the global pressure is small with $h$. In fact, follows (6.3), one gets

$$\begin{aligned}&\int \limits _{Q_T} |p_{{\delta t,h}}(t,x) - {\tilde{p}}_{{\delta t,h}}(t,x) | \mathrm {d}x \mathrm {d}t\\&\quad =\sum _{n=1}^{N}\delta t\sum _{K \in \mathcal {T}_h}\sum _{\sigma _D\in \mathcal {E}_K}\int _{K\cap D} \left| p_{{\delta t,h}}(t,x) - {\tilde{p}}_{{\delta t,h}}(t,x) \right| \mathrm {d}x \\&\quad = \sum _{n=1}^{N}\delta t\sum _{K \in \mathcal {T}_h}\sum _{\sigma _D\in \mathcal {E}_K}\int \limits _{K\cap D} \left| p_{{\delta t,h}}(t,x) - p_{{\delta t,h}}(t,Q_D) \right| \mathrm {d}x \\&\quad = \sum _{n=1}^{N}\delta t\sum _{K \in \mathcal {T}_h}\sum _{\sigma _D\in \mathcal {E}_K}\int \limits _{K\cap D} \left| \nabla p_{{\delta t,h}}(t,x) \cdot (x-Q_D) \right| \mathrm {d}x \\&\quad \le h\left( \sum _{n=1}^{N}\delta t\left\| p_{{\delta t,h}}\right\| ^2_{X_h} + C \right) \le C h. \end{aligned}$$

Finally, we consider the case where $\alpha = g$ in the same manner. $\square$

We now give the space translate estimate for ${\tilde{\mathbf{U }}}_{\alpha ,{\delta t,h}}$ given by (3.5).

Lemma 6

$\mathrm {(Space~translate~of }$ ${\tilde{\mathbf{U }}}_{\alpha ,{\delta t,h}})$ Under the assumptions (A1)–(A7). Let $p_{\alpha ,{\delta t,h}}$ be a solution of (3.6)–(3.9). Then, the following inequality hold:

$$\begin{aligned} \int \limits _{\Omega ^{'}\times (0,T)}\left| {\tilde{\mathbf{U }}}_{\alpha ,{\delta t,h}}(t,x+y) - {\tilde{\mathbf{U }}}_{\alpha ,{\delta t,h}}(t,x)\right| \mathrm {d}x \mathrm {d}t\le \omega (\left| y\right| ), \end{aligned}$$

(6.6)

for all $y \in \mathbb {R}^d$ with $\Omega '=\{x \in \Omega , \, [x,x+y]\subset \Omega \}$ and $\omega (\left| y\right| ) \rightarrow 0$ when $\left| y\right| \rightarrow 0$.

Proof

For $\alpha = l$ and from the definition of $\mathbf U _{l,{\delta t,h}}$, one gets

$$\begin{aligned}&\int \limits _{(0,T)\times \Omega ^{'}}\left| {\tilde{\mathbf{U }}}_{l,{\delta t,h}}(t,x+y) - {\tilde{\mathbf{U }}}_{l,{\delta t,h}}(t,x)\right| \mathrm {d}x\mathrm {d}t\\&= \int \limits _{(0,T)\times \Omega ^{'}} \left| \left( \rho _l({\tilde{p}}_{l,{{\delta t,h}}}) {\tilde{s}}_{l,{\delta t,h}}\right) (t,x+y) - \left( \rho _l({\tilde{p}}_{l,{{\delta t,h}}}) {\tilde{s}}_{l,{\delta t,h}}\right) (t,x)\right| \mathrm {d}x\mathrm {d}t\\&\le E_1+E_2, \end{aligned}$$

where $E_1$ and $E_2$ defined as follows

$$\begin{aligned} E_1&= \rho _M \int \limits _{(0,T)\times \Omega ^{'}} \left| {\tilde{s}}_{l,{\delta t,h}}(t,x+y) - {\tilde{s}}_{l,{\delta t,h}}(t,x) \right| \mathrm {d}x\mathrm {d}t,\end{aligned}$$

(6.7)

$$\begin{aligned} E_2&= \int \limits _{(0,T)\times \Omega ^{'}} \left| \rho _l({\tilde{p}}_{l,{{\delta t,h}}}(t,x+y)) -\rho _l({\tilde{p}}_{l,{{\delta t,h}}}(t,x)) \right| \mathrm {d}x\mathrm {d}t. \end{aligned}$$

(6.8)

To handle with the space translation on saturation, we use again the fact that $\mathcal {B}^{-1}$ is a Hölder function, then

$$\begin{aligned} E_1 \le \rho _M C \int \limits _{(0,T)\times \Omega ^{'}} \left| \mathcal {B}({\tilde{s}}_{l,{\delta t,h}}(t,x+y) )- \mathcal {B}( {\tilde{s}}_{l,{\delta t,h}}(t,x))\right| ^\theta \mathrm {d}x\mathrm {d}t \end{aligned}$$

and by application of the Cauchy-Schwarz inequality, we deduce

$$\begin{aligned} E_1\le C \left( \,\,\int \limits _{(0,T)\times \Omega ^{'}} \left| \mathcal {B}({\tilde{s}}_{l,{\delta t,h}}(t,x+y) )- \mathcal {B}({\tilde{s}}_{l,{\delta t,h}}(t,x))\right| \mathrm {d}x\mathrm {d}t\right) ^\theta . \end{aligned}$$

According to [14]), let $y \in \mathbb {R}^d$, $x \in \Omega '$, and $L \in N(K)$. We define a function $\beta _\sigma (x)$ for each $\sigma \in \mathcal {F}_h^\text {int}$ by

$$\begin{aligned} \beta _{\sigma }=\left\{ \begin{array}{ll} 1, &{}\quad \text {if the line segment }\,\, [x,x+y]\,\, \text { intersects }\, \sigma ,\\ 0, &{}\quad \text {otherwise}. \end{array}\right. \end{aligned}$$

We observe that (see [14] for more details) $\int _{\Omega '}\beta _{\sigma _{{D,E}}}(x) \,dx \le |\sigma _{{D,E}}| |y|.$ Now, denote that

$$\begin{aligned} E_1&\le C \left( \sum _{n=1}^{N}\delta t\sum _{\sigma _{{D,E}} \in \mathcal {F}_h^\text {int}} \Big |\mathcal {B}(s_{l,E}) - \mathcal {B}(s_{l,D})\Big | \int \limits _{\Omega '}\beta _{\sigma _{{D,E}}}(x)\mathrm {d}x\right) ^\theta \\&\le C \left( \left| y\right| \sum _{n=1}^{N}\delta t\sum _{\sigma _{{D,E}} \in \mathcal {F}_h^\text {int}} \left| \sigma _{{D,E}}\right| \Big |\mathcal {B}(s_{l,E}) - \mathcal {B}(s_{l,D})\Big |\right) ^\theta . \end{aligned}$$

Let us write $\left| \sigma _{{D,E}}\right| = (d_{{D,E}}|\sigma _{{D,E}}|)^{\frac{1}{2}} \left( \frac{|\sigma _{{D,E}}|}{d_{{D,E}}}\right) ^{\frac{1}{2}}$. Obviously $d_{D,E}\le \frac{\text {diam}(K_{D,E})}{d}$, and $\left| \sigma _{D,E}\right| \le \frac{\text {diam}(K_{D,E})^{d-1} }{d-1}$, thus by the regularity shape assumption (3.1), we have

$$\begin{aligned} \exists \; C_{te} > 0 , \quad \forall h, \; \forall D\in \mathcal {D}_h\; \forall E\in \mathcal N(D) \qquad \left| \sigma _{D,E}\right| d_{D,E}\le C_{te} \left| K\right| . \end{aligned}$$

(6.9)

Applying again the Cauchy-Schwarz inequality, using (6.9), (4.1) and the fact that the discrete gradient of the function $\mathcal {B}$ is bounded (5.12) to obtain

$$\begin{aligned} E_1 \le C \left| y\right| ^\theta . \end{aligned}$$

(6.10)

To treat the space translate of $E_2$, we use the fact that the map $\rho _l^\prime$ is bounded and the relationship between the gas pressure and the global pressure, namely : $p_l=p-\bar{p}$ defined in (2.6), then we have

$$\begin{aligned} E_2&\le \max _\mathbb {R}|\rho _l^\prime |\int \limits _{(0,T)\times \Omega ^{'}} \left| {\tilde{p}}_{l,{{\delta t,h}}}(t,x+y) - {\tilde{p}}_{l,{{\delta t,h}}}(t,x)\right| \mathrm {d}x\mathrm {d}t\\&\le \max _\mathbb {R}|\rho _l^\prime | \int \limits _{(0,T)\times \Omega ^{'}} \left| {\tilde{p}}_{{\delta t,h}}(t,x+y) - {\tilde{p}}_{{\delta t,h}}(t,x)\right| \mathrm {d}x\mathrm {d}t \\&\quad + \max _\mathbb {R}|\rho _l^\prime | \int \limits _{(0,T)\times \Omega ^{'}}\left| \bar{p}({\tilde{s}}_{l,{\delta t,h}}(t,x+y)) -\bar{p}( {\tilde{s}}_{l,{\delta t,h}}(t,x)) \right| \mathrm {d}x\mathrm {d}t. \end{aligned}$$

Furthermore one can easily show that $\bar{p}$ is a $C^1([0,1];\mathbb {R})$, it follows, there exists a positive constant $C>0$ such that

$$\begin{aligned} \mathbf{E}_2&\le C \int \limits _{(0,T)\times \Omega ^{'}}|{\tilde{p}}_{{\delta t,h}}(t,x+y) - {\tilde{p}}_{{\delta t,h}}(t,x) | \mathrm {d}x \mathrm {d}t\\&\quad + C \int \limits _{(0,T)\times \Omega ^{'}} |{\tilde{s}}_{l,{\delta t,h}}(t,x+y) - {\tilde{s}}_{l,{\delta t,h}}(t,x)| \mathrm {d}x \mathrm {d}t. \end{aligned}$$

The last term in the previous inequality is proportional to $E_1$, and consequently it remains to show that the space translate on the global pressure is small with $y$. In fact

$$\begin{aligned} \int \limits _{(0,T)\times \Omega ^{'}} |{\tilde{p}}_{{\delta t,h}}(t,x+y) - {\tilde{p}}_{{\delta t,h}}(t,x) | \mathrm {d}x \mathrm {d}t&\le \sum _{n=1}^{N}\delta t\sum _{\sigma _{{D,E}}} |p_{E}^{n} - p_{D}^{n} |\int \limits _{\Omega '}\beta _{\sigma _{{D,E}}}(x)\mathrm {d}x \\&\le \left| y\right| \sum _{n=1}^{N}\delta t\sum _{\sigma _{{D,E}}} \left| \sigma _{{D,E}}\right| |p_{E}^{n} - p_{D}^{n}|. \end{aligned}$$

Finally, using (4.1) and the fact that the discrete gradient of global pressure is bounded (5.5), we deduce that

$$\begin{aligned} \int \limits _{(0,T)\times \Omega ^{'}}\left| {\tilde{\mathbf{U }}}_{l,D}(t,x+y)-{\tilde{\mathbf{U }}}_{l,D}(t,x)\right| \mathrm {d}x \le C (\left| y\right| +\left| y\right| ^\theta ), \end{aligned}$$

(6.11)

for some constant $C>0$.

In the same way, we prove the space translate for $\alpha = g$. $\square$

We give below a time translate estimate for ${\tilde{\mathbf{U }}}_{\alpha ,{\delta t,h}}$ given by (3.5).

Lemma 7

$(\mathrm{{Time~translate~of }}~{\tilde{\mathbf{U }}}_{\alpha ,{\delta t,h}})$ Under the assumptions (A1)–(A7). Let $p_{\alpha ,{\delta t,h}}$ be a solution of (3.6)–(3.9). Then, there exists a positive constant $C>0$ depending on $\Omega$, $T$ such that the following inequality hold:

$$\begin{aligned} \int \limits _{\Omega \times (0,T-\tau )}\left| {\tilde{\mathbf{U }}}_{\alpha ,{\delta t,h}}(t+\tau ,x) - {\tilde{\mathbf{U }}}_{\alpha ,{\delta t,h}}(t,x)\right| ^2\,dx \,dt \le \tilde{\omega }(\tau ), \end{aligned}$$

(6.12)

for all $\tau \in (0,T)$. Here $\tilde{\omega }:\mathbb {R}^+\rightarrow \mathbb {R}^+$ is a modulus of continuity, i.e. $\lim _{\tau \rightarrow 0}\tilde{\omega }(\tau )=0$.

We state without proof the above lemma on time translate of ${\tilde{\mathbf{U }}}_{\alpha ,{\delta t,h}}$. Following [14] and [17], the proof is a direct consequence of (4.1) and the estimations (5.5) and (5.12), then we omit it.

7 Convergence and study of the limit

Using the a priori estimates of the previous section and the Kolmogorov relative compactness theorem, we show in this section that the approximate solutions $p_{\alpha ,h,\delta t}$ converge strongly in $L^1(Q_T)$ to a function $p_\alpha$ and we prove that $p_{\alpha }$ is a weak solution of the continuous problem.

7.1 Strong convergence in $L^1(Q_T)$ and convergence almost everywhere in $Q_T$

Theorem 2

(Strong convergence in $L^1(Q_T)$) Up to a subsequence, the following convergences hold:

$$\begin{aligned}&{\tilde{\mathbf{U }}}_{\alpha ,{\delta t,h}} \textit{ and } \mathbf U _{\alpha ,{\delta t,h}} \longrightarrow \mathbf U _\alpha&\textit{ strongly in }L^1(Q_T) \textit{ and a.e. in } Q_T,\end{aligned}$$

(7.1)

$$\begin{aligned}&{\tilde{s}}_{\alpha ,{\delta t,h}} \textit{ and } s_{\alpha ,{\delta t,h}}\longrightarrow s_\alpha&\textit{ almost everywhere in }Q_T,\end{aligned}$$

(7.2)

$$\begin{aligned}&{\tilde{p}}_{\alpha ,{\delta t,h}} \textit{ and } p_{\alpha ,{\delta t,h}} {\longrightarrow } p_\alpha&\textit{ almost everywhere in } Q_T. \end{aligned}$$

(7.3)

Furthermore, $\mathcal {B}(s_\alpha )$ and $p_\alpha$ belong in $L^1(0,T;H^1_{\Gamma _l}(\Omega ))$ and

$$\begin{aligned}&0\le s_\alpha \le 1 \textit{ a.e. in } Q_T,\end{aligned}$$

(7.4)

$$\begin{aligned}&\mathbf U _\alpha =\phi \rho _\alpha (p_\alpha )s_\alpha \textit{ a.e. in } Q_T. \end{aligned}$$

(7.5)

Proof

Observe that from Lemmas 6 and 7 and Kolmogorov’s compactness criterion ([5, Theorem IV.25], [14, Theorem 14.1]), we deduce that ${\tilde{\mathbf{U }}}_{\alpha ,{\delta t,h}}$ is relatively compact in $L^1(Q_T)$. This ensures the following strong convergences of a subsquence of ${\tilde{\mathbf{U }}}_{\alpha ,{\delta t,h}}$

$$\begin{aligned} \rho _\alpha ({\tilde{p}}_{\alpha ,{\delta t,h}}) {\tilde{s}}_{\alpha ,{\delta t,h}} \longrightarrow l_\alpha \quad \text { in } L^1(Q_T) \text { and a.e. in } Q_T, \end{aligned}$$

and due to the Lemma 5, we deduce that $\mathbf U _{\alpha ,{\delta t,h}}$ converges to the same $l_\alpha$.

Denote by $u_\alpha = \rho _\alpha (p_\alpha )s_\alpha$. Define the map $\mathbb {A} : \mathbb {R}^+ \times \mathbb {R}^+ \mapsto \mathbb {R}^+ \times [0,\mathcal {B}(1)]$ defined by

$$\begin{aligned} \mathbb {A}(u_l,u_g) = (p,\mathcal {B}(s_l)) \end{aligned}$$

(7.6)

where $u_\alpha$ are solutions of the system

$$\begin{aligned}&u_l(p,\mathcal {B}(s_l)) = \rho _l(p-\bar{p}(\mathcal {B}^{-1}(\mathcal {B}(s_l))))\mathcal {B}^{-1}(\mathcal {B}(s_l))\\&u_g(p,\mathcal {B}(s_l)) = \rho _g(p-\tilde{p}(\mathcal {B}^{-1}(\mathcal {B}(s_l))))(1-\mathcal {B}^{-1}(\mathcal {B}(s_l)). \end{aligned}$$

Note that $\mathbb {A}$ is well defined as a diffeomorphism [20, 21] and [6]. As the map ${\mathbb {A}}$ defined in (7.6) is continuous, we deduce

$$\begin{aligned}&p_{\delta t,h}\longrightarrow p \quad \text { a.e. in } Q_T,\\&\mathcal {B}(s_{l,{\delta t,h}}) \longrightarrow \mathcal {B}^{*}\quad \text { a.e. in } Q_T. \end{aligned}$$

Then, as $\mathcal {B}^{-1}$ is continuous, this leads to the desired estimate (7.2)

$$\begin{aligned} s_{l,{\delta t,h}} \longrightarrow s_l = \mathcal {B}^{-1}(\mathcal {B}^{*})\quad \text { a.e. in } Q_T. \end{aligned}$$

Consequently and due to the relationship between the pressure of each phase and the global pressure defined in (2.6), then the convergences (7.3) hold

$$\begin{aligned} p_{\alpha ,{\delta t,h}} \longrightarrow p_\alpha \quad \text { a.e. in } Q_T. \end{aligned}$$

Moreover, due to the space translate estimate on the saturation and the global pressure (6.7)–(6.8), [14, Theorem 3.10] gives that $\mathcal {B}(s_\alpha )$ and $p\in L^1(0,T;H^1_{\Gamma _l}(\Omega ))$. The identification of the limit in (7.5) follows from the previous convergence. $\square$

7.2 Proof of theorem 1

In order to achieve the proof of Theorem 1 and show that $p_\alpha$ is a weak solution of the continuous problem, it remains to pass to the limit as $({\delta t,h})$ goes to zero in the formulations (3.7)–(3.8). For this purpose, we introduce

$$\begin{aligned} \mathcal G := \{ \psi \in \mathcal C^{2,1}(\Omega \times [0,T]), \psi = 0 \text { on }\partial \Omega \times [0,T], \psi (\cdot ,T) = 0\}. \end{aligned}$$

(7.7)

Let $T$ be a fixed positive constant and $\psi \in \mathcal G$. Set $\psi _D^n:=\psi (t^n,Q_D)$ for all $D\in \mathcal {D}_h$ and $n\in [0,N]$.

For the discrete liquid equation, we multiply the equation (3.7) by $\delta t\psi _D^{n}$ and sum the result over $D\in \mathcal {D}_h^\text {int}$ and $n\in \{1,\ldots ,N\}$. This yields

$$\begin{aligned} \mathfrak {C}_1 +\mathfrak {C}_2+\mathfrak {C}_3 = 0, \end{aligned}$$

where

$$\begin{aligned} \mathfrak {C}_1&= \sum _{n=1}^{N}\sum _{D\in \mathcal {D}_h} \left| D\right| \phi _D\left( \rho _l(p^{n}_{l,D})s^{n}_{l,D} -\rho _l(p^{n-1}_{l,D})s^{n-1}_{l,D}\right) \psi _D^{n},\\ \mathfrak {C}_2&= \sum _{n=1}^{N}\delta t\sum _{D\in \mathcal {D}_h} \sum _{E\in \mathcal N(D)}\rho ^{n}_{l,{D,E}} \Lambda _{D,E}G_l(s^{n}_{l,D},s^{n}_{l,E};\delta _{D|E}^{n}(p_{l})) \psi _D^{n},\\ \mathfrak {C}_3&= \sum _{n=1}^{N}\delta t\sum _{D\in \mathcal {D}_h} \sum _{E\in \mathcal N(D)}\left| D\right| \left( \rho _l(p_{l,D}^{n}) s_{l,D}^{n} f_{P,D}^{n}\psi _D^{n}- \rho _l(p_{l,D}^{n}) (s^I_{l,D})^{n}f_{I,D}^{n}\psi _D^{n}\right) . \end{aligned}$$

using $\psi _D^{n} = 0$ for all $D\in \mathcal {D}_h^\text {ext}$ and $n\in \{0,\ldots ,N\}$. We now show that each of the above terms converges to its continuous version as $h$ and $\delta t$ tend to zero.

Firstly, for the evolution term. Making summation by parts in time and keeping in mind that $\psi (T=t^{N},Q_D) = \psi _D^{N}=0$. For all $D\in \mathcal {D}_h$, we get

$$\begin{aligned} \mathfrak {C}_1 =&- \sum _{n=1}^{N}\sum _{D\in \mathcal {D}_h} \left| D\right| \phi _D\rho _l(p^{n}_{l,D}) s^{n}_{l,D} \left( \psi _D^{n}- \psi _D^{n-1} \right) - \sum _{D\in \mathcal {D}_h} \left| D\right| \phi _D\rho _l(p^{0}_{l,D})s^{0}_{l,D} \psi _D^0\\ =&-\sum _{n=1}^{N}\sum _{D\in \mathcal {D}_h} \int \limits _{t^{n-1}}^{t^{n}} \int \limits _{D} \phi _D\rho _l(p^{n}_{l,D}) s^{n}_{l,D} \partial _t \psi (t,Q_D)\mathrm {d}x\mathrm {d}t\\&- \sum _{D\in \mathcal {D}_h} \int \limits _{D} \phi _D\rho _l(p^{0}_{l,D})s^{0}_{l,D}\psi (0,Q_D) \mathrm {d}x. \end{aligned}$$

Since $\phi _h\rho _l(p_{l,{\delta t,h}}) s_{l,{\delta t,h}}$ and $\phi _h\rho _l(p^0_{l,{\delta t,h}})s^0_{l,{\delta t,h}}$ converge almost everywhere respectively to $\phi \rho _l(p_l)s_l$ and $\phi \rho _l(p^0_l)s^0_l$, and as a consequence of Lebesgue dominated convergence theorem, we get

$$\begin{aligned} \mathfrak {C}_1 \longrightarrow - \int \limits _{Q_T}\phi \rho _l(p_l)s_l\partial _t \psi (t,x) \mathrm {d}x \mathrm {d}t - \int \limits _{\Omega }\phi \rho _l(p^0_l)s^0_l\psi (0,x)\mathrm {d}x, \text { as } h, \delta t\rightarrow 0. \end{aligned}$$

Now, let us focus on convergence of the degenerate diffusive term to show

$$\begin{aligned} \mathfrak {C}_2 \longrightarrow - \int \limits _{Q_T}\rho _l(p_l)M_l(s_l)\nabla p_l\cdot \nabla \psi \mathrm {d}x \mathrm {d}t, \text { as } h, \delta t\rightarrow 0. \end{aligned}$$

(7.8)

Since the discrete gradient of each phase is not bounded, it is not possible to justify the pass to the limit in a straightforward way. To do this, we use the feature of global pressure and the auxiliary pressures defined in (2.6) and the discrete energy estimates in Proposition 1 and corollary 1.

We rewrite $\mathfrak {C}_2$ as

$$\begin{aligned} \mathfrak {C}_2 = \mathfrak {C}_{2,1}+\mathfrak {C}_{2,2} \end{aligned}$$

with, by using the definition (2.6),

$$\begin{aligned}&\mathfrak {C}_{2,1} = - \sum _{n=1}^{N}\delta t\sum _{D\in \mathcal {D}_h} \sum _{E\in \mathcal N(D)}\Lambda _{D,E}\rho ^{n}_{l,{D,E}}M_l(s^{n}_{l,{D,E}}) \delta _{D|E}^{n}(p) \psi _D^{n},\\&\mathfrak {C}_{2,2} = \sum _{n=1}^{N}\delta t\sum _{D\in \mathcal {D}_h} \sum _{E\in \mathcal N(D)}\Lambda _{D,E}\rho ^{n}_{l,{D,E}}M_l(s^{n}_{l,{D,E}}) \delta _{D|E}^{n}(\bar{p}(s_l)) \psi _D^{n}. \end{aligned}$$

Let us show that

$$\begin{aligned} \mathfrak {C}_{2,1} \longrightarrow - \int \limits _{Q_T} \mathbf{\Lambda }(x) \; \rho _l(p_l)M_l(s_l) \nabla p \cdot \nabla \psi \, \mathrm {d}x \mathrm {d}t \quad \text { as } {\delta t,h}\rightarrow 0. \end{aligned}$$

(7.9)

For each couple of neighbours $D$ and $E$ we denote $s_{l,min}^{n}$ the minimum of $s_{l,D}^{n}$ and $s_{l,E}^{n}$ and we introduce

$$\begin{aligned} \mathfrak {C}_{2,1}^{*} = - \sum _{n=1}^{N}\delta t\sum _{D\in \mathcal {D}_h} \sum _{E\in \mathcal N(D)}\Lambda _{D,E}\; \rho ^{n}_{l,{D,E}}M_l(s_{l,min}^{n}) \delta _{D|E}^{n}(p) \psi _D^{n} \end{aligned}$$

(7.10)

We now show

$$\begin{aligned} \mathfrak {C}_{2,1}^{*} \longrightarrow - \int \limits _{Q_T} \mathbf{\Lambda }(x) \; \rho _l(p_l)M_l(s_l) \nabla p \cdot \nabla \psi \, \mathrm {d}x \mathrm {d}t \quad \text { as } {\delta t,h}\rightarrow 0 \end{aligned}$$

(7.11)

as ${\delta t,h}\rightarrow 0$. Define $\overline{s}_{\alpha ,{\delta t,h}}$ and $\underline{s}_{\alpha ,{\delta t,h}}$ by

$$\begin{aligned} \overline{s}_{\alpha ,{\delta t,h}}|_{(t^n,t^{n}]\times K_{{D,E}}}:=\max \{s_{\alpha ,D}^{n},s_{\alpha ,E}^{n}\},\quad \underline{s}_{\alpha ,{\delta t,h}}|_{(t^n,t^{n}]\times K_{{D,E}}}:=\min \{s_{\alpha ,D}^{n},s_{\alpha ,E}^{n}\} \end{aligned}$$

Remark that

$$\begin{aligned} \mathfrak {C}_{2,1}^{*}&= - \sum _{n=1}^{N}\delta t\sum _{D\in \mathcal {D}_h} \sum _{E\in \mathcal N(D)}\Lambda _{D,E}\; \rho ^{n}_{l,{D,E}}M_l(s_{l,min}^{n}) \delta _{D|E}^{n}(p) \psi _D^{n} \\&= - \sum _{n=1}^{N}\delta t\sum _{D\in \mathcal {D}_h} \sum _{E\in \mathcal {D}_h}\Lambda _{D,E}\; \rho ^{n}_{l,{D,E}}M_l(s_{l,min}^{n}) p_E^{n} \psi _D^{n} \\&= - \sum _{n=1}^{N}\delta t\sum _{D\in \mathcal {D}_h} \sum _{E\in \mathcal {D}_h}\rho ^{n}_{l,{D,E}}M_l(s_{l,min}^{n}) \sum _{K\in \mathcal {T}_h} \left( \mathbf{\Lambda }(x) \nabla \varphi _E, \nabla \varphi _D\right) _{0,K} p_E^{n} \psi _D^{n} \\&= \sum _{n=1}^{N}\delta t\sum _{K\in \mathcal {T}_h} \int _{K} \mathbf{\Lambda }(x) \rho _l(p_{l,h}^{n}) M_l(\underline{s}_{l,h}^{n}) \nabla p_h^{n} \cdot \nabla \left( \sum _{D\in \mathcal {D}_h} \psi (t^{n},Q_D) \varphi _D(x) \right) \mathrm {d}x. \end{aligned}$$

We will show the validity of two passages to the limit. We begin by defining:

$$\begin{aligned} \mathfrak {D}_1 =\mathfrak {C}_{2,1}^{*}- \sum _{n=1}^{N}\delta t\sum _{K\in \mathcal {T}_h} \int \limits _{K} \mathbf{\Lambda }(x) \rho _l(p_{l,h}^{n}) M_l(\underline{s}_{l,h}^{n}) \nabla p_h^{n} \cdot \nabla \psi (t^{n},x ) \mathrm {d}x. \end{aligned}$$

We then estimate

$$\begin{aligned} \left| \mathfrak {D}_1\right| \le C h, \end{aligned}$$

using the estimate (5.5), for more details see [9, Theorem 15.3] and [17, Sect. 6.2]. Then,

$$\begin{aligned} \mathfrak {D}_1\longrightarrow 0 \quad \text { as } h\rightarrow 0. \end{aligned}$$

We next show that

$$\begin{aligned}&\sum _{n=1}^{N}\delta t\sum _{K\in \mathcal {T}_h} \int \limits _{K} \mathbf{\Lambda }(x) \rho _l(p_{l,h}^{n}) M_l(\underline{s}_{l,h}^{n}) \nabla p_h^{n} \cdot \nabla \psi (t^{n},x ) \mathrm {d}x \longrightarrow \nonumber \\&\quad \int \limits _{0}^{T} \int \limits _\Omega \mathbf{\Lambda }(x) \rho _l(p_l) M_l(s_l) \nabla p(t,x) \cdot \nabla \psi (t,x) \; \mathrm {d}x \mathrm {d}t \end{aligned}$$

(7.12)

as ${\delta t,h}\rightarrow 0$. We see that both $p_h^{n}(x)$ and $\psi (t^{n},x)$ are constant in time, so that we can easily introduce an integral with respect to time into the first term of (7.12). We further add and subtract

$$\begin{aligned} \sum _{n=1}^{N}\int \limits _{t^{n-1}}^{t^{n}} \int \limits _\Omega \mathbf{\Lambda }(x) \rho _l(p_{l,h}^{n}) M_l(\underline{s}_{l,h}^{n}) \nabla p_h^{n} \cdot \nabla \psi (t,x ) \; \mathrm {d}x \mathrm {d}t \end{aligned}$$

and introduce

$$\begin{aligned}&\mathfrak {D}_2 := \sum _{n=1}^{N}\int \limits _{t^{n-1}}^{t^{n}} \sum _{K\in \mathcal {T}_h} \int _{K} \mathbf{\Lambda }(x) \rho _l(p_{l,h}^{n}) M_l(\underline{s}_{l,h}^{n})\nabla p_h^{n} \cdot \left( \nabla \psi (t^{n},x) - \nabla \psi (t,x) \right) \; \mathrm {d}x \mathrm {d}t, \\&\mathfrak {D}_3 := \int \limits _{0}^{T} \sum _{K\in \mathcal {T}_h} \int _{K} \mathbf{\Lambda }(x) \rho _l(p_{l,{\delta t,h}}) M_l(\underline{s}_{l,{\delta t,h}}) \nabla p_{\delta t,h}(t,x) \cdot \nabla \psi (t,x )\\&\qquad - \int \limits _{0}^{T} \int _\Omega \mathbf{\Lambda }(x) \; \rho _l(p_l)M_l(s_l) \nabla p(t,x) \cdot \nabla \psi (t,x) \; \mathrm {d}x \mathrm {d}t \end{aligned}$$

where $p_{\delta t,h}$ is given by (3.4). Clearly, (7.12) is valid when $\mathfrak {D}_2$ and $\mathfrak {D}_3$ tend to zero as ${\delta t,h}\rightarrow 0$. We first estimate $\mathfrak {D}_2$. We have, for $t \in (t^{n-1},t^{n}]$,

$$\begin{aligned} \left| \nabla \psi (t^{n},x) - \nabla \psi (t,x)\right| \le g(\delta t), \end{aligned}$$

where $g$ satisfies $g(\delta t)>0$ and $g(\delta t) \longrightarrow 0$ as $\delta t\rightarrow 0$. Thus

$$\begin{aligned} \left| \mathfrak {D}_2 \right| \le C g(\delta t) \sum _{n=1}^{N}\delta t\sum _{K\in \mathcal {T}_h} \left| \nabla p_h^{n}|_K\right| \left| K\right| \le C g(\delta t) T^{\frac{1}{2}} \left| \Omega \right| ^{\frac{1}{2}} \end{aligned}$$

using the Cauchy-Schwarz inequality and the estimate (5.5).

We now turn to $\mathfrak {D}_3$. We easily notice that we cannot use the Green theorem for $p_h^{n}$ on $\Omega$, since $p_h^{n}\notin H^1(\Omega )$. So, we are thus forced to apply it on each $K\in \mathcal {T}_h$.

To show that $\mathfrak {D}_3 \longrightarrow 0$ as ${\delta t,h}\rightarrow 0$, we begin by showing that

$$\begin{aligned} \int \limits _{0}^{T} \sum _{K\in \mathcal {T}_h} \int \limits _{K} \left( \nabla p_{{\delta t,h}}(t,x) - \nabla p(t,x) \right) \cdot \mathbf{w}(t,x) \; \mathrm {d}x \mathrm {d}t \longrightarrow 0 \end{aligned}$$

(7.13)

as ${\delta t,h}\rightarrow 0$ for all $\mathbf{w}\in (\mathcal {C}^1(\overline{Q_T}))^d$. To this purpose, we use the a priori estimate (5.5) and [17, Sect. 6.2]. Using the density of the set $[\mathcal {C}^1(\overline{Q_T})]^d$ in $[L^2(Q_T)]^d$ and (7.13), we will conclude a weak convergence of $\nabla p_{\delta t,h}$ (piecewise constant is space and time) to $\nabla p$.

We now finally conclude that $\mathfrak {D}_3\longrightarrow 0$ as ${\delta t,h}\rightarrow 0$. To do that, we begin by showing that $\underline{s}_{l,{\delta t,h}} \rightarrow s_l$ a.e on $Q_T$. Define $\overline{s}_{\alpha ,{\delta t,h}}$ and $\underline{s}_{\alpha ,{\delta t,h}}$ by

$$\begin{aligned} \overline{s}_{\alpha ,{\delta t,h}}|_{(t^n,t^{n}]\times K_{{D,E}}}:=\max \{s_{\alpha ,D}^{n},s_{\alpha ,E}^{n}\},\quad \underline{s}_{\alpha ,{\delta t,h}}|_{(t^n,t^{n}]\times K_{{D,E}}}:=\min \{s_{\alpha ,D}^{n},s_{\alpha ,E}^{n}\} \end{aligned}$$

By the monotonicity of $\mathcal {B}$, we have

$$\begin{aligned}&\int \limits _{0}^{T}\int \limits _{\Omega }\left| \mathcal {B}(\overline{s}_{l,{\delta t,h}})- \mathcal {B}(\underline{s}_{l,{\delta t,h}})\right| ^2\mathrm {d}x \mathrm {d}t \\&\quad \le \sum _{n=1}^{N}\delta t\sum _{D\in \mathcal {D}_h} \sum _{E\in \mathcal N(D)}\int \limits _{K_{D,E}} \left( \mathcal {B}(s_{l,E}^{n}) - \mathcal {B}(s_{l,D}^{n}) \right) ^2 \mathrm {d}x\\&\quad \le \sum _{n=1}^{N}\delta t\sum _{D\in \mathcal {D}_h} \sum _{E\in \mathcal N(D)}\int \limits _{K_{D,E}}\left| \nabla \mathcal {B}(s_{l,h}^{n})|_{K_{D,E}}\right| ^2 d_{D,E}^2 \; \mathrm {d}x \\&\quad \le \sum _{n=1}^{N}\delta t\sum _{D\in \mathcal {D}_h} \sum _{E\in \mathcal N(D)}\left| \nabla \mathcal {B}(s_{l,h}^{n})|_{K_{D,E}}\right| ^2 d_{D,E}^2 \left| {K_{D,E}}\right| \\&\quad \le h^2 \sum _{n=1}^{N}\delta t\sum _{\sigma _{D,E}\in \mathcal {F}_h^\text {int}} \left| \nabla \mathcal {B}(s_{l,h}^{n})|_{K_{D,E}}\right| ^2 \left| {K_{D,E}}\right| \\&\quad \le h^2 \sum _{n=1}^{N}\delta t\left\| \mathcal {B}(s_{l,h}^n)\right\| _{X_h}^2 \le C h^2 \end{aligned}$$

where we have used the estimate (5.12).

Since $\mathcal {B}^{-1}$ is continuous, we deduce up to a subsequence

$$\begin{aligned} \left| \underline{s}_{\alpha ,\mathcal {D}_m}-\overline{s}_{\alpha ,\mathcal {D}_m}\right| \rightarrow 0 \text { a.e. on } Q_T. \end{aligned}$$

(7.14)

Moreover, we have $\underline{s}_{\alpha ,{\delta t,h}}\le s_{\alpha ,{\delta t,h}}\le \overline{s}_{\alpha ,{\delta t,h}}$ and $s_{\alpha ,{\delta t,h}}\rightarrow s_\alpha$ a.e. on $Q_T$. Consequently, and due to the continuity of the mobility function $M_l$ we have

$$\begin{aligned} M_l(\underline{s}_{l,{\delta t,h}})\rightarrow M_l(s_l) \end{aligned}$$

(7.15)

a.e on $Q_T$ and in $L^p(Q_T)$ for $p<+\infty$.

Finally, we further add and subtract $\int _{0}^{T} \int _\Omega \mathbf{\Lambda }(x) \; \rho _l(p_l)M_l(s_l) \nabla p_{\delta t,h}(t,x) \cdot \nabla \psi (t,x) \; \mathrm {d}x \mathrm {d}t$ to $\mathfrak {D}_3$ and using (7.3), (7.15), the a priori estimate (5.5), the weak convergence of $\nabla p_{\delta t,h}$ to $\nabla p$ (7.13), to conclude that $\mathfrak {D}_3\longrightarrow 0$ as ${\delta t,h}\rightarrow 0$. Altogether, combining (7.10) and (7.12) gives

$$\begin{aligned} \mathfrak {C}_{2,1}^{*} \longrightarrow - \int \limits _{Q_T} \mathbf{\Lambda }(x) \; \rho _l(p_l)M_l(s_l) \nabla p \cdot \nabla \psi \, \mathrm {d}x \mathrm {d}t \quad \text { as } {\delta t,h}\rightarrow 0. \end{aligned}$$

It remains to show that

$$\begin{aligned} \left| \mathfrak {C}_{2,1}-\mathfrak {C}_{2,1}^{*}\right| \longrightarrow 0 \quad \text { as } {\delta t,h}\rightarrow 0. \end{aligned}$$

(7.16)

Remark that

$$\begin{aligned} \left| M_l(s_{l,{D,E}}^{n})\delta _{D|E}^{n}(p)-M_l(s_{l,min}^{n})\delta _{D|E}^{n}(p)\right| \le C \left| s_{l,E}^{n}-s_{l,D}^{n}\right| \left| \delta _{D|E}^{n}(p)\right| . \end{aligned}$$

Consequently

$$\begin{aligned} \left| \mathfrak {C}_{2,1}-\mathfrak {C}_{2,1}^{*}\right| \le C\sum _{n=1}^{N}\delta t\sum _{K\in \mathcal {T}_h} \int \limits _{K} \left| s_{l,E}^{n}\!-\!s_{l,D}^{n}\right| \nabla p_h^{n} \cdot \nabla \left( \sum _{D\in \mathcal {D}_h} \psi (t^{n},Q_D) \varphi _D(x) \right) \mathrm {d}x \end{aligned}$$

Applying the Cauchy-Schwarz inequality, and thanks to the uniform bound (5.5) and the convergence (7.14), we establish (7.16).

In order to pass to the limit in $\mathfrak {C}_{2,2}$, we prove firstly that

$$\begin{aligned} \Vert \delta _{D|E}^{n}(\Gamma (s_l)) - \sqrt{M_l(s^{n}_{l,{D,E}})} \delta _{D|E}^{n}(\bar{p}(s_l))\Vert _{L^2(Q_T)} \rightarrow 0\quad \text { as } {\delta t,h}\rightarrow 0, \end{aligned}$$

where $\Gamma (s_l)=\int _{0}^{s_l}\sqrt{M_l(z)}\frac{\mathrm {d}\bar{p}}{\mathrm {d}s_l}(z) \mathrm {d}z$.

In fact, remark that there exist $a\in [s_{l,D},s_{l,E}]$ such as:

$$\begin{aligned} |\delta _{D|E}^{n}(\Gamma (s_l)) - \sqrt{M_l(s^{n}_{l,D,E})}\delta _{D|E}^{n}(\bar{p}(s_l))|&= |\sqrt{M_l(a)} - \sqrt{M_l(s^{n}_{l,D,E})}|| \delta _{D|E}^{n}(\bar{p}(s_l))|\\&\le C |\delta _{D|E}^{n}(\bar{p}(s_l))| \le C \left| s_{l,E}^{n}-s_{l,D}^{n}\right| \\&\le C \left| \mathcal {B}(s_{l,E}^{n}) - \mathcal {B}(s_{l,D}^{n})\right| ^\theta , \end{aligned}$$

since $\mathcal {B}^{-1}$ is an Hölder function. Thus we get,

$$\begin{aligned}&\Vert \delta _{D|E}^{n}(\Gamma (s_l)) - \sqrt{M_l(s^{n}_{l,D,E})} \delta _{D|E}^{n}(\bar{p}(s_l))\Vert ^2_{L^2(Q_T)}\\&\quad = \sum _{n=1}^{N}\delta t\sum _{D\in \mathcal {D}_h} \sum _{E\in \mathcal N(D)}|K_{D,E}| | \delta _{D|E}^{n}(\Gamma (s_l)) - \sqrt{M_l(s^{n}_{l,D,E})} \delta _{D|E}^{n}(\bar{p}(s_l))|^2\\&\quad \le \sum _{n=1}^{N}\delta t\sum _{D\in \mathcal {D}_h} \sum _{E\in \mathcal N(D)}|K_{D,E}|^{1-\theta } |K_{D,E}|^{\theta } \left| \mathcal {B}(s_{l,E}^{n}) - \mathcal {B}(s_{l,D}^{n})\right| ^{2\theta }, \end{aligned}$$

and using the Cauchy-Schwarz inequality and the estimate, we deduce

$$\begin{aligned}&\Vert \delta _{D|E}^{n}(\Gamma (s_l)) - \sqrt{M_l(s^{n}_{l,D,L})} \delta _{D|E}^{n}(\bar{p}(s_l))\Vert ^2_{L^2(Q_T)}\\&\quad \le \left( \sum _{n=1}^{N}\delta t\sum _{D\in \mathcal {D}_h} \sum _{E\in \mathcal N(D)}|K_{D,E}|\right) ^{1-\theta }\\&\qquad \times \left( \sum _{n=1}^{N}\delta t\sum _{D\in \mathcal {D}_h} \sum _{E\in \mathcal N(D)}|K_{D,E}| \left| \mathcal {B}(s_{l,E}^{n}) - \mathcal {B}(s_{l,D}^{n})\right| ^2\right) ^{\theta }\\&\quad \le \left( \sum _{n=1}^{N}\delta t\sum _{D\in \mathcal {D}_h} \sum _{E\in \mathcal N(D)}|K_{D,E}|\right) ^{1-\theta }\\&\qquad \times \left( \sum _{n=1}^{N}\delta t\sum _{D\in \mathcal {D}_h} \sum _{E\in \mathcal N(D)}|K_{D,E}| \left| \nabla \mathcal {B}(s_{l,h}^{n})|_{K_{D,E}}\right| ^2 d_{D,E}^2\right) ^{\theta }\\&\quad \le C h^{2\theta } \left( \sum _{n=1}^{N}\delta t\sum _{\sigma _{D,E}\in \mathcal {F}_h^\text {int}} |K_{D,E}| \left| \nabla \mathcal {B}(s_{l,h}^{n})|_{K_{D,E}}\right| ^2 \right) ^{\theta }\\&\quad \le C h^{2\theta } \left( \sum _{n=1}^{N}\delta t\left\| \nabla \mathcal {B}(s_{l,h}^{n})\right\| _{X_h}^2 \right) ^{\theta } \le C h^{2\theta } \end{aligned}$$

which shows that $\Vert \delta _{D|E}^{n}(\Gamma (s_l)) - \sqrt{M_l(s^{n}_{l,{D,E}})} \delta _{D|E}^{n}(\bar{p}(s_l))\Vert ^2_{L^2(Q_T)} \rightarrow 0$ as $h\rightarrow 0$. And from (5.13) in Corollary 1, we deduce that there exists a constant $C>0$ where the following inequalities hold:

$$\begin{aligned} \sum _{n=1}^{N}\delta t\left\| \Gamma (s_{l,h}^{n})\right\| _{X_h}^2 \le C. \end{aligned}$$

(7.17)

Furthermore, by (7.2), we have

$$\begin{aligned} \Gamma (s_{l,{\delta t,h}}) \longrightarrow \Gamma (s_{l}) \quad \text { in } L^2(Q_T). \end{aligned}$$

In the same manner of (7.13), we prove a weak convergence of $\nabla \Gamma (s_{l,{\delta t,h}})$ (piecewise constant function is space and time) to $\nabla \Gamma (s_l)$. As consequence $\sqrt{M_l(s_{l,{\delta t,h}})}\nabla \bar{p}(s_{l,{\delta t,h}})$ converges to $\nabla \Gamma (s_l)$ in $L^2(Q_T)$, and

$$\begin{aligned} \mathfrak {C}_{2,2}&\longrightarrow - \int \limits _{0}^{T}\int _{\Omega } \rho _l(p_{l}) \sqrt{M_l(s_{l})} \nabla \Gamma (s_{l})\cdot \nabla \psi \mathrm {d}x \mathrm {d}t\end{aligned}$$

(7.18)

$$\begin{aligned}&= - \int \limits _{0}^{T}\int \limits _{\Omega } \rho _l(p_{l}) M_l(s_{l}) \nabla \bar{p}(s_{l})\cdot \nabla \psi \mathrm {d}x \mathrm {d}t. \end{aligned}$$

(7.19)