Fixed-Point and STILS Method to Solve a Coupled System of Transport Equations

In this paper, a coupled system of two transport equations is studied. The techniques are a ﬁ xed-point and Space-Time Integrated Least Square (STILS) method. The nonstationary advective transport equation is transformed to a “ stationary ” one by integrating space and time. Using a variational formulation and an adequate Poincare inequality, we prove the existence and the uniqueness of the solution. The transport equation with a nonlinear feedback is solved using a ﬁ xed-point method.


Introduction
This work is motivated by the crystal dissolution and precipitation model in saturated porous medium [1]. In [1], the authors present a macroscopic model describing ions transport by fluid flow in a porous medium undergoing dissolution and precipitation reactions. Such models received much attention during the past years (e.g., [2][3][4]).
All these papers deal with the upscaled formulation of the phenomena. A rigorous justification, starting from a wellposed microscopic (pore-scale) model and applying a suitable upscaling, has been given for important classes of problems. For instance, [5] presents homogenization as a method for upscaling and contains an overview with particular emphasis on porous media flow including chemical reactions. In this respect, we also mention [6], where the reaction rates and isotherms are linear (see also [7] or [8]), where nonlinear cases as well as multivalued interface conditions are analyzed.
In [9], the authors study the pore-scale analogue of the model proposed in [1], which is built on Stokes flow in the pores, transport of dissolved ions by convection and diffusion, and dissolution-precipitation reactions on the surface of the porous skeleton. They use regularisation techniques and a fixed-point argument to obtain existence of a weak solution in general domains. The results obtained are a rigorous justification of the macroscopic model in [1].
In this paper, we give a mathematical analysis of the macroscopic model in [1] using the fixed-point theorem and the STILS method to solve the transport equations.
The least squares method is widely used to solve partial differential equations. We can refer to [10,11] for application on elasticity and fluid mechanics problems. Some general mathematical results have been obtained for this method in the case of first-order time-dependent conservations laws. With the space-time objects below, the STILS method transforms a nonstationary problem into a "stationary" problem by integrating space and time. This "stationary" problem is of advective form. For instance, in [12], an equivalence between the advective formulation and that of anisotropic diffusion is established.
The STILS method is originated to [13,14]. In [13,14], a least squares method is used to solve a 2D stationary firstorder conservation equation with regularity assumptions on the advection velocity. A comparison between the least squares solution and the renormalized solution in the sense of [15] for some equations is given in [16]. The STILS method leads to some numerical schemes which are much simpler than the usual ones (like the streamline diffusion method, the characteristic method, and the discontinuous finite element method with flux limiter). Some numerical examples are presented in [17][18][19].
In this paper, we make use of both the fixed-point theorem and the so called STILS method to solve nonlinear transport equations in the model described below. This paper is organized as follows. Section 2 provides a brief presentation of the model proposed in [1]. Assumptions and preliminary results useful to the resolution are also presented. Section 3 is devoted to an existence and uniqueness theorem. The fixed-point theorem and the STILS method are both used to deal with the above mentioned model.

Model Equations and Problem Description
2.1. Model Equations. It is useful to recall the model equations suggested in [1] without going into the details. We assume having two species M 1 and M 2 , for example, ions, say M 1 being a cation and M 2 an anion. In addition, there may be a crystalline solid M 12 present at the porous skeleton. M 1 and M 2 may precipitate at the surface of the porous skeleton to form M 12 , and conversely, the crystalline solid may dissolve. The stoichiometry of the reaction is supposed to be as follows: n and m denote positive numbers. Let c i , i = 1, 2, be the molar concentration of M i in the solution relative to the water volume, and let c 12 be the molar concentration of M 12 relative to the mass of the porous skeleton. The particle M 12 is attached to the surface of the porous skeleton and thus is immobile. The conservation of the corresponding total masses leads to the partial differential equations where the water content θ is supposed to be constant and not affected by the reaction (1), ρ is the bulk density, D is the diffusion/dispersion tensor, and q * is the specific discharge vector. If we define then equations (2) and (3) imply that the quantity c verifies Another equation for c 12 results from a description of the precipitation and dissolution processes. Following the detailed discussion in [1], we have where r d and r P = k p rðc 1 , c 2 Þ are, respectively, the dissolution and precipitation rates and k * the reaction velocity. r is a nonlinear smooth nonnegative function depending on c 1 and c 2 . A typical example is leading to Summarizing the discussion done in about precipitationdissolution reaction, we have for the crystalline solid the equation where which means 0 ≤ w ≤ 1 for c 12 = 0 and w = 1 for c 12 > 0.
H is the set-valued Heaviside function defined by We make now some assumptions which lead to the model we study in this paper. Equation (4) gives Then, in this model, we consider equations (2), (5), and (9).
If the dispersive transport is negligible compared to the advective transport, it is reasonable to tend D to zero. This assumption cancels the corresponding terms in (2) and (5). We assume also that c 12 > 0 anywhere; it means w = 1 in (9). The velocity of the solute transport q = q * /θ in (2) is the gradient of the hydraulic potential. Setting the model equations have the following form: In this paper, we study the coupled system (14)- (16) in addition to the appropriate boundary conditions.
where nðxÞ is the outer normal to ∂Ω at x. Let Let H m ðΩÞ be the classical Hilbert space of order m, and We also define the following Hilbert space V: equipped with the graph norm: Let The problem consists in finding c, c 1 : satisfying the following partial differential equation system Equations (14) and (16) give system ðP2Þ. In systems ðP1Þ and ðP2Þ, initial conditions and inflow boundary conditions are given for c and c 1 . The technique is to solve first ðP1Þ and after ðP2Þ. With the solutions c and c 1 and equation (16), we obtain c 12 .

Preliminary
Results. The STILS method will be used to solve the problem ðP1Þ like in [12,20]. The method leads to a variational formulation problem, and we use the classical Lax-Milgram theorem to find the solution. So we give the following lemma to prove the bilinear map coercivity.

Lemma 1 (curved Poincaré inequality). There exists a constant
The proof is in [20]. Let HðX, x, tÞ be a polynomial with respect to X with bounded coefficients h i defined in Ω T .
Let a i , i = 0, ⋯, n be defined by ðP2Þ is also a transport problem with a nonlinear feedback. We use both STILS method and fixed-point theorem to solve the problem. So we need ðc k 1 Þ k∈ℕ to be bounded in L ∞ ðΩ T Þ; hence, we give the following lemma.

Lemma 2.
Let T be a positive real number and u 0 ∈ DðΩ T Þ a positive function such that Let ðu k Þ k∈ℕ be the sequence of functions defined by Then, we have Proof. Assume that there exists k ∈ ℕ * such that We know that the velocity field q is bounded in its time component and has a constant dot product (equal to 1, too) with the vector ð0, ⋯, 0, 1Þ. This ensures (see Proposition 7 3 Journal of Applied Mathematics in [12]) that Ω T is filled by the characteristics in the following ways: There exists an S > 0 such that for almost each point ðx, tÞ of Ω T , there exists an integral curve that connect ðx, tÞ to the space-time inflow boundary: On this integral curve (characteristic), system (29) gives Hence, thanks to (28). Moreover, since Ω T is filled by characteristics, we have Finally, we conclude that for any n ∈ ℕ, sup Ω T ðju n jÞ ≤ sup Ω T ðu 0 Þ. This ends the proof of the lemma. Proof. The proof will be done in two steps.

Existence and Uniqueness Theorem
Step one. In the system ðP1Þ, we assume that the divergence of the velocity q is zero; then, we can use the result in [12] or in [21,22] cited in [12] to prove the existence of the boundary trace on Γ T of a function in V.
So, using the extension d ∈ V of g in system ðP1Þ, this later is equivalent to searching w in Hða, with F = −ð∂d/∂tÞ − ð1/θÞq · ∇d. Now, like in [12], we define the following convex quadratic form J: The Gateau derivative of J is Hence, w is the solution of (37) if and only if w satisfies Using the Lax-Milgram theorem, we have the solution of the variational problem (40). Lemma 1 is used to prove the bilinear application coercivity.
Step two. This last step is devoted to the problem ðP2Þ: We set which is a polynomial function with respect to c 1 with coefficients depending on c. Then, ðP3Þ becomes This is a transport equation with a nonlinear feedback G ðc 1 Þ due to the chemical reactions (dissolution-precipitation). To solve it, we first split (42) into the following systems: In order to solve the nonlinear system (43), we use the fixed-point theory. For T > 0 and c 0 ∈ DðΩ T Þ satisfying (28), let ðc k Þ k∈ℕ be the sequence of functions defined by Using Lemma 2, we prove that ðc k Þ k∈ℕ is bounded in L ∞ ðΩ T Þ.

4
Journal of Applied Mathematics Let us show that ðc k 1 Þ k∈ℕ is a Cauchy sequence in V and that its limit c 1 is the solution of (42).
Let p and q be integers such that Equations (47) and (48) give By a variational formulation, (49) give We choose w = c Hence, Since the function G is regular enough, we have So using the Cauchy-Schwarz inequality in the right hand side of (52), we get Hence, Let τ = max 0≤s≤γ jG′ðsÞj; we have Using the curved Poincaré inequality of Lemma 1, we get So, if T < 1/2τ, the sequence ðc k 1 Þ k∈ℕ is a Cauchy sequence in V which is a Hilbert space then converges to c 1 ∈ V.
Let us now prove that the limit c 1 of ðc k 1 Þ k∈ℕ is the solution of (42). For this purpose, we write Since a · ∇ t c 1 − a · ∇ t c k 1 tends to zero in L 2 ðΩ T Þ and a · ∇ t c k 1 − Gðc k−1 1 Þ = 0, it remains to be proved that Gðc k−1 1 Þ − G ðc 1 Þ tends to zero. We have We know that G ′ is bounded and ðc k−1 1 Þ tends to c 1 in V so Gðc k−1 1 Þ − Gðc 1 Þ tends to zero. Then, c 1 is the solution of the first equation of (42). The boundary and initial conditions are obtained by the continuity of the trace mapping.

Conclusion
The existence and uniqueness solutions of a coupled transport equations have been proved by a fixed-point and STILS method. Free divergence of the transport velocity is considered in this work, but the method used can be extended to no free divergence case. The techniques used in this paper can be applied in other coupled nonstationary systems. The spatial operator in N dimension, for example, can be replaced by an operator in N + 1 dimension, where the N + 1 component is the time derivative. The obtained result can be improved adding a concrete example like solving a salt-wedge intrusion model or a coupled system arising in epidemiology. All those things and the numerical simulation will be done in the future work.

Data Availability
There are no data supporting this study.

Conflicts of Interest
The author declares that there are no conflicts of interest regarding the publication of this work.