A Fully Discretized Finite Element Approximation for an Incompressible Flow in Porous Media

In this paper, we will study the fully discretized finite element approximation for an incompressible flow in porous media. The model consists of the heat equation, the equation for the concentration and the equations of motion under the Darcy law. The model is rewritten using the stream function-vorticity formulation. The Stability of the fully discrete problem is established. Optimal a priori error estimates are given.


Introduction
Within the last decades, a significant number of works has been devoted to the reaction front propagation in porous media (see for instance [1,2,3] and the references therein). The vast majority of works used the velocity-pressure formulation in their study via the Darcy equation. However, the stream function-vorticity formulation is particulary used to perform the numerical simulation of an incompressible fluid flow in porous media [4,5]. Among the advantages of this formulation is to reduce the unknows of the numerical problem. In this paper, we are interested in studying the finite element of stream function-vorticity formulation for propagating reaction front in porous media. The model considered is a system of reactiondiffusion equations coupled with the hydrodynamic under the Darcy-Boussinesq approximation in the open bounded convex domain Ω ⊂ R 2 [6,7,8]: , where T is the temperature, C is the concentration, u is the velocity, p is the pressure, λ is the thermal diffusivity, η is the diffusion, µ is the viscosity, K is the permeability, E is the activation energy, R is the universal gas constant, α is the Arrhenius pre-exponential factor, T 0 is a mean value of temperature, g is the gravity, γ is the upward unit vector and β is the coefficient of the thermal expansion of the fluid.
The boundary conditions are of Dirichlet-Neumann type for the temperature, the concentration and of impermeability where Γ 1 and Γ 2 are disjoined opens parts of ∂Ω such that Γ 1 ∪ Γ 2 = ∂Ω.
Due to the incompressibility of the fluid, we introduce the stream function ψ defined by: u = rot(ψ), by introducing also the vorticity ω = curl (u), the problem (P) becomes: Because of there is no flow of fluid through the boundary (impermeability boundary), we will have the following zero-flux condition: ψ| ∂Ω = 0. (1. 2) The existence and uniqueness of solutions has been studied in [9], while the finite elements approximations of the semidiscretized problem has been established in [10]; the authors studied the existence and the uniqueness of the semi-discrete solution and give some error estimates on the problem unknowns. In this work, attention is focused to the fully discretized problem. First, we will study the stability of the numerical scheme. Next, we will give some optimal error estimates on vorticity, stream function, concentration, pressure and temperature.
2 The numerical stability of the problem 2.1 The variational formulation of the problem Before giving the problem in its variational form; first, we will describe the functional framework. We set X = L 2 (0, t, H 1 0 (Ω)), W = L 2 (0, t, H 1 0,Γ1 (Ω)) and M = L 2 (0, t, L 2 (Ω)). (2.1) The variational form of the continued problem (P) can be written in the following form: The existence of the weak solution of problem (P n v ) has been established in Reference [9]. We can see clearly that the Universal Journal of Computational Mathematics 3 (3): 27-43, 2015   29 parameters and the functions f and g of the problem verify the following conditions: The reals µ p , η and λ are strictly positives, where T i is the temperature of the unburned mixture. We introduce now the constant of Friedrichs-Poincaré which depends on the geometry of the domain Ω:

The fully semi-discrete problem
In order to give the semi-discrete problem, we will need the following spaces: (Ω) onto X h , such that: for all Examples of such spaces verifying these conditions is given in [11] and [12]. For convenience, for all (T, ϕ, C, ψ) ∈ W 4 , we introduce the forms defined by: (2.5) The fully semi-discretized problem can be writhen as follows: Let k = ∆t be a small parameter, the fully discrete problem is given by By classical arguments [11,13], we have the following lemma: Lemma 2.1. For all n ∈ I N * , the problem admits a unique solution In the sequel of this paper, for each n positive integer, we denoted by C n h , T n h , ψ n h and ω n h the discrete solution of the problem (P n h ). In the following, we set: The main theorem of the paper is the following: If the solution of the problem (P n h ) admits the following regularity: Then, we have the following error estimates: for the stream function, for the vorticity, for the temperature and for the concentration. Here 0 < σ ≤ 1 and w = 8k

Study of stability
The main result of this subsection is to prove stability of our fully-discrete approximate scheme. For this we need some technical lemmas. First we have Proof. First of all, let us notice that By choosing C n+1 h as test function in the first equation of the problem (P n h ) and using 3.1, it follows

Now, by summing over time, we obtain
Also, for the temperature, we have the following Proof. By choosing T n+1 h as test function in the second equation of the problem (P n h ) and using (3.1) we have By summing over time and using Lemma 2.2, We conclude

First we have
Lemma 2.4. For any local solution ω n h of the problem (P n h ), we have the estimate: Proof. By choosing v h = ω n h , as test function in the first equation of the problem (P n h ), we have: By using the triangular inequality, we get:

Now, by summing over time, we obtain
However, using lemma 2.3, get: Finally, for the velocity, the following result holds: Lemma 2.5. For any local solution ψ n h of the problem (P n h ), we have the estimate: Proof. Let ψ n h be the solution of the problem (P n h ). From [14,15] we have, .
(Ω) we have: where Π h is the projection operator defined from W 1, 4 3 (Ω) onto X h , such as, By choosing v h = Π h v as test function in the last equation of the problem (P n h ), we get: Due to the embedding of W 1, 4 3 (Ω) onto L 2 (Ω), we have: and due to the propriety of the operator Π h , we have: . However, using lemma 2.4, we have:

The error estimates
In this subsection, we will prove some error estimates on speed, on the pressure, on the temperature and on the concentration and we will use the following identity First, we have the following technical estimate: For any ω n and ω n h solutions of the problems (P n h ) and (P n h ) respectively, we have: Proof. For ω n and ω n h respectively, solutions of the continuous and the semi-discrete problem, we have: where κ is any element of a regular family of affine meshes τ h . Let ω n h be the approximation of ω n in X h defined by: Hence, for all 0 < σ < 1 and ω n ∈ H σ (Ω) we have the following result: where h κ is the diameter of the element κ. By using the equality (3.2) and setting χ n h = ω n h − ω n h , we obtain: Therefore, we have: However, by using the equality (3.1), we can write: However, for all 0 < σ ≤ 1 and ω n ∈ H 1 (0, T, H σ (κ)), we have By using (3.4) and summing over time, we get: by using (3.3) and (3.4), we obtain: Finally, using the Jensen's inequality, we get: We conclude, using the triangular inequality we have the following error estimate for the stream function Lemma 3.2. For any ψ n and ψ n h solution of the problem (P n v ) and (P n h ) respectively, we have: Proof. Let ψ n and ψ n h be the solution of the problem (P n v ) and (P n h ) respectively. From [14,15] we have: . (3.6) However where Π h is the projection operator verifying(2.6). Moreover, we have Then we get, By choosing v h = Π h v as test function in the last equation of the problem (P n h ), we obtain: Then we get and On the other hand, due to the embedding of W 1, 4 3 (Ω) into L 2 (Ω), we have: .
In addition, , we get: From the inequalities (3.6), (3.7) and (3.8), we conclude that However, we can write: For the temperature estimate, we will need the following lemmas:

. For all constants θ 3 , θ 4 independents of k and h, we have
Proof. We set η n = r h T n − T n h . By using the triangular inequality, we have It leads to Lemma 3.4. We have the following a priori estimate: Proof. By applying the development of Taylor with remainder integral, we obtain Proof. For T n and T n h solution of the problem (P n h ) and (P n v ), from the tree following equalities: and (∂ t T n , η n ) = −λj(T n , η n ) − a 1 (ψ n , T n , η n ) + Z(C n , T n , η n ), where η n = r h T n − T n h , we obtain However using (3.1), the equalities (3.9) and (3.10), yields to 1 2k

A Fully Discretized Finite Element Approximation for an Incompressible Flow in Porous Media
We have also by using the Lemma 3.3, Lemma 3.4 and the equality (3.11), we have 1 2k Finally, we deduce 1 2k Now, we are able to state the following: Lemma 3.6. We assume that (k ≤ 1 4 ), H 1 and H 4 are verified. Then we have For the proof of this lemma, we recall the following lemma, known as the Gronwal lemma [16]: Lemma 3.7. Let a n , b n and c n three positive sequences, c n not decreasing sequence, Assume that We have the following result: a n ≤ c n exp(νn) .

From the triangular inequality
The estimate error on concentration derive from the two technical lemmas: Lemma 3.8. For all constants θ 8 , θ 9 , independents of h and k, we have Proof. The proof is similar to the proof of the Lemma 3.3.

A Fully Discretized Finite Element Approximation for an Incompressible Flow in Porous Media
Lemma 3.9. We have the following estimate: Proof. The proof is similar to the proof of the Lemma 3.4.
Proof. We set ϵ n = r h C n − C n h . First of all, we have Therefore, we get: Also whence, j(r h C n − C n h , C n − r h C n ) = 0, by using the identity (3.1), it follows 1 2k We have as well, using the Holder inequality From where 1 2k Lemma 3.11. Assuming that (k ≤ 1 4 ), H 2 and H 3 are verified, the following estimate holds ,Ω . Proof. By choosing θ 10 = θ 11 = θ 12 = 1 12 and the inequality in Lemma 3.10 becomes 1 2k

A Fully Discretized Finite Element Approximation for an Incompressible Flow in Porous Media
While multiplying by 2k, using the assumptions (k ≤ 1 4 ) and (6(αρ) 2 < η), we obtain It leads to Using the discrete lemma of Gronwall ) exp(kn).
By using the triangular inequality, we get We are able now to establish the error estimate on the stream function: Lemma 3.12. supposing that H 5 is verified, then for θ 4 > 0 and θ 9 > 0, we have the following error estimate 2 )) (kn(h 2σ + k 2 ) + k 2 n 2 (h 2σ + k 2 )).
Proof. According to the Lemmas 3.11 and 3.6, we have By summing the inequality (3.12) and (3.13), we obtain and assume that H 5 is verified then w < 1, it follows ) .

Conclusion
The fully discretized finite element approximations of reaction front propagation within a porous matrix is studied in this paper. The problem is modelled by a system of equations, coupling the reaction-diffusion equations with the hydrodynamic equations under Darcy law. Due to the incompressibility of the fluid, we have consider the Darcy-Boussinesq approximation. The problem is rewritten using the stream function-vorticity formulation. After choosing the appropriate functional framework for our variational problem, we have proved the existence result for the fully-discrete problem. Furthermore, we have established an optimal a priori estimates on the temperature, the concentration, the stream function and the vorticity.