On the Stream Function-Vorticity Finite Element Formulation for Incompressible Flow in Porous Media

Stream function-vorticity finite element formulation for incompressible flow in porous media is presented. The model consists of the heat equation, the equation for the concentration, and the equations of motion under the Darcy law. The existence of solution for the discrete problem is established. Optimal a priori error estimates are given.


Introduction
The stream function-vorticity formulation is widely used to perform the numerical simulation of an incompressible fluid flow in porous media [1,2]. The main advantage of the formulation is a reduction of the numerical problem unknowns with the fact that the continuity equation remains satisfied. 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 reaction-diffusion equations coupled with the hydrodynamic under the Darcy-Boussinesq approximation in the open bounded convex domain Ω ⊂ R 2 [3][4][5]: where is the temperature, is the concentration, is the velocity, is the pressure, is the thermal diffusivity, is the diffusion, is the viscosity, is the permeability, is where ( ) = exp(− / ), ( ) = ( / ), and = / .
Because there is no flow of fluid through the boundary (impermeability boundary), we will have the following zeroflux condition: The paper is organized as follows. We present the variational formulation of the problem in the next section. We establish the existence result in Section 3. A priori error estimates are given in Section 4. We conclude in the last section.

The Existence Result
3.1. The Semidiscrete Problem. In order to give the semidiscrete problem, we will need the following spaces: where ℎ is a strictly positive constant. We assume that the spaces ℎ , ℎ , and ℎ satisfy the following assumptions.
(1) For all 0 < ≤ 1, there exists a linear continuous (2) For all 0 < ≤ 1, there exists a linear continuous Examples of such spaces verifying these conditions are given in [6,7]. The discretized form of the problem (P V ) is given as follows: and ( ℎ , ℎ ) ∈ ( 1 (0, , ℎ )) 2 such that: With the initial conditions, In the sequel, we assume for simplicity that The main result of the paper is written as follows.

Existence of Semidiscrete Solutions.
In order to prove the existence result of the problem (P ℎ ), we need the following lemmas; first for the concentration, we have the following.

Lemma 2.
For any local solution ℎ of the problem (P ℎ ), one has the a priori estimate Proof. By choosing ℎ = ℎ , as test function in the third equation of the problem (P ℎ ), we have Integrating the last equality and noticing that 1 ( ℎ , ℎ , ℎ ) is positive, we obtain By using the two last inequalities, we obtain the result of the lemma.
Also, for the temperature, we have the following result.

Lemma 3.
For any local solution ℎ of the problem (P ℎ ), one has the a priori estimate Proof. By choosing ℎ = ℎ , as test function in the second equation of the problem (P ℎ ), we have Via integration of the last inequality, it follows that Thus, using inequality (23), we get It leads to From (28) and (29), it follows that For the vorticity, we have the following.

Lemma 4.
For any local solution ℎ of the problem (P ℎ ), one has the estimate Proof. By choosing V ℎ = ℎ , as test function in the first equation of the problem (P ℎ ), we have Then, by using triangular inequality, we get While integrating the last inequality, we obtain However, using Lemma 3, we have It leads to Proof. Let ℎ be the solution of the problem (P ℎ ). From [8,9], we have However, where Π ℎ is the projection operator defined from 1,4/3 (Ω) onto ℎ , such as By choosing V ℎ = ℎ , as test function in the last equation of the problem (P ℎ ), we get Therefore, Due to the embedding of 1,4/3 (Ω) onto 2 (Ω), we have and also, due to the propriety of the operator Π ℎ , we have We conclude that Then, we get ℎ 2 (0, ,( 1,4 (Ω))) ≲ ℎ 2 (0, , 2 (Ω)) .
Using Lemma 4, we conclude that Now, we are able to prove the main theorem of this section more precisely. Theorem 6. The problem (P ℎ ) admits at least a solution Proof. Indeed, it is obvious that the problem (P ℎ ) admits a local solution in the interval (0, ℎ ). For ℎ which is rather small, Lemmas 2, 3, 4, and 5 show that this solution can be defined on the interval (0, ) for > 0.
(77) Also, we have the following error estimate for temperature.

Lemma 9. If we assume that the hypothesis ( 1 ) is verified, then one has
.

(78)
Proof. For solution of the problem (P) and ℎ solution of the problem (P ℎ ), we have Therefore, using the propriety of the operator ℎ , we have So, By setting ℎ = ℎ − ℎ , we obtain Then, we get (Ω) We have also From (83) and (84), we get (Ω) Then, we deduce (Ω) 8
Finally, if we set we obtain the following estimate: 4 (Ω)) .
In addition, we have the following error estimate on concentration.

Conclusion
The propagation of reaction front in porous media is modelled by a system of equations, coupling hydrodynamic equations and the reaction-diffusion equations. We have taken into account Darcy-Boussinesq approximation. We have adopted the stream function-vorticity formulation of the Darcy equation and we have chosen the appropriate functional framework for our variational problem. We have proved the existence result for the semidiscrete solution. Furthermore, we have established an optimal a priori estimate on the temperature, the concentration, the stream function, and the vorticity.