Mixed Finite Element Approximation of Reaction Front Propagation in Porous Media

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Propagation in Porous Media

Karam Allali, Siham Binna

To cite this version:

Karam Allali, Siham Binna. Mixed Finite Element Approximation of Reaction Front Propagation in

Porous Media. International Journal of Engineering and Mathematical Modelling, ORB Academic

Publisher, 2015, 2 (3), pp.79-109. �hal-01186731�

Reaction Front Propagation in Porous Media

Karam Allali

, Siham Binna

1Department of Mathematics, FSTM, University Hassan II-Casablanca, PO Box 146, Mohammadia, Morocco.

Email: allali@fstm.ac.ma

•Corresponding author.

ABSTRACT

Mixed finite element approximation of reaction front propagation model in porous media is presented.

The model consists of system of reaction-diffusion equations coupled with the equations of motion under the Darcy law. The existence of solution for the semi-discrete problem is established. The stability of the fully-discrete problem is analyzed. Optimal error estimates are proved for both semi-discrete and fully-discrete approximate schemes.

KEYWORDS

A priori error estimates — Darcy law — Mixed finite element — Porous media

1. Introduction

A fluid flow through porous media is of great interest in many scientific and engineering applications such as groundwater pollution, oil recovery and polymerization [?,?,?]. The influence of convective instability on reaction front propagation in porous media is studied in [?,?]. The mixed finite element approximation of reaction front propagation with fully Navier-Stokes equations is studied in [?]. In this paper, we are interested in the study of the mixed finite element approximation of the model describing the reaction front propagation in porous media. For this purpose, we will consider a model coupling system of motion equations with heat and concentration equations. We will assume that the fluid is incompressible, so the model considered will be under Darcy-Boussinesq approximation [?,?,?] in the open bounded open domainΩ⊂R^d(d=2,3):

(P)























∂_tT−λ∆T+u.∇T =Cg(T),

∂_tC−η∆C+u.∇C=−Cg(T),

∂tu+µ

Ku+∇p=f(T), div u=0,

whereTis the temperature,Cis the concentration,uis the velocity,pis the pressure,λ is the thermal diffusivity, ηis the diffusion,µis the viscosity,Kis the permeability,gis the reaction source term, given by the Arrhenius law [?]:

g(T) =αexp(−_RT^E ) andf is the gravity force taken under Boussinesq approximation [?]:

f(T) =β(T−T0)gγ,

here,Eis the activation energy,Ris the universal gas constant,αis the Arrhenius pre-exponential factor,T₀is a mean value of temperature,gis the gravity,γis the upward unit vector andβ 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 imperme-

ability type for the normal component of velocity:

(

T|_Γ1=C|_Γ1=0, ∂T

∂n|_Γ2=∂C

∂n|_Γ2=0, and u·n|_∂Ω=0, u|_t=0=u₀, T|_t=0=T₀, and C|_t=0=C₀,

whereΓ1andΓ2are disjoined opens parts of∂Ωsuch thatΓ1∪Γ2=∂Ω.

The paper is organized as follows. We present the semi-discrete problem in the next section. We establish the existence of the discrete solution and we obtain the error estimates on speed, pressure, temperature and concentration in section 3. The fully-discrete problem is also under consideration and similar results are obtained in section 4. We conclude in the last section.

2. Presentation of the semi-discrete problem

2.1 The problem in variational form

In order to give the variational formulation of the problem(P), let us specify the functional framework in which we carry out our study.

We set:

X=L²(0,t,H₀(div,Ω)), W =L²(0,t,H_0,Γ¹

1(Ω)) and M=L²(0,t,L²₀(Ω)), whereH₀(div,Ω) =

u∈(L²(Ω))^d,div(u)∈L²(Ω)andu·n|_∂Ω=0 , in which, we define the following norm:

kωk_div,Ω= (kωk²_(L2(Ω))^d+kdiv(ω)k²_L2(Ω))¹². The variational formulation of the problem(P)is written as follows:

Find (u,p,T,C)∈X×M×W², such that ∀(v,q,φ,ξ)∈X×M×W², we have:

(Pv)























(∂_tu,v) +µ_p(u,v)−b(p,v) = (f(T),v), ∀v∈X,

(∂_tT,φ) +λj(T,φ) +a₁(u,T,φ)−Z(C,T,φ) =0, ∀φ∈W, (∂_tC,ξ) +ηj(C,ξ) +a1(u,C,ξ) +Z(C,T,ξ) =0, ∀ξ ∈W, b(q,u) =0, ∀q∈M,

wherea₁,b, j,Zare the forms, such that, for all(u,v)∈X²,(T,φ,C,ψ)∈W⁴,p∈M, we have b(p,v) =

Z

Ω

pdiv(v)dx, a₁(u,T,φ) =1 2

Z Ω

(u∇)Tφdx− Z

Ω

(u∇)φT dx

,

j(T,ψ) = Z

Ω

∇T∇ψdx and Z(C,T,ψ) = Z

Ω

Cg(T)ψdx, withX=n

u∈X;^∂u

∂t ∈L²(0,t,(H₀(div,Ω))^∗o

, W=n

T ∈W;^∂T

∂t ∈L²(0,t,(H_0,Γ¹

1(Ω))^∗)o

and µp=^µ_K.Here (H_0,Γ¹

1(Ω))^∗and(H₀(div,Ω))^∗are the dual spaces ofH_0,Γ¹

1(Ω)andH₀(div,Ω)respectively.

From the definition of the functions f,gand the physical parameters of the problem(P), we have the following:















The reals µp, η and λ are strictly positives, g∈W^1,∞(R), kg⁰k_L∞(Ω)= Eα

RT_i²,g≥0 and kgk_L∞(Ω)=α, f ∈W^1,∞(R), f(T₀) =0 and

∀(T₁,T₂)∈(H_0,Γ¹

1(Ω))², k f(T₁)−f(T₂)k_L2(Ω)≤ ρ βgk∇(T₁−T₂)k_L2(Ω),

whereT_iis the temperature of the burned mixture andρis the constant of Friedrichs-Poincar´e which is related to the geometry domain.

2.2 The mixed formulation

In order to give the semi-discrete problem, we will need the following spaces:

X_h⊂H₀(div,Ω),M_h⊂L²₀(Ω) and W_h⊂H_0,Γ¹

1(Ω),

wherehis a strictly positive constant. Throughout the paper the following notation is used: For eachϑ,ζ; ϑ≤C⁰ζ ⇐⇒ϑ.ζ ; where the constantC⁰is independent of the mesh size and the solutions. We assume that the spacesX_h,M_handW_hsatisfy the following conditions (see [?,?,?,?]):

1. For all 0<σ≤1, the following diagram is commutative [?], H^σ(div,Ω)∩H₀(div,Ω) ^div //

R_h

L²₀(Ω)∩H^σ(Ω)

r_h

X_h

div //M_h

(2.1)

whereR_handr_hare projection operators verifying the following [?],

∀u∈H^σ(div,Ω)∩H₀(div,Ω), kR_hu−uk_div,Ω.h^σkuk_div,σ,

∀q∈H^σ(Ω)∩L²₀(Ω), kr_hq−qk_0,Ω.h^σkqk_σ,Ω. 2. The inf-sup condition:

∀q_h∈M_h,∃v_h∈X_h, (div v_h,q_h)_0,Ω≥νkq_hk_0,Ωkv_hk_div,Ω. 3. For all 0<σ≤1; there exist a linear continuous operatori_hfromH^1+σ(Ω)∩H_0,Γ¹

1(Ω))ontoW_hsuch that:

k∇i_hTk_L2(Ω).k∇Tk_L2(Ω),

∀T ∈H^1+σ(Ω)∩H_0,Γ¹

1(Ω), kT−i_hTk_1,Ω.h^σ|T|_1+σ,Ω. (2.2) whereνis a constant independent of mesh size andH^σ(div,Ω) =

u∈(H^σ(Ω))^d;div(u)∈H^σ(Ω) is the space with the normkωk_div,σ= (kωk²

(H^σ(Ω))^d +kdiv(ω)k²_Hσ(Ω))¹².

An example of such spaces verifying those conditions is given as follows:





 X_h=

u_h∈H₀(div,Ω)∩(C⁰(Ω))^d;∀K ∈τh;u_h|K ∈RT₀(K) , W_h=n

T_h∈H_0,Γ¹

1(Ω)∩C⁰(Ω);∀K ∈τh;T_h|K ∈P1(K)o , M_h=

p_h∈L²₀(Ω);∀K ∈τ_h;p_h|K ∈P0(K) .

HereK is an element of the quasi-uniform meshes familyτ_h,RT₀(K)is the Raviart-Thomas space [?,?] and P0(K),P1(K)are the polynomials spaces of total degree 0 and 1 respectively.

The discrete form of the problem(Ph)is given as follows:

Findu_h∈C¹(0,t,X_h), p_h∈C⁰(0,t,M_h) and (C_h,T_h)∈(C¹(0,t,W_h))²,such that,

(Ph)























(∂_tu_h,v_h) +µp(u_h,v_h)−b(p_h,v_h) = (f(T_h),v_h), ∀v_h∈X_h,

(∂_tT_h,φh) +λj(T_h,φh) +a₁(u_h,T_h,φh)−Z(C_h,T_h,φ_h) =0, ∀φ_h∈W_h, (∂_tC_h,ξ_h) +ηj(C_h,ξ_h) +a₁(u_h,C_h,ξ_h) +Z(C_h,T_h,ξ_h) =0, ∀ξ_h∈W_h, b(q_h,u_h) =0, ∀q_h∈M_h,

with the boundary conditions:







T_h|_Γ1 =C_h|_Γ1=0, ∂T_h

∂n|_Γ2=∂C_h

∂n |_Γ2=0, and u_h·n|_∂Ω=0, u_h|t=0=u⁰_h, T_h|t=0=T_h⁰, and C_h|t=0=C_h⁰.

Remark2.1. From the commutativity of the diagram (??), it follows thatdiv(X_h)⊂M_h, from where we have

div(u_h) =div(u) =0, (2.3)

whereuandu_hare, respectively, solution of the problem(Pv)and(Ph).

3. A priori error estimates

3.1 The existence of solution

The main result of the section is written as follows

Theorem 3.1. The problem(Ph)admits a unique solution. Moreover, for u, p, T and C solutions of the problem (Pv)and for u_h, p_h, T_hand C_hsolutions of the problem(Ph), we have

ku−u_hk_L2(0,t,H₀(div,Ω))+kT−T_hk_L2(0,t,H_0,Γ¹

1(Ω))+kC−C_hk_L2(0,t,H_0,Γ¹

1(Ω))+kp−p_hk_L2(0,t,L²₀(Ω))

.h^σ

kT k_L2(0,t,H^1+σ(Ω))+kTk_H1(0,t,H^σ(Ω))+kCk_L2(0,t,H^1+σ(Ω))

+kCk_H1(0,t,H^σ(Ω))+kuk_L2(0,t,H^σ(div,Ω))+kuk_H1(0,t,H^σ(Ω))

+kpk_H1(0,t,H^σ(Ω))

,

under the following conditions:



















δ₁=1−

√

10α ρ²N_CE

RT_i²λ > 0, (H₁)

δ2=1−2√

10(α ρ²)²N_CE δ1ηRT_i²λ

> 0, (H₂)

4√

15ρgβNa

µ_pλ

2√

2α ρ²N_C η +N_T

!

≤δ₁δ₂, (H₃)

where























NT=sup

kT_hk_L∞(0,t,H1

0,Γ1(Ω)),kT k_L∞(0,t,H1 0,Γ1(Ω))

, N_C=sup

kC_hk_L∞(0,t,H_0,Γ¹

1(Ω)),kCk_L∞(0,t,H_0,Γ¹

1(Ω))

, N_u=sup ku_hk_L∞(0,t,H₀(div,Ω)),kuk_L∞(0,t,H₀(div,Ω))

,

Na= sup

u∈H₀(div,Ω),(v,w)∈H_0,Γ¹

1(Ω)²

a₁(u,v,w) kvk_H

0,Γ1

1(Ω)kuk_div,Ωkwk_H1 0,Γ1(Ω)

and0<σ≤1.

In order to prove the existence result of the problem(Ph), we need the following lemmas; first for the concentration, we have

Lemma 3.2. For any local solution C_hof the problem(Ph), we have the a priori estimate:

kC_hk²

L^∞(0,t,L²(Ω))+2ηkC_hk²

L²(0,t,H_0,Γ¹

1(Ω))≤2kC_h⁰k²

L²(Ω).

Proof. By choosingξh=C_h, as test function in the third equation of the problem(Ph), we have 1

2 d

dt kC_hk²_L2(Ω)+ηk∇C_hk²_L2(Ω)+Z(C_h,T_h,C_h) =0.

Integrating the last equality, and noticing thatZ(C_h,T_h,C_h)is positive, we obtain

kC_h(t=s)k_L²2(Ω)+2ηkC_hk²

L²(0,t,H_0,Γ¹

1(Ω))≤kC_h⁰k²_L2(Ω), it yields

2ηkC_hk²

L²(0,t,H_0,Γ¹

1(Ω))≤kC_h⁰k²_L2(Ω) (3.1)

and

kC_hk²_L∞(0,t,L2(Ω))≤kC_h⁰k²_L2(Ω).

By summing the two last inequalities, the lemma holds.

Also, for the temperature, we have the following result:

Lemma 3.3. For any local solution T_hof the problem(Ph), we have the a priori estimate:

kT_hk²

L^∞(0,t,L²(Ω))+λ kT_hk²

L²(0,t,H_0,Γ¹

1(Ω))≤^α2ρ2

λ η kC⁰_hk²

L²(Ω)+2kT_h⁰k²

L²(Ω).

Proof. By choosingφh=T_h, as test function in the second equation of the problem(Ph), by using H ¨older and Young inequalities, we get

1 2

d

dt kT_hk²_L2(Ω)+λ k∇T_hk²_L2(Ω) = Z(C_h,T_h,T_h)

≤ ρ αkC_hk_L2(Ω)k∇T_hk_L2(Ω)

≤ (α ρ)²

2λ kC_hk²_L2(Ω)+λ

2 k∇T_hk²_L2(Ω). Via integration of the last inequality, it follows

kT_h(t=s)k_L²2(Ω)+λkT_hk²

L²(0,t,H_0,Γ¹

1(Ω))≤(α ρ)²

λ kC_hk²_L2(0,t,L²(Ω))+kT_h⁰k²_L2(Ω). Thus, using (??), we get

kT_h(t=s)k_L²2(Ω)+λkT_hk²

L²(0,t,H_0,Γ¹

1(Ω))≤α²ρ²

2λ η kC_h⁰k²_L2(Ω)+kT_h⁰k²_L2(Ω). (3.2) It leads to

kT_hk²_L∞(0,t,L2(Ω))≤α²ρ²

2λ η kC_h⁰k²_L2(Ω)+kT_h⁰k²_L2(Ω). (3.3) From (??) and (??), the lemma holds.

Finally, for the speed, we have the following result:

Lemma 3.4. For any local solution u_hof the problem(Ph), we have the estimate:

ku_hk²

L^∞(0,t,(L²(Ω))^d)+µ_pku_hk²

L²(0,t,H₀(div,Ω)) ≤ ^2(βgα)2ρ4

λ²η µ_p kC⁰_hk²

L²(Ω)

+ ^{4(β ρg)2}

λ µp kT_h⁰k²

L²(Ω)

+ 2ku⁰_hk²_(L2(Ω))d .

Proof. Similarly to the previous lemma, by choosingv_h=u_h, as test function in the first equation of the problem (Ph), we have

1 2

d

dt ku_hk²_(L2(Ω))^d +µ_pku_hk²_(L2(Ω))^d≤ ρ βgk∇T_hk_L2(Ω)ku_hk_(L2(Ω))^d

≤^{(ρ βg)}_2µ ²

p k∇T_hk²

L²(Ω)+^µ₂^p ku_hk²

(L²(Ω))^d . Then, we get

1 2

d

dt ku_hk²_(L2(Ω))^d +µp

2 ku_hk²_(L2(Ω))^d≤(ρ βg)²

2µ_p k∇T_hk²_L2(Ω). From (??), we obtain

1 2

d

dt ku_hk²_(L2(Ω))d +µp

2 ku_hk²_div,Ω≤(ρ βg)²

2µ_p k∇T_hk²

L²(Ω).

By integrating the last inequality, we obtain ku_h(t=s)k²_(L2(Ω))d +µ_pku_hk²

L²(0,t,H₀(div,Ω))≤(ρ βg)² µ_p kT_hk²

L²(0,t,H_0,Γ¹

1(Ω))+ku⁰_hk²_(L2(Ω))d . Thus, using Lemma??, it follows

ku_h(t=s)k²

(L²(Ω))^d +µ_pku_hk²

L²(0,t,H₀(div,Ω)) ≤ (βgα)²ρ⁴ λ²η µ_p kC_h⁰k²

L²(Ω)

+ 2(β ρg)² λ µp

kT_h⁰k²_L2(Ω)+ku⁰_hk²_(L2(Ω))d . It leads to

ku_hk²

L^∞(0,t,(L²(Ω))^d) ≤ (βgα)²ρ⁴ λ²η µp

kC_h⁰k²_L2(Ω)

+ 2(β ρg)² λ µp

kT_h⁰k²_L2(Ω)+ku⁰_hk²_(L2(Ω))^d (3.4) and

µ_pku_hk²_L2(0,t,H0(div,Ω)) ≤ (βgα)²ρ⁴ λ²η µp

kC⁰_hk²_L2(Ω)

+ 2(β ρg)² λ µ_p kT_h⁰k²

L²(Ω)+ku⁰_hk²

(L²(Ω))^d. (3.5)

By summing (??) and (??), we conclude the lemma.

The existence result is given by the following theorem:

Theorem 3.5. The problem(Ph)admits at least a solution:

(u_h,p_h,C_h,T_h)∈H¹(0,t,X_h)×L²(0,t,M_h)× H¹(0,t,W_h)2

.

Proof. Indeed, it is obvious that the problem(Ph)admits a local solution in the interval(0,t_h), with smallt_h. We conclude form Lemma??, Lemma??and Lemma??that this solution can be extended to the interval(0,t) witht>0.

Remark3.1. The existence of the continuous variational problem(Pv)can be established similarly as for the problem(Ph).

3.2 The Error Estimates

In this section, we prove the error estimates on speed, on pressure, on temperature and on concentration. Finally, we will state the main theorem of the section. In the sequel, we assume that there exists 0<σ≤1, such that







p∈L²(0,t,H^σ(Ω)),u∈L²(0,t,H^σ(div,Ω))∩H¹(0,t,H^σ(Ω)) and

T, C∈L²(0,t,H^1+σ(Ω))∩H¹(0,t,H^σ(Ω)).

First, we have the following a priori error estimate on the speed:

Lemma 3.6. For any solution u, u_hof the problem(Pv)and(Ph), respectively, and for anyθ₀>0, we have











√µ_pku−u_hk_L2(0,t,H₀(div,Ω)). h^σ

kuk_L2(0,t,H^σ(div,Ω))+kuk_H1(0,t,H^σ(Ω))

+kpk_L2(0,t,H^σ(Ω))

+θ₀kp−p_hk_L2(0,t,L²(Ω))

+

√

√3ρgβ

µp kT−T_hk_L2(0,t,H_0,Γ¹

1(Ω)).

Proof. Letuandu_hbe the solution of the continuous and the discretized problem(Pv)and(Ph), respectively.

We have

(∂_tu−∂tu_h,v_h) +µp(u−u_h,v_h)−b(p−p_h,v_h) = (f(T)−f(T_h),v_h), ∀v_h∈X_h. (3.6) By settingv_h=R_hu−u_h, we obtain

(R_h∂tu−∂tu_h,v_h) +µp(R_hu−u_h,R_hu−u_h) =−(∂_tu−R_h∂tu,v_h) + (f(T)−f(T_h),v_h) +µ_p(R_hu−u,R_hu−u_h) +b(p−p_h,u−u_h) +b(p−p_h,R_hu−u).

However

b(p−p_h,u−u_h) =b(p−r_hp,u−u_h).

Using holder inequality, it yields to 1

2 d

dt kR_hu−u_hk²_(L2(Ω))^d +µ_pkR_hu−u_hk²_(L2(Ω))^d

≤

k∂tu−R_h∂tuk_(L2(Ω))^d +ρgβk∇(T−T_h)k_L2(Ω)

× kR_hu−u_hk_(L2(Ω))^d+kp−r_hpk_L2(Ω)kdiv(u−u_h)k_L2(Ω)

+µpku−R_huk_(L2(Ω))^dkR_hu−u_hk_(L2(Ω))^d

+kp−p_hk_L2(Ω)kdiv(R_hu−u)k_L2(Ω). Using the young inequality, it follows

1 2

∂

∂t kR_hu−u_hk²

(L²(Ω))^d +µp

2 kR_hu−u_hk²

(L²(Ω))^d

≤ 3

2µ_pk∂tu−R_h∂tuk²_(L2(Ω))^d +3(ρgβ)²

2µ_p k∇(T−T_h)k²_L2(Ω)

+θ₀²

2 kp−p_hk²_L2(Ω)+ 1

2θ₀²kdiv(R_hu−u)k²_L2(Ω) (3.7)

+3µ_p

2 kR_hu−uk²_(L2(Ω))^d+1

2kp−r_hpk²_L2(Ω)

+1

2 kdiv(u−u_h)k²_L2(Ω),

whereθ0>0. Therfore, using (??), the properties of the operatorsR_handr_hand integrating (??), it follows

√

µpkR_hu−u_hk_L2(0,t,(L²(Ω))^d) . h^σ

kuk_L2(0,t,H^σ(div,Ω))+kuk_H1(0,t,H^σ(Ω))

+ kpk_L2(0,t,H^σ(Ω))

+θ₀kp−p_hk_L2(0,t,L²(Ω))

+

√3ρgβ

õp

kT−T_hk_L2(0,t,H_0,Γ¹

1(Ω)). From the triangular inequality, we get

ku−u_hk_L2(0,t,(L²(Ω))^d).h^σkuk_H1+σ(Ω)+kR_hu−u_hk_H1

0,Γ1(Ω). (3.8)

Finally, using (??) and (??),we conclude

√

µpku−u_hk_L2(0,t,H0(div,Ω)) . h^σ

kuk_L2(0,t,H^σ(div,Ω))+kuk_H1(0,t,H^σ(Ω))

+ kpk_L2(0,t,H^σ(Ω))

+

√3ρgβ

√µ_p kT−T_hk_L2(0,t,H_0,Γ¹

1(Ω))

+ θ0kp−p_hk_L2(0,t,L²(Ω)).

On the other hand, we have the following a priori error estimate on pressure:

Lemma 3.7. For p and p_h, respectively, solution of the problem(Pv)and(Ph), we have











kp−p_hk_L2(0,t,L²(Ω)). (ν)⁻¹k∂tu−∂tu_hk_L2(0,t,(H₀(div,Ω))^∗)

+^µp

ν ku−u_hk_L2(0,t,H₀(div,Ω))

+^ρgβ

ν kT−T_hk_L2(0,t,H_0,Γ¹

1(Ω))

+h^σkpk_L2(0,t,H^σ(Ω)). Proof. According to (??), we have

b(r_hp−p_h,v_h) =b(r_hp−p,v_h) +b(p−p_h,v_h) =b(r_hp−p,v_h)

+ (∂_tu−∂_tu_h,v_h) +µp(u−u_h,v_h)−(f(T)−f(T_h),v_h), ∀v_h∈X_h. Therefore, using Holder inequality, it holds

b(r_hp−p_h,v_h)≤ kr_hp−pk_L2(Ω)kv_hk_div,Ω+ k∂tu−∂_tu_hk_(H

0(div,Ω))^∗

+µ_pku−u_hk_(L2(Ω))^d +ρgβ k∇(T−T_h)k_L2(Ω)

kv_hk_(L2(Ω))^d . Then

b(r_hp−p_h,v_h)≤

kr_hp−pk_L2(Ω)+k∂tu−∂_tu_hk_(H

0(div,Ω))^∗ (3.9)

+µpku−u_hk_(L2(Ω))^d +ρgβ k∇(T−T_h)k_L2(Ω)

kv_hk_div,Ω. Whence

kp−p_hk_L2(Ω)≤kp−r_hpk_L2(Ω)+1 ν sup

v_h∈X_h

b(r_hp−p_h,v_h)

kv_hk_div,Ω . (3.10)

From (??) and (??), we deduce

kp−p_hk_L2(0,t,L²(Ω)).h^σkpk_L2(0,t,H^σ(Ω))+(ν)⁻¹k∂_tu−∂_tu_hk_L2(0,t,(H₀(div,Ω))^∗)

+µp

ν ku−u_hk_L2(0,t,H₀(div,Ω))+ρgβ

ν kT−T_hk_L2(0,t,H_0,Γ¹

1(Ω)).

Lemma 3.8. For u and u_h, respectively, solution of the problem(Pv)and(Ph), the following estimate holds











k∂tu−∂_tu_hk_L2(0,t,(H₀(div,Ω))^∗). h^σ

kuk_H1(0,t,H^σ(Ω))+kpk_L2(0,t,H^σ(Ω))

+ρgβkT−T_hk_L2(0,t,H_0,Γ¹

1(Ω))

+µ_pku−u_hk_L2(0,t,H₀(div,Ω)). Proof. We introduce now the operatorπh:V→V_h such that

_∂

∂t(v−πhv,v_h) =0, ∀v_h∈V_h,

kπ_h(v)k_L2(Ω)≤ρ¯kvk_L2(Ω), ∀v∈H¹(0,t,V), (3.11) whereV_h={v_h∈X_h; ^R_Ωdiv(v_h) q=0∀q∈M_h}andV is the subspace ofH₀(div,Ω)with divergence-free elements. For simplicity in the sequel, we will note byρthe maximum ofρand ¯ρ.

We have

k∂tu−∂_tu_hk_(H

0(div,Ω))^∗≤k∂tu−πh∂tuk_(H

0(div,Ω))^∗ +kπh∂tu−∂tu_hk_(H

0(div,Ω))^∗. From which we obtain

k∂_tu−∂_tu_hk_{(H0(div,Ω))}^∗≤k∂_tu−π_h∂_tuk_{(H0(div,Ω))}^∗ +sup

v_h∈V_h

∂

∂t(π_hu−u_h,v_h)

kv_hk_div,Ω . (3.12)

From (??) and (??), it follows

∂

∂t(π_hu−u_h,v_h) =(∂_tπhu−∂_tu_h,v_h) = (∂_tu−∂_tu_h,v_h)

=−µp(u−u_h,v_h) +b(p−p_h,v_h) + (f(T)−f(T_h),v_h)

≤ρgβ k∇(T−T_h)k_L2(Ω)kv_hk_(L2(Ω))^d +µ_pku−u_hk_L2(Ω)k v_hk_(L2(Ω))^d

+kp−r_hpk_L2(Ω)kdiv v_hk_L2(Ω). Then

sup

v_h∈V_h

∂

∂t(π_hu−u_h,v_h)

kv_hk_div,Ω ≤ρgβ k∇(T−T_h)k_L2(Ω)+µ_pku−u_hk_L2(Ω)

+h^σkpk_L2(0,t,H^σ(Ω)). (3.13)

Using the inequality (??), the property of the operatorπhandr_hand by integrating (??), it follows

k∂tu−∂tu_hk_L2(0,t,(H0(div,Ω))^∗).h^σ

kuk_H1(0,t,H^σ(Ω))+kpk_L2(0,t,H^σ(Ω))

+ρgβ kT−T_hk_L2(0,t,H_0,Γ¹

1(Ω))+µ_pku−u_hk_L2(0,t,H₀(div,Ω)).

We are now able to establish the following

Lemma 3.9. For u and u_h, respectively, solution of the problem(Pv)and(Ph)we have the following error estimate:











√

µpku−u_hk_L2(0,t,H₀¹(Ω)). h^σ

kuk_L2(0,t,H^σ(div,Ω))+kuk_H1(0,t,H^σ(Ω))

+kpk_L2(0,t,H^σ(Ω))

+^2θ0µp

ν ku−u_hk_L2(0,t,H0(div,Ω))

+ρgβ

2θ0

ν +q

3 µp

kT−T_hk_L2(0,t,H_0,Γ¹

1(Ω)). whereθ0is any strictly positif real.

Proof. From Lemma??and Lemma??, we have

kp−p_hk_L2(0,t,L²(Ω)) . h^σ(kuk_H1(0,t,H^σ(Ω))+kpk_L2(0,t,H^σ(Ω))) (3.14) + 2ρgβ

ν kT−T_hk_L2(0,t,H_0,Γ¹

1(Ω))+2µ_p

ν ku−u_hk_L2(0,t,H₀(div,Ω)). We deduce from Lemma??and the inequality (??), the following:

√µ_pku−u_hk_L2(0,t,H₀¹(Ω)).h^σ

kuk_L2(0,t,H^σ(div,Ω))+kuk_H1(0,t,H^σ(Ω))+kpk_L2(0,t,H^σ(Ω))

+ρgβ 2θ₀ ν +

s 3 µ_p

!

kT−T_hk_L2(0,t,H_0,Γ¹

1(Ω))

+2θ₀µp

ν ku−u_hk_L2(0,t,H₀(div,Ω)).

Also, we have the following error estimate for temperature:

Lemma 3.10. We assume that the hypothesis(H₁)is verified. We have















kT−T_hk_L2(0,t,H_0,Γ¹

1(Ω)). h^σ

kTk_H1(0,t,H^σ(Ω))+kT k_L2(0,t,H^1+σ(Ω))

+

√ 5α ρ²

δ₁λ kC−C_hk_L2(0,t,H_0,Γ¹

1(Ω))

+

√5N_aNT

δ₁λ ku−u_hk_L2(0,t,H₀(div,Ω)). Proof. ForT solution of the problem(Pv)andT_hsolution of the problem(Ph), we have

(∂_tT−∂_tT_h,φ_h)+λj(T−T_h,φ_h) +a₁(u−u_h,T,φ_h) +a₁(u_h,T−T_h,φ_h)

=Z(C,T,φh)−Z(C_h,T_h,φh).

Therefore, using the propriety of the operatori_h, we havej(i_hT−T_h,φ_h) =0, then, by settingφh=i_hT−T_h, we obtain

1 2

d

dt ki_hT−T_hk_L²2(Ω)+λk∇(i_hT−T_h)k²_L2(Ω)=−(∂_tT−i_h∂tT,φh) (3.15)

−a₁(u−u_h,T,φh)−a₁(u_h,T−i_hT,i_hT−T_h) +Z(C,T,φ_h)−Z(C_h,T_h,φh),

From the H ¨older inequality and the embedding of the spaceH¹(Ω)intoL⁴(Ω), we will use in the sequel the following result [?]:

For(u,v,w)∈(H_0,Γ¹

1(Ω))³, we have

| Z

Ω

u(x)v(x)w(x)dx| ≤2¹²kuk¹²

L²(Ω)k∇uk¹²

L²(Ω)kvk_L2(Ω)kwk¹²

L²(Ω)k∇wk¹²

L²(Ω), (3.16)

it yields to

Z(C,T,φ_h)−Z(C_h,T_h,φ_h) =Z(C−C_h,T,φ_h) +Z(C_h,T−T_h,φ_h) (3.17)

≤α ρ²kC−C_hk_H1

0,Γ1(Ω)kφ_hk_H1 0,Γ1(Ω)

+√

2N_Cα E

RT_i²ρ²kT−T_hk_H1

0,Γ1(Ω)kφhk_H1 0,Γ1(Ω).

Then (??) and (??) yield to 1

2 d

dt ki_hT−T_hk²_L2(Ω)+λk∇(i_hT−T_h)k²_L2(Ω)

≤

ρk∂tT−i_h∂tT k_L2(Ω)+N_aN_T ku−u_hk_div,Ω +N_aN_Uk∇(T−i_hT)k_L2(Ω)

k∇(i_hT−T_h)k_L2(Ω)

+

α ρ²kC−C_hk_H1

0,Γ1(Ω)+√

2N_Cα E

RT_i²ρ²kT−T_hk_H1 0,Γ1(Ω)

× k∇(i_hT−T_h)k_L2(Ω). Whence, using the young inequality, we obtain

1 2

d

dt ki_hT−T_hk²

L²(Ω)+λ

2 k∇(i_hT−T_h)k²

L²(Ω)

≤5ρ²

2λ k∂_tT−i_h∂_tT k²_L2(Ω)+5N_a²N_U²

2λ k∇(T−i_hT)k²_L2(Ω)

+5α²ρ⁴

2λ k∇(C−C_h)k²_L2(Ω)+5N_a²N_T²

2λ ku−u_hk²_div,Ω +10N_C²α²E²ρ⁴

2λR²T_i⁴ k∇(T−T_h)k²_L2(Ω). By integrating and multiplying the last inequality by 2

λ, we get ki_hT−T_hk²

L²(0,t,H_0,Γ¹

1(Ω)).

h^σkTk_H1(0,t,H^σ(Ω))+h^σkT k_L2(0,t,H^1+σ(Ω))

2

+5α²ρ⁴

λ² kC−C_hk²

L²(0,t,H_0,Γ¹

1(Ω))

+5N_a²N_T²

λ² ku−u_hk²_L2(0,t,H₀(div,Ω))

+10N_C²(αE)²ρ⁴ R²T_i⁴λ²

kT−T_hk²

L²(0,t,H_0,Γ¹

1(Ω)). However

kT−T_hk_H1

0,Γ1(Ω).h^σkTk_H1+σ(Ω)+ki_hT−T_hk_H1

0,Γ1(Ω). (3.18)

Whence (??) allow us to obtain the following estimate:

1−

√10N_CαEρ² RT_i²λ

!

kT−T_hk_L2(0,t,H_0,Γ¹

1(Ω))

.h^σ

kTk_H1(0,t,H^σ(Ω))+kT k_L2(0,t,H^1+σ(Ω))

+

√ 5α ρ²

λ kC−C_hk_L2(0,t,H_0,Γ¹

1(Ω))

+

√5N_aN_T

λ ku−u_hk_L2(0,t,H₀(div,Ω)). Finally, by setting

δ1=1−

√10N_CαEρ² RT_i²λ

> 0, the lemma holds.