A TWO-LEVEL STABILIZED OSEEN ITERATIVE METHOD FOR STATIONARY CONDUCTION-CONVECTION EQUATIONS

FOR STATIONARY CONDUCTION-CONVECTION EQUATIONS

PENGZHAN HUANG

Communicated by the former editorial board

Two-level stabilized Oseen iterative finite element method for the stationary conduction-convection equations is given. Moreover, the decouple technique is applied to this scheme, which can save much computational time. This two-level stabilized Oseen method involves solving a conduction-convection problem using stabilized Oseen iterative method on a coarse grid firstly, then seek a fine grid solution by solving a simply linearized problem. The stability and iterative error estimates are analyzed, which show that the presented method is stable and has good precision. Numerical results are given to support the developed theory analysis and demonstrate the efficiency of the given method.

AMS 2010 Subject Classification: 65N30, 65N12.

Key words: two-level stabilized Oseen iterative, conduction-convection equa- tions, stability, iterative error estimate.

1. INTRODUCTION

Let Ω be a bounded domain in R² with boundary ∂Ω. The stationary conduction-convection equations are

−ν∆u+ (u· ∇)u+∇p=λjT in Ω, divu= 0 in Ω,

−∆T+λu· ∇T = 0 in Ω, u= 0, T =T0 on ∂Ω, (1)

whereu= (u₁(x), u₂(x)) represents the velocity vector,p=p(x) the pressure, T = T(x) the temperature, ν > 0 the viscosity, λ > 0 the Grashoff number andj= (0,1) the two-dimensional vector. The development of an efficient and effective computational method for investigating this problem has practical significance.

In this paper, by combining a novel stabilization technique of He and Li [1]

and the two-level discretization method of He and Wang [2] with the Oseen iter- ative method, we propose a two-level stabilized Oseen iterative finite element method for the stationary conduction-convection equations. This two-level

MATH. REPORTS16(66),2(2014), 285–293

scheme only involves solving a linearized conduction-convection problem of m times on the coarse mesh and a simply linearized considered problem on the fine mesh. On the other hand, we solve the equation in a decoupling way which can save a lot of CPU time. The result (27) is very interesting. It suggests a stopping criterion for the nonlinear iteration, which is a most useful tool in practice.

2.PRELIMINARIES

For the mathematical setting of problem (1), we introduce the following Hilbert spaces:

X =H₀¹(Ω)², W =H¹(Ω), W0=H₀¹(Ω), Y =L²(Ω)², M =L²₀(Ω) =

q ∈L²(Ω) : Z

Ω

q dx= 0

. Next, let the closed subset H ofY be given by

H={v∈Y : divv= 0, v·n|_∂Ω= 0}.

We denote the Stokes operator byA=−P∆, wherePis theL²-orthogonal projection of Y onto H.

It is well known [3] that

kvk₀ ≤c₁k∇vk₀, kvk_0,4 ≤c₂k∇vk₀, ∀v∈X(orW₀), (2)

where c (with or without a subscript) denotes a generic positive constant, which is independent of the mesh parameter h, but may depend on Ω and other parameters introduced in this paper. We define the continuous bilinear formsa(·,·) andd(·,·) onX×XandX×M and a trilinear form onX×X×X, respectively, by

a(u, v) =ν(∇u,∇v), d(v, q) = (q,divv), ∀v∈X, q ∈M, b(u;v, w) = ((u· ∇)v, w) +1

2((divu)v, w) = 1

2b1(u;v, w)−1

2b1(u;w, v),

∀u, v, w∈X,

where b1(u;v, w) = ((u· ∇)v, w).For fixedu, note that b(u;v, w) is the skew- symmetric part of b₁(u;v, w). We also define the continuous bilinear forms

¯

a(·,·) and the trilinear form ¯b(·;·,·) onW×W andX×W×W, respectively, by a(T, s) = (∇T,¯ ∇s),

¯b(u;T, s) = ((u· ∇)T, s) +1

2((divu)T, s) = 1

2¯b1(u;T, s)−1

2¯b1(u;s, T),

∀u∈X, T, s∈W.

The trilinear forms b(·;·,·) and ¯b(·;·,·) satisfy:

b(u;v, w) =−b(u;w, v), |b(u;v, w)| ≤Nk∇uk₀k∇vk₀k∇wk₀, (3)

∀u, v, w ∈X,

¯b(u;T, s) =−¯b(u;s, T), |¯b(u;T, s)| ≤N¯k∇uk₀k∇Tk₀k∇sk₀, (4)

∀u∈X, T, s∈W, whereN = sup

u,v,w∈X

|b(u,v,w)|

k∇uk0k∇vk0k∇wk0, and N¯ = sup

u∈X, T ,s∈W

|¯b(u,T ,s)|

k∇uk0k∇Tk0k∇sk0. With the above notations, the variational formulation of problem (1) reads as follows: find (u, p, T)∈(X, M, W) such that for all (v, q, s)∈(X, M, W0), (5)

(B((u, p); (v, q)) +b(u;u, v) =λ(jT, v),

¯

a(T, s) +λ¯b(u;T, s) = 0,

whereB((u, p); (v, q)) =a(u, v)−d(v, p) +d(u, q),which satisfies the continuity property and inf-sup condition.

Moreover, we need a further assumption on T₀ provided in [4].

(A). Assume that ∂Ω ∈ C^k,α(k ≥ 0, α > 0), then, for T0 ∈ C^k,α(∂Ω), there exists an extension of T0∈C₀^k,α(R²) (still marked as T0) such that

kT₀k_k,q ≤ε, k≥0, 1≤q ≤ ∞, whereε >0 is a sufficiently small constant.

The following existence and uniqueness of solution of (5) are recalled.

Theorem 2.1 ([4]). Assume that A = 2ν⁻¹λ(1 + 3c₁)kT₀k₁ and B = 2k∇T₀k₀+c⁻²₁ (1+3c1)kT₀k₁.Letδ1 andδ2 be two constants and 0< δ1, δ2≤1, such that

ν⁻¹N A≤1−δ1, δ₁⁻¹ν⁻¹λ²c²₁N B¯ ≤1−δ2.

Then, under the assumption of(A), there exists a unique solution(u, p, T)∈ (X, M, W),satisfying

k∇uk₀ ≤A, k∇Tk₀ ≤B.

3. STABILIZED OSEEN ITERATIVE FINITE ELEMENT METHOD

Let T_~ = {K} be a regular triangulation of Ω, indexed by a parameter

~= max

K∈T_~{~K :~K = diam(K)}, (~=Horh, H h).We consider the finite element spaces

X_~ = n

v_~ ∈X∩C⁰( ¯Ω)² :v_~|_K ∈P1(K)², ∀K ∈ T_~o , M_~ =n

q_~∈M∩C⁰( ¯Ω) :q_~|_K ∈P₁(K), ∀K∈ T_~o ,

W_~ =n

s_~ ∈W ∩C⁰( ¯Ω) :s_~|_K∈P₁(K), ∀K ∈ T_~o ,

whereP1(K) is the set of all polynomials onKof degree less than 1. We define W_0~ =W_~∩W₀.And the subspace V_~ of X_~ is given by

V_~={v_~∈X_~ :d(v_~, q_~) = 0, ∀q_~ ∈M_~}.

Let P_~ :Y →V_~ denote the L²-orthogonal projection defined by (P_~v, v_~) = (v, v_~), ∀v∈Y, v_~∈V_~.

Moreover, a discrete analogue A_~ = −P_~∆_~ of the Stokes operator A is defined through the condition that (−∆_~u_~, v_~) = (∇u_~,∇v_~) for allu_~, v_~ ∈ X_~.Besides, we have the following results [5]:

|b(u_~;v_~, w)|+|b(v_~;u_~, w)|+|b(w;u_~, v_~)| ≤c₃kA_~u_~k₀k∇v_~k₀kwk₀, (6)

|b(u_~;v_~, w)|+|b(v_~;u_~, w)|+|b(w;u_~, v_~)| ≤1

2c₄kA_~v_~k

1 2

0kv_~k

1 2

0k∇u_~k₀kwk₀ (7)

+1

2c₄kA_~v_~k

1 2

0k∇v_~k

1 2

0ku_~k

1 2

0k∇u_~k

1 2

0kwk₀,

for all u_~, v_~ ∈ V_~, w ∈ Y. Note that (X_~, M_~) does not satisfy the discrete inf-sup condition

sup

v∈X_~

d(v, q)

k∇vk₀ ≥β₁kqk₀, ∀q ∈M_~,

where the constantβ1>0 is independent of~. In order to fulfill this condition, a stabilized bilinear term [1] is used:

B_~((u_~, p_~); (v, q)) =a(u_~, v)−d(v, p_~) +d(u_~, q) +G_~(p_~, q), whereG_~(p_~, q) can be defined by

G_~(p_~, q) = (p_~−Π_~p_~, q−Π_~q), and Π_~ is aL²-projection operator, which is defined by

(p, q_~) = (Π_~p, q_~), ∀p∈L²(Ω), q_~ ∈R_~.

Here, R_~ ⊂L²(Ω) denotes the piecewise constant space associated with the triangulation T_~.

The corresponding discrete variational formulation of (5) for the conduction- convection equations is recast: find (u_~, p_~, T_~)∈(X_~, M_~, W_~) such that

(8)

(B_~((u_~, p_~); (v, q)) +b(u_~;u_~, v) =λ(jT_~, v), ∀(v, q)∈(X_~, M_~),

¯

a(T_~, s) +λ¯b(u_~;T_~, s) = 0, ∀s∈W_0~.

Theorem 3.1. For all (u_~, p_~),(v, q) ∈ (X_~, M_~), the bilinear form B_~((·,·); (·,·)) satisfies the continuity property and the weak coercivity property

|B_~((u_~, p_~); (v, q))| ≤c(k∇u_~k₀+kp_~k₀)(k∇vk₀+kqk₀), (9)

sup

(v,q)∈(X_~,M_~)

|B_h((u_~, p_~); (v, q))|

k∇vk₀+kqk₀ ≥β2(k∇u_~k₀+kp_~k₀), (10)

where β₂>0 is independent of~.

Theorem 3.2 ([4]). Let ε be a sufficiently small positive constant such that

25c²₁N λ

3ν² ε≤1, 32c²₁N λ¯ ²

3ν ε≤1, k∇T₀k₀ ≤ ε

4, kT₀k₀≤ c₁ε 4 . (11)

Then, under the assumption (A),(u_~, T_~) satisfies the following stability:

k∇u_~k₀ ≤ 5c²₁λ

3ν ε, k∇T_~k₀≤ε, kA_~u_~k₀≤

c1c²₄ 15N² + 2

c²₁λε ν . (12)

Algorithm 1. Two-level stabilized Oseen iterative scheme.

Step I:Find a global coarse grid iterative solution (uⁿ_H, pⁿ_H, T_Hⁿ)∈(XH,MH,WH) (n= 1,2, . . . , m) such that

(13)

(B_H((uⁿ_H, pⁿ_H); (v, q))+b(uⁿ⁻¹_H ;uⁿ_H, v) =λ(jT_Hⁿ, v), ∀(v, q)∈(X_H, M_H),

¯

a(T_Hⁿ, s)+λ¯b(uⁿ⁻¹_H ;T_Hⁿ, s) = 0, ∀s∈W0H.

Step II:Find a fine grid correction (umh, pmh, Tmh)∈(Xh, Mh, Wh) such that (14)

(Bh((umh, pmh); (v, q)) +b(u^m_H;u^m_H, v) =λ(jTmh, v), ∀(v, q)∈(Xh, Mh),

¯

a(T_mh, s) +λ¯b(u^m_H;T_mh, s) = 0, ∀s∈W_0h.

Remark 3.1. In this article, letu⁰_H = 0, T_H⁰ = 0 be the given initial guess.

In the given method, we findT_Hⁿ firstly, then we solve the momentum equation foruⁿ_H and pⁿ_H in the step I. In the step II, we also deal with the equations in a decoupling way as the same as the step I, which can save much CPU time.

4. STABILITY AND ERROR ESTIMATE

In this section, we derive the stability and iterative errors for the two-level stabilized Oseen iterative finite element methods for the stationary conduction- convection equations.

Theorem 4.1. Under the assumptions of Theorem 3.2, (uⁿ_H, T_Hⁿ) (n = 1,2, . . . , m),and (u_mh, T_mh) defined by (13) and (14), respectively, satisfies

k∇uⁿ_Hk₀≤ 5c²₁λ

3ν ε, k∇T_Hⁿk₀ ≤ε; k∇u_mhk₀ ≤ 5c²₁λ

3ν ε, k∇T_mhk₀ ≤ε.

(15)

Proof. Here, we only give the proof of the first two inequalities of (15).

The proof of the other two is similar, so we omit it.

It is obvious that the first two inequalities of (15) hold forn= 0.Assuming that they hold forn=J−1,we want to prove that they hold forn=J.Taking (v, q) = (u^J_H, p^J_H) ∈ (X_H, M_H) in the first equation of (13) with n = J and T_H^J =ω^J_H +T0, we get

νk∇u^J_Hk₀≤c²₁λk∇ω_H^Jk₀+c₁λkT₀k₀. (16)

Setting s=ω_H^J in the second equation of (13) withn=J and using (4), we obtain

k∇ω_H^Jk₀ ≤λN¯k∇u^J−1_H k₀k∇T₀k₀+k∇T₀k₀≤ 5c²₁λ²N¯

3ν εk∇T₀k₀+k∇T₀k₀ ≤ 3 4ε.

(17)

From the above inequality, we arrive at

k∇T_H^Jk₀ ≤ k∇ω_H^Jk₀+k∇T₀k₀ ≤ε.

(18)

Applying (17), we have

k∇u^J_Hk₀ ≤ 5c²₁λ 3ν ε.

Combining the above inequality and (18) shows that the first two inequal- ities of (15) are valid for n=J.

Theorem4.2. Setting(eⁿ, ηⁿ, ξⁿ) = (uH−uⁿ_H, pH−pⁿ_H, TH−T_Hⁿ). Under the assumptions of Theorem 3.2, it is valid that

k∇eⁿk₀ ≤ c²₁λ

3ν2ⁿ⁻³ε, kηⁿk₀ ≤ β₂⁻¹c²₁λ

2ⁿ⁻² ε, k∇ξⁿk₀ ≤ 1 2ⁿ⁺¹ε.

(19)

Proof. Subtracting (13) from (8), we have

B_H((eⁿ, ηⁿ); (v, q)) +b(eⁿ⁻¹;u_H, v) +b(uⁿ⁻¹_H ;eⁿ, v) =λ(jξⁿ, v), (20)

¯

a(ξⁿ, s) +λ¯b(eⁿ⁻¹;T_H, s) +λ¯b(uⁿ⁻¹_H ;ξⁿ, s) = 0.

(21)

Taking (v, q, s) = (eⁿ, ηⁿ, ξⁿ) in the above two equations, and using (2)–

(4) and Theorem 4.1, we arrive at

νk∇eⁿk₀ ≤Nk∇eⁿ⁻¹k₀k∇u_Hk₀+c²₁λk∇ξⁿk₀, (22)

k∇ξⁿk₀ ≤λN¯k∇eⁿ⁻¹k₀k∇T_Hk₀, (23)

for all n >1.Combining (23) with Theorem 3.2, (22) can be rewritten k∇eⁿk₀ ≤ 5N c²₁λε

3ν² k∇eⁿ⁻¹k₀+N c¯ ²₁λ²ε

ν k∇eⁿ⁻¹k₀≤ 1

2k∇eⁿ⁻¹k₀. (24)

For n = 1, using the given initial guess in Remark 3.1 and (23), and taking (v, q) = (e¹, η¹) in (20), we get

k∇e¹k₀ ≤ν⁻¹Nk∇u_Hk²₀+ν⁻¹c²₁λk∇ξ¹k₀ (25)

≤ν⁻¹N25c⁴₁λ²ε²

9ν² +ν⁻¹c²₁λ²N ε¯ 5c²₁λε

3ν ≤ 4c²₁λε 3ν ,

which together with (24) give k∇eⁿk₀ ≤ _3ν2^c21n−3^λ ε. Next, we deduce from (23) and (12) that

k∇ξⁿk₀ ≤λN ε¯ ·c²₁λε 3ν · 1

2ⁿ⁻⁴ ≤ ε 2ⁿ⁺¹. Furthermore, from (10) and (20), we obtain

kηⁿk₀ ≤β₂⁻¹N(k∇u_Hk₀k∇eⁿ⁻¹k₀+k∇uⁿ⁻¹_H k₀k∇eⁿk₀) +β⁻¹₂ c²₁λk∇ξⁿk₀ (26)

≤ β₂⁻¹c²₁λ 2ⁿ⁻² ε.

Thus, we finish the proof of the theorem.

Next, we consider the following error estimates in the error factor uⁿ_H − uⁿ⁻¹_H .

Theorem4.3. Under the assumptions of Theorem4.2,(uⁿ_H, pⁿ_H, T_Hⁿ) (n= 1,2, . . . , m) defined by(13) satisfies

n→∞lim(kuⁿ_H −uⁿ⁻¹_H k₀+k∇(uⁿ_H −uⁿ⁻¹_H )k₀) = 0, (27)

k∇eⁿk₀+kηⁿk₀+k∇ξⁿk₀≤ckuⁿ_H −uⁿ⁻¹_H k₀+ c 2ⁿ⁺¹. Proof. By using (2) and the triangle inequality, we obtain kuⁿ_H−uⁿ⁻¹_H k₀+k∇(uⁿ_H−uⁿ⁻¹_H )k₀ ≤(c1+ 1)k∇(uⁿ_H −uⁿ⁻¹_H )k₀

≤(c₁+ 1)(k∇eⁿk₀+k∇eⁿ⁻¹k₀).

Lettingn→ ∞in the above inequality and using Theorem 4.2, we arrive at the first equation of (27).

Next, takingv=eⁿ and q =ηⁿin (20), we get

a(eⁿ, eⁿ) +G_H(ηⁿ, ηⁿ) +b(eⁿ;u_H, eⁿ) +b(eⁿ⁻¹−eⁿ;u_H, eⁿ) =λ(jξⁿ, eⁿ).

(28)

Hence, we deduce from (3), (6) and Theorem 3.2 that (ν−5c²₁N λε

3ν )k∇eⁿk₀ ≤ c₃c²₁λε ν

c₁c²₄ 15N² + 2

kuⁿ_H −uⁿ⁻¹_H k₀+c²₁λ ε 2ⁿ⁺¹. Moreover, we get

k∇eⁿk₀≤ckuⁿ_H −uⁿ⁻¹_H k₀+ c 2ⁿ⁺¹. Also, using (23) and the first inequality in (26), we have

kηⁿk₀+k∇ξⁿk₀ ≤ckuⁿ_H−uⁿ⁻¹_H k₀+ c 2ⁿ⁺¹. Therefore, we complete the proof of the theorem.

Using a similar argument for Theorem 4.2, we easily have the following two theorems.

Theorem4.4. Let (e^m_hH, η^m_hH, ξ^m_hH) = (u_h−u^m_H, p_h−p^m_H, T_h−T_H^m).Under the assumptions of Theorem 3.2, it holds that

k∇e^m_hHk₀≤ c²₁λ

3ν2^m−3ε, kη^m_hHk₀ ≤ β₂⁻¹c²₁λ

2^m−2 ε, k∇ξ_hH^m k₀≤ 1 2^m+1ε.

(29)

Theorem 4.5. Let (e_h, η_h, ξ_h) = (u_h−u_mh, p_h−p_mh, T_h−T_mh). Under the assumptions of Theorem 3.2, it is valid that

k∇e_hk₀≤ 23c²₁λ

135ν2^m−3ε, kη_hk₀ ≤ 23β₂⁻¹c²₁λ

2^m−3135ε, k∇ξ_hk₀ ≤ 1 2^m−327ε.

(30)

5. NUMERICAL EXPERIMENTS

In this section, we present some numerical experiments with a physical model of square cavity stationary flow. In this problem, computations are carried out in the domain Ω = [0,1]×[0,1]. T = 0 on the left and bottom boundary,∂T /∂n= 0 on the top side,T = 4y(1−y) on the rest of the boundary and zero Dirichlet conditions on velocity are imposed. The mesh consists of triangular elements that are obtained by dividing Ω into sub-squares of equal size and then drawing the diagonal in each sub-square. Here, we set Grashoff number λ= 1.

To show the effective of Algorithm 1, in Table 1, we give the computa- tional time of Algorithm 1 without decouple and the one-level stabilized Oseen finite element method with decouple. For different viscosities, we take the fixed tolerance 10⁻⁶ in these methods. From Table 1, we can see that our algorithm takes the least CPU time.

Table 1

CPU time for differentν

methods H h ν CPU-time

Algorithm 1 ₃₀¹ ₅₀¹ ₁₀₀₀¹ 15.109

Algorithm 1 without decouple ₃₀¹ ₅₀¹ 67.532

One-level stabilized Oseen ₄₀¹ 45.719

Algorithm 1 ₆₀¹ ₁₀₀¹ ₂₀₀₀¹ 209.969

Algorithm 1 without decouple ₆₀¹ ₁₀₀¹ 429.204

One-level stabilized Oseen ₈₀¹ 473.859

6.CONCLUSIONS

In this work, we have presented and analyzed a two-level stabilized Oseen iterative finite element method based on decouple technique for solving the stationary conduction-convection equations. Numerical results illustrated the high efficiency of the proposed method.

Acknowledgments. The author is greatful to the editor and the anonymous referees for their helpful work on the manuscript. This work was supported by the NSF of Xinjiang Province (Grant No. 2013211B01).

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[3] V. Girault and P.A. Raviart,Finite element method for Navier–Stokes equations: theory and algorithms. Springer, Berlin, 1987.

[4] Z.D. Luo, Theory Bases and Applications of Finite Element Mixed Methods. Science Press, Beijing, 2006. (in Chinese)

[5] Y.N. He and J. Li, Convergence of three iterative methods based on the finite element discretization for the stationary Navier–Stokes equations. Comput. Methods Appl. Mech.

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Received 3 March 2012 Xinjiang University,

College of Mathematics and System Sciences, Urumqi 830046,

P.R. China hpzh007@yahoo.cn