A STABILIZED NONCONFORMING QUADRILATERAL FINITE ELEMENT METHOD FOR THE GENERALIZED STOKES EQUATIONS

Transcription

1 INTERNATIONAL JOURNAL OF NUMERICAL ANALYSIS AND MODELING Volume 9, Number 2, Pages c 2012 Institute for Scientific Computing and Information A STABILIZED NONCONFORMING QUADRILATERAL FINITE ELEMENT METHOD FOR THE GENERALIZED STOKES EQUATIONS ZHEN WANG, ZHANGXIN CHEN, AND JIAN LI Abstract. In this paper, we study a local stabilized nonconforming finite element method for the generalized Stokes equations. This nonconforming method is based on two local Gauss integrals, and uses the equal order pairs of mixed finite elements on quadrilaterals. Optimal order error estimates are obtained for velocity and pressure. Numerical experiments performed agree with the theoretical results. Key words. Generalized Stokes equations, nonconforming quadrilateral finite elements, optimal error estimates, inf-sup condition, numerical experiments, stability 1. Introduction Much attention has recently been attracted to using the equal order finite element pairs (e.g., P 1 P 1 the linear function pair and Q 1 Q 1 the bilinear function pair) for the fluid mechanics equations, particularly for the Stokes and Navier-Stokes equations [1, 10, 11, 12]. While they do not satisfy the inf-sup stability condition, these element pairs offer simple and practical uniform data structure and adequate accuracy. Many stabilization techniques have been proposed to stabilize them such as penalty [7, 8], pressure projection [1, 10], and residual [15] stabilization methods. Among these methods, the pressure projection stabilization method is a preferable choice in that it is free of stabilization parameters, does not require any calculation of high-order derivatives or edge-based data structures, and can be implemented at the element level. As formulated in [1, 10, 11, 14], it is based on two local Gauss integrals. Nonconforming finite elements [4] are popular for the discretization of partial differential equations since they are simple and have small support sets of basis functions. These elements on triangles have been studied in the context of the pressure projection stabilization method [9]. However, due to a technical reason, the nonconforming finite elements on quadrilaterals have not been studied for this stabilization method. In this paper, an argument is introduced to study this class of nonconforming finite elements for the stabilization method of the generalized Stokes equations. As examples, the nonconforming rotated element span{1,x,y,x 2 y 2 } [3, 13] and the element span{1,x,y,((3x 2 5x 4 ) (3y 2 5y 4 ))} proposed by Douglas et al. [6] will be analyzed. After a stability condition is proven for the pressure projection stabilization method, optimal order error estimates are obtained for velocity and pressure. Numerical experiments will be performed to check the theoretical results derived. Received by the editors January 1, 2011 and, in revised form, March 1, Mathematics Subject Classification. 35Q10, 65N30, 76D05. This research was supported in part by the NSF of China (No and ), Natural Science Basic Research Plan in Shaanxi Province of China (Program No. SJ08A14) and NSERC/AERI/Foundation CMG Chair and icore Chair Funds in Reservoir Simulation. 449

2 450 Z. WANG, Z. CHEN, AND J. LI An outline of this paper is given as follows: In the second section, we introduce some basic notation and the generalized Stokes equations. Then, in the third section, the nonconforming quadrilateral finite elements and the local stabilization method are given. In the fourth section, a stability result is shown. Optimal order error estimates are derived in the fifth section. Finally, numerical experiments are presented in the sixth section. 2. Preliminaries We consider the following generalized Stokes problem: σu ν u+ p = f in Ω, (2.1) u = 0 in Ω, u = 0 on Ω, where Ω represents a polygonal convex domain in R 2 with a Lipschitz-continuous boundary Ω, u(x) = (u 1 (x),u 2 (x)) the velocity vector, p(x) the pressure, f(x) the prescribed force, ν > 0 the viscosity, and σ 0 a nonnegative real number. For a time dependent problem, for example, σ can represent a time step. To introduce a weak formulation of (2.1), set X = (H0(Ω)) 1 2, Y = (L 2 (Ω)) 2, { } M = q L 2 (Ω) : qdx = 0. Below the standard notation is used for the Sobolev space W m,r (Ω), with the norm m,r and the seminorm m,r, m,r 0. We will write H m (Ω) for W m,2 (Ω) and m for m,2 when r = 2. The spaces (L 2 (Ω)) m, m = 1,2,4, are endowed with the L 2 (Ω)-scalar product (, ) and L 2 (Ω)-norm 0, respectively, as appropriate. Also, the space X is equipped with the scalar product ( u, v) and the norm u 1, u,v X. Because of the norm equivalence between 1 and 1 on X, we sometimes use the same notation for them. We define the continuous bilinear forms: a(u,v) = σ(u,v)+ν( u, v) u, v X, d(v,p) = ( v,p) v X, p M. Now, the variational formulation of problem (2.1) is to find a pair (u,p) X M such that (2.2) B((u,p),(v,q)) = (f,v) (v,q) X M, Ω where B((u,p),(v,q)) = a(u,v) d(v,p) d(u,q). The bilinear form d(, ) satisfies the inf-sup condition [4]: d(v, q) sup β q 0, q M, 0 v X v 1 where β is a positive constant depending only on the domain Ω.

3 A STABILIZED NONCONFORMING QUADRILATERAL FINITE ELEMENT Nonconforming Quadrilateral Finite Elements Let K h be a quasi-regular partition of Ω into convex quadrilaterals. Set Ω = K j, Γ j = Ω K j, Γ jk = Γ kj = K j K k, K j K h. j We denote the centers of Γ j and Γ jk by c j and c jk, respectively. Let K be the reference square [ 1,1] [ 1,1] in the (ξ,η)-plane. On this reference element, we define the nonconforming rotated element [3, 13] (3.1) X( K) = span{1,ξ,η,ξ 2 η 2 } or the element [6] (3.2) X( K) = span{1,ξ,η,((3ξ 2 5ξ 4 ) (3η 2 5η 4 ))}. For every convex quadrilateral K K h with vertices (x i,y i ), i = 1,2,3,4, there is a unique bilinear mapping F K : K K: (x,y) = F K (ξ,η) = ( ( FK 1 ) 4 ) 4,F2 K = x i φ i, y i φ i, i=1 i=1 where φ 1 = 1 4 (1 ξ)(1 η), φ 2 = 1 4 (1+ξ)(1 η), φ 3 = 1 4 (1+ξ)(1+η), φ 4 = 1 4 (1 ξ)(1+η). Then we define the element on the quadrilateral K K h : X(K) = {v : v = v F 1 K, v X( K)}, where X( K) is defined by either (3.1) or (3.2). With the above notation, we construct the following velocity-pressure finite element spaces: { } X h = v Y : v Kj X(K j ) X(K j ), [v] ds = 0, v ds = 0 j,k, Γ jk Γ j M h = { q M : q Kj Q 1 (K j ) j }, where [v] = v Γjk v Γkj denotes the jump of the function v across the interface Γ jk. When X( K) is defined by (3.2), the space X h can equivalently be defined as follows [6]: X h = { v Y : v Kj X(K j ) X(K j ), v(c jk ) = v(c kj ), v(c j ) = 0 j,k }. Define the energy norm v 1,h = j v 2 1,K j + j v 2 0,K j 1/2, v X h. The two finite element spaces X h and M h satisfy the approximation property: For (v,q) (H 2 (Ω) X) (H 1 (Ω) M), there are two approximations v I X h and q I M h such that (3.3) v v I 0 +h( v v I 1,h + q q I 0 ) Ch 2 ( v 2 + q 1 ), where (and below) C > 0 is a generic constant independent of the mesh size h.

4 452 Z. WANG, Z. CHEN, AND J. LI Set (, ) j = (, ) Kj,, j = (, ) Kj, 0,j = 0,Kj, and 0,j = 0,Kj. Then the discrete bilinear forms are given as follows: a h (u,v) = σ j (u,v) j +ν j ( u, v) j, u, v X h, d h (v,q) = j ( v,q) j. v X h, q M h. For the nonconforming vector space X h, we define the local operator Π j : H 1 (K j ) X(K j ) by (3.4) (v Π j v) ds = 0. K j This local operator satisfies [4]: (3.5) (3.6) v Π j v 1,j Ch i v i+1,j, v H i+1 (K j ), i = 0,1, Π j v 1,j C v 1,j, v H 1 (K j ). The global operator Π h : X X h is now defined by Π h v j = Π j v v X. The operator Π h has the following properties: (3.7) d h (v Π h v,q h ) = 0 q h W h, Π h v 1,h C v 1 v X, where W h L 2 (Ω) denotes the piecewise constant space associated with K h. As a result, the discrete inf-sup condition holds [4]: d h (v,q) sup β q 0, q W h, 0 v X h v 1,h where β > 0 is independent of h. As noted, the X h M h pair does not satisfy the inf-sup condition. However, following [9], we can add a simple local, effective stabilization term G h (, ): G h (p,q) = { } pq dx pq dx, p,q L 2 (Ω), K j,2 K j,1 K j K h where K j,i pq dx indicates an appropriate Gauss integral over K j that is exact for polynomials of degree i (i = 1,2), and pq is a polynomial of degree not greater than two. Thus, for all test functions q M h, the trial function p M h must be piecewiseconstantwheni = 1. Consequently, wedefinethe L 2 projectionoperator π h : L 2 (Ω) W h by (3.8) (p,q h ) = (π h p,q h ) p L 2 (Ω), q h W h. The projection operator π h has the following properties [9]: (3.9) and (3.10) π h p 0 C p 0 p π h p 0 Ch p 1 p L 2 (Ω), p H 1 (Ω). Now, using (3.8 ), we can define the bilinear form G h (, ) as follows: (3.11) G h (p,q) = (p π h p,q) = (p π h p,q π h q).

5 A STABILIZED NONCONFORMING QUADRILATERAL FINITE ELEMENT 453 Finally, the nonconforming finite element approximation of problem (2.1) is to find a pair (u h,p h ) (X h,m h ) such that (3.12) where B h ((u h,p h ),(v h,q h )) = (f,v h ) (v h,p h ) (X h,m h ), B h ((u h,p h ),(v h,q h )) = a h (u h,v h ) d h (v h,p h ) d h (u h,q h ) G h (p h,q h ) is a bilinear form defined on (X h,m h ) (X h,m h ). In the subsequent two sections we will establish stability and convergence results for method (3.12). 4. Stability The stability result will come from the next theorem. Theorem 4.1. The bilinear form B h ((, ),(, )) satisfies the continuous property (4.1) B h ((u h,p h ),(v h,q h )) C( u h 1,h + p h 0 )( v h 1,h + q h 0 ) (u h,p h ), (v h,q h ) (X h,m h ), and the coercive property (4.2) B h ((u h,p h ),(v h,q h )) sup β( u h 1,h + p h 0 ) 0 (v h,q h ) (X h,m h ) v h 1,h + q h 0 (u h,p h ) (X h,m h ), where β is a positive constant depending only on Ω. Proof. By the continuous property of the bilinear forms a(, ), d(, ), and G h (, ), we can easily obtain the continuous property of B h ((, ),(, )) so it suffices to show the coercive property. For all p h L 2 (K j ), there exists w ( H 1 (K j ) ) 2 such that [4] ( w,p h ) j = p h 2 0,j, w 1,j C 0 p h 0,j. Setting w h = Π h w, we see that (4.3) w h 1,h C 1 p h 0. Choosing (v h,q h ) = (u h ǫw h, p h ) for some constant ǫ > 0 yet to be determined, we have (4.4) From (4.3) it follows that (4.5) B h ((u h,p h ),(u h ǫw h,p h )) = a h (u h,u h ) ǫa h (u h,w h )+ǫd h (w h,p h )+G h (p h,p h ). ǫa h (u h,w h ) max{σ,ν} 2γ ǫc 2 u h 2 1,h + max{σ,ν}γ ǫc 3 p h 2 0 2, where γ > 0 is another constant to be determined. Note that, by (3.7), and, by (3.7) and (4.3), p h 2 0 = (p h, w) = (p h π h p h, w)+(π h p h, w) = (p h π h p h, w)+(π h p h, w h ) = (p h π h p h, (w w h ))+(p h, w h ), (p h π h p h, (w w h )) p h π h p h 0 (w w h ) 0 C 4 G 1/2 h (p h,p h ) p h p h C2 4G h (p h,p h ).

6 454 Z. WANG, Z. CHEN, AND J. LI Consequently, we obtain 1 (4.6) 2 p h 2 0 C 5G h (p h,p h )+(p h, w h ). Combining (4.4) (4.6) gives Choosing B h ((u h,p h ),(u h ǫw h,p h )) min{σ,ν} u h 2 1,h max{σ,ν} ǫc 2 u h 2 1,h 2γ max{σ,ν}γ ǫc 3 p h ǫ p h 2 0 C 5 ǫg(p h,p h )+G(p h,p h ) γ = we see that (4.7) ( 2 min{σ,ν} max{σ,ν} ǫc 2 ) u h 2 1,h 2γ ǫ(1 max{σ,ν}γc 3) p h 2 0 +(1 C 5ǫ)G(p h,p h ). 1 2max{σ,ν}C 3, { min{σ, ν} ǫ = min 2max{σ,ν} 2, C 2 C 3 1 2C 5 B h ((u h,p h ),(u h ǫw h,p h )) C 6 ( u h 1,h + p h 0 ) 2. Clearly, using (4.3) and the triangle inequality, we have (4.8) u h ǫw h 1,h + p h 0 C 7 ( u h 1,h + p h 0 ). Finally, setting β = C 6 /C 7, combining (4.7) and (4.8) yields (4.2). [] 5. Error Estimates The next lemma will be used [2]. Lemma 5.1. For any φ, w X X h, w (5.1),φ j j j Ch w 2 φ 1,h, w X (H 2 (Ω)) 2, (5.2) q,φ n j j Ch q 1 φ 1,h, q H 1 (Ω). Set j B h ((u,p),(v,q)) = B h ((u,p),(v,q))+g h (p,q). We introduce the projection operators (R h,q h ) : (X,M) (X h,m h ) by (5.3) B h ((R h (v,q),q h (v,q)),(v h,q h )) = B h ((v,q),(v h,q h )) (v h,q h ) (X h,m h ). which is well defined by Theorem 4.1 and satisfies the next approximation property, whose proof follows [9]. Lemma 5.2. It holds that, for (v,q) ( X (H 2 (Ω)) 2,M H 1 (Ω) ), (5.4) }, v R h (v,q) 0 +h( v R h (v,q) 1,h + q Q h (v,q) 0 ) Ch 2 ( v 2 + q 1 ).

7 A STABILIZED NONCONFORMING QUADRILATERAL FINITE ELEMENT 455 (5.5) Proof. Using (5.3), we see that B h ((v R h (v,q),q Q h (v,q)),(v h,q h )) = G h (q,q h ) (v h,q h ) (X h,m h ). Setting E = v Π h v and (w,r) = (Π h v R h (v,q),q I Q h (v,q)), where q I is the interpolation of q in M h, we have (5.6) B h ((w,r),(v h,q h )) = B h ((E,q q I )),(v h,q h )) G h (q,q h ) (v h,q h ) (X h,m h ). From Theorem 4.1 and the continuous property of G h (, ), we obtain (5.7) β( w 1,h + r 0 ) B h ((w,r),(v h,q h )) sup, (v h,q h ) (X h,m h ) v h 1,h + q h 0 G h (q,q h ) Ch q 1 ( v h 1,h + q h 0 ). From (3.3), (3.5), and the continuous property of B h (, ), it follows that B h ((E,q q I )),(v h,q h )) C( E 1,h + q q I 0 )( v h 1,h + q h 0 ) (5.8) Ch( v 2 + q 1 )( v h 1,h + q h 0 ). Combining (5.6) (5.8) gives (5.9) Therefore, we obtain (5.10) w 1,h + r 0 Ch( v 2 + q 1 ). v R h (v,q) 1,h + q Q h (v,q) 0 ( v Π h v 1,h + w 1,h )+( q q I 0 + r 0 ) Ch( v 2 + q 1 ). Next, we consider the following dual problem, with (ζ,τ) = (v R h (v,q),q Q h (v,q)): σφ ν Φ+ Ψ = ζ in Ω, (5.11) Φ = 0 in Ω, Φ = 0 on Ω. Because of the convexity of the domain Ω, the solution of this problem satisfies the regularity property: (5.12) Φ 2 + Ψ 1 c ζ 0. Multiplying the first and second equations of (5.11) by ζ and τ, respectively, integrating the resulting equations over Ω, and using (5.5) with (v h,q h ) = (Π h Φ,Ψ I ), we see that ζ 2 0 = a h(ζ,φ) d h (ζ,ψ) d h (Φ,τ) ν Φ,ζ + ζ n j,ψ j (5.13) = a h (ζ,φ Π h Φ) d h (ζ,ψ Ψ I ) d h (Φ Π h Φ,τ) G h (Ψ Ψ I,τ) +G h (Ψ,τ) G h (q,ψ I ) ν Φ,ζ + ζ n j,ψ j. From the continuous property of the bilinear terms a(, ), d(, ), and G h (, ), the approximation properties (3.3) and (3.5), the estimate (5.10), and the regularity

8 456 Z. WANG, Z. CHEN, AND J. LI (5.12), we have a h (ζ,φ Π h Φ) d h (ζ,ψ Ψ I ) d h (Φ Π h Φ,τ) G h (Ψ Ψ I,τ) (5.14) C( ζ 1,h + τ 0 )( Φ ΠΦ 1,h + Ψ Ψ I 0 ) Ch 2 ( v 2 + q 1 ) ζ 0, and (5.15) G h (Ψ,τ) G h (q,ψ I ) Ch Ψ 1 τ 0 +Ch 2 q 1 Ψ 1 Ch 2 ( v 2 + q 1 ) ζ 0. In addition, it follows from Lemma 5.1, (5.10), and (5.12) that Φ ν,ζ (5.16) ζ n j,ψ j Ch2 ( v 2 + q 1 ) ζ 0. Finally, combining (5.13) (5.16) gives (5.4). [] We are now in a position to obtain the error estimates for method (3.12). Theorem 5.3. Let (u,p) and (u h,p h ) be the respective solutions of (2.1) and (3.12). Then (5.17) u u h 1,h + p p h 0 Ch( u 2 + p 1 ). Proof: Let (ζ,τ) = (R h (u,p) u h,q h (u,p) p h ). Using (5.3) and a similar argument as for (5.15) and (5.16), we see that B h ((ζ,τ),(v h,q h )) = B h ((u,p),(v h,q h )) B h ((u h,p h ),(v h,q h )) = ν u,v h v h n j,p j Ch( u 2 + p 1 ) v h 1,h. Consequently, by the triangle inequality, (4.1), (4.2), and Lemma 5.2, we obtain { u u h 1,h + p p h 0 C u R h (u,p) 1,h + p Q h (u,p) 0 (5.18) + 1 } B h ((ζ,τ),(v h,q h )) sup β (v h,q h ) (X h,m h ) v h 1,h + q h 0 Ch( u h 2 + p h 0 ), which implies the desired result (5.17). [] Theorem 5.4. Let (u,p) and (u h,p h ) be the respective solutions of (2.1) and (3.12). Then (5.19) u u h 0 Ch 2 ( u 2 + p 1 ). Proof. AsintheproofofLemma5.2,setting(ζ,τ) = (u u h,p p h ), multiplying the first and second equations of (5.11) by ζ and τ, respectively, and integrating the resulting equations over Ω, we see that (5.20) ζ 2 0 = a h (ζ,φ) d h (ζ,ψ) d h (Φ,τ) ν Φ,ζ + ζ n j,ψ j.

9 A STABILIZED NONCONFORMING QUADRILATERAL FINITE ELEMENT 457 Using (2.1) and (3.12) with (v h,q h ) = (Π h Φ,Ψ I ), it follows that ζ 2 0 = a h (ζ,φ Π h Φ) d h (ζ,ψ Ψ I ) d h (Φ Π h Φ,τ) G h (Ψ Ψ I,τ) +G h (Ψ,τ) G h (p,ψ I ) ν Φ,ζ + ζ n j,ψ j (5.21) +ν u,π h Φ Π h Φ n j,p j. Applying an analogous argument as for (5.14) (5.16), we have (5.22) a h (ζ,φ Π h Φ) d h (ζ,ψ Ψ I ) d h (Φ Π h Φ,τ) G h (Ψ Ψ I,τ) Ch 2 ( u 2 + p 1 ) ζ 0, G h (Ψ,τ) G h (p,ψ I ) Ch 2 ( u 2 + p 1 ) ζ 0, Φ ν,ζ ζ n j,ψ j Ch2 ( u 2 + p 1 ) ζ 0. Note that, by the regularity of the solution u and p, ν u,π h Φ j j j Π h Φ n j,p j = ν u,π h Φ Φ j j j j j (Π h Φ Φ) n j,p j. As a consequence, it follows from Lemma 5.1, (3.5), and (5.12) that (5.23) u ν,π h Φ Π h Φ n j,p j j j Ch( u 2 + p 1 ) Π h Φ Φ 1,h j j Ch 2 ( u 2 + p 1 ) ζ 0. Finally, combining (5.20) (5.23) yields (5.19). [] 6. Numerical Experiments In this section we present numerical experiments to check the numerical theory developed in the previous sections. In all the experiments, Ω = (0,1) 2 is the unit square in R 2, the exact solution for the velocity u = (u 1,u 2 ) and the pressure p is given as follows: (6.1) p = cos(πx 1 )cos(πx 2 ), u 1 = 2πsin 2 (πx 1 )sin(πx 2 )cos(πx 1 ), u 2 = 2πsin(πx 1 )sin 2 (πx 2 )cos(πx 1 ), and the right-hand side f(x) is determined by (2.1) through this exact solution. Furthermore, the nonconforming finite element given in (3.2) is used. When σ = 0 and ν = 0.1, the convergence results are given in Table 1. We now fix ν = 0.1 and vary σ = 0.1,1,10,100; the results are given in Table 2. These convergence results fully agree with the theoretical results we have obtained in the previous two sections. Table 1. Convergence results.

10 458 Z. WANG, Z. CHEN, AND J. LI 1/h u u h 0 u 0 u u h 1 u 1 p p h 0 p 0 u L 2rate u H 1rate p L 2rate Table 2. Convergence results for different σ σ u L 2rate u H 1rate p L 2rate u L 2rate u H 1rate p L 2rate References [1] P. Bochev and C. R. Dohrmann, A computational study of stabilized low-order C 0 finite element approximations of Darcy equations, Comput. Mech. 38 (2006), [2] Z. Cai, J. Douglas, Jr., and X. Ye, A stable nonconforming quadrilateral finite element method for the stationary Stokes and Navier-Stokes equations, Calcolo 36 (1999), [3] Z. Chen, Projection finite element methods for semiconductor device equations, Computers Math. Applic. 25 (1993), [4] Z. Chen, Finite Element Methods and Their Applications, Springer-Verlag, Heidelberg and New York, [5] Z. Chen, G. Huan, and Y. Ma, Computational Methods for Multiphase Flows in Porous Media, in the Computational Science and Engineering Series, Vol. 2, SIAM, Philadelphia, [6] J. Douglas, Jr., J. E. Santos, D. Sheen, and X. Ye, Nonconforming Galenkin methods based on quadrilateral elements for second order elliptic problems, RAIRA Modél. Math. Anal. Numér 33 (1999), [7] R. Falk, An analysis of the penalty method and extrapolation for the stationary Stokes equations, in Advances in Computer Methods for Partial Differential Equations, R. Vichnevetsky, ed., AICA, 1975, [8] T. J. R. Hughes, W. Liu, and A. Brooks, Finite element analysis of incompressible viscous flows by the penalty function formulation, J. Comput. Phys. 30 (1979), [9] J. Li and Z. Chen, A new local stabilized nonconforming finite element method for the Stokes equations, Computing 82 (2008), [10] J. Li and Y. He, A new stabilized finite element method based on local Gauss integration techniques for the Stokes equations, J. Comp. Appl. Math. 214 (2008), [11] J. Li, Y. He, and Z. Chen, A new stabilized finite element method based on a local Gauss integration technique for the transient Navier-Stokes equations, Comp. Meth. Appl. Mech. Eng. 197 (2007), [12] J. Li, Y. He, and Z. Chen, Performance of several stabilized finite element methods for the Stokes equations based on the lowest equal-order pairs, Computing 86 (2009), [13] R. Rannacher and S. Turek, Simple nonconforming quadrilateral Stokes element, Numer. Methods for Partial Differential Equations 8 (1992), [14] L. Shen, J. Li, and Z. Chen, Analysis of a stabilized finite volume method for the transient Stokes equations, International Journal of Numerical Analysis and Modeling 6 (2009), [15] D. Silvester, Stabilized mixed finite element methods, Numerical Analysis Report, No. 262, 1995.

11 A STABILIZED NONCONFORMING QUADRILATERAL FINITE ELEMENT 459 Center for Computational Geoscienes, Faculty of Science, Xi an Jiaotong University, Xi an,710049, P.R. China. Center for Computational Geoscienes and Mathematics, Faculty of Science, Xi an Jiaotong University, Xi an,710049, P.R. China. Department of Chemical & Petroleum Engineering, Schulich School of Engineering, University of Calgary, 2500 University Drive N.W. Calgary, Alberta, Canada T2N 1N4. College of Mathematics and Information Science, Shaanxi Normal University, Xi an , P.R. China. The Department of Mathematics, Baoji University of Arts and Sciences, Baoji , P.R. China.