基于带约束非协调旋转Q_1元的有限体积元法
|
摘要
本文从带约束非协调旋转Q1元(即CN RQ1元)出发,先后构造了求解Stokesl司题的CNRQ1-P0元有限体积元法和求解平面弹性问题的CNRQ1元有限体积元法.
对于求解Stokes问题的CNRQ1-P0元有限体积元格式,我们选取了原始剖分上的带约束的非协调旋转Q1元为速度的试探函数空间(Uh),对偶剖分上的分片常数空间为速度的检验函数空间(Vh),宏单元上的分片常数为压力的近似函数空间(Ph).在分析了此方法的正定性和稳定性之后,我们证明了速度的H1误差和L2误差估计,压力的L2误差以及速度的H1误差和L2误差的超收敛性.同时给出了支持理论结果的数值算例.
对于求解平面弹性问题的CNRQ1元有限体积元法,我们选取了原始剖分上的带约束的非协调旋转Q1元为位移的试探函数空间(Uh),对偶剖分上的分片常数空间为位移的检验函数空间(Vh).同时我们证明了位移向量的能量模误差,以及能量模误差的超收敛性.最后,我们针对不同的Lame参数λ给出了数值算例结果,验证了我们理论的可靠性.
Finite volume method (FVM) is an important numerical discretization technique for solving partial differential equations, especially for those arising from physical conserva-tion laws including mass, momentum and energy. In general, FVM has both simplic-ity in implementation and local conservation property, so it has enjoyed great popular-ity in many fields, such as computational fluid dynamics, computational aerodynamics, petroleum engineering and so on. About some recent development of FVM, readers can refer to the monographs [46,49,61,79,88,92,93,112,118] for details.
In2003, Park and Sheen [98] have introduced a P1-nonconforming quadrilateral element which has only three degrees of freedom on each quadrilateral. Later, Man and Shi [93] proposed the P1-nonconforming quadrilateral FVM for the elliptic problem by using a dual partition of overlapping type. Following the line of the finite element in [98], In2005, Hu and Shi [73] presented a constrained nonconforming rotated Q1(CNRQ1) element and applied it to the second order elliptic problem. In [73], the authors also point out that the CNRQ1element and the P1-nonconforming element are equivalent on a rectangle since the constraint and the continuity are the same, however, the two elements are different on a general quadrilateral. Afterwards, Hu, Man and Shi [72] and Mao and Chen [94] investigated and analyzed the CNRQ1-P0finite element method for the Stokes problem on rectangular meshes. The application of the CNRQ1element to the nearly incompressible planar elasticity problem can be referred to [72,95]. Meanwhile, Mao and Chen [94] and Liu and Yan [90] discussed the supconvergence of the finite element scheme for the Stokes problem on rectangular meshes.
The main work of this paper is as follows:
1. Finite volume element method for the Stokes problems.
Let Ω be a bounded, convex and open polygon of R2with boundary (?)Q. We consider the following Stokes equations with the homogeneous Dirichlet boundary condition-△u+▽p=f, in Ω (9) divu=0, in Ω,(10) u=0, on (?)Ω,(11) where u=(u1,u2) represents the velocity vector, p is the pressure, and f indicates a prescribed body force.
Let Th={K} which is called primal partition be a strictly convex and quadrilateral partition of Ω. The dual partition is nonoverlapping barycenter quadrilateral partition. Choose CNRQ1function space Uh as the trial function space and the piecewise constant space Vh as the test function space for the velocity. For the pressure, we select the piecewise constant space(Ph) on the macro-elements as discrete space. The FVEM for the Stokes problem (2.1)-(2.3) investigated in this paper is:find a pair (uh,ph)∈Uh×Ph, such that ah(uh,vh)+bh(vh,ph)=(f,vh),(?)vh∈Vh,(12) b'h(uh,qh)=0,(?)qh∈Ph.(13)
(1). Stability.
Suppose the primal partition is quasi-uniform and regular. Each quadrilateral in Th satisfies quasi-parallel quadrilateral condition. Then, we obtain the following results:
Lemma1. For sufficiently small h, there exists a constant C0>0, we have ah(uh,Γhuh)≥C0|uh|1,h2,(?)uh∈Uh.
Lemma2. The pair (Uh,Ph) satisfies an uniform inf-sup condition, i.e., there holds
(2). Error estimates.
We prove the errors in the H1norm and L2norm for the velocity, the error in the L2norm for the pressure, and the superconvergence of the velocity in the broken H1norm and the pressure in the L2norm..
Theorem1. Let the pair (u,p)∈(H01(Ω)∩H2(Ω))2×(L02(Ω)∩H1(Ω)) be the solution of (9)-(11); and (uh,Ph)∈Gh×be the solution of (12)-(13). There exists a positive constant C independent of h such that|u-uh|1,h+||p-Ph||0≤Ch(||u||2+||p||1).
Theorem2. Let (u,p)∈(H01(Ω)∩H2(Ω))2×(L02(Ω)∩H1(Ω)) and (uh,ph)∈Uh×P be the solutions of (9)-(11) and (12)-(13), respectively. If f∈H1(Ω)2, then||u-uh|0≤Ch2(||u||2+||p||1+||f||1).
Theorem3. Let (u,p)∈(H01(Ω)∩H3(Ω))2×(L02(Ω)∩H1(Ω)) and (uh,ph)∈Uh×Ph be the solutions of (9)-(11) and (12)-(13), respectively. There exists a positive constant C independent of h such that|Πhu-uh|1,h+||Jhp-Ph||0≤Ch2(||u||3+||p||2).(15)
Let Π2hvh∈Q2(T) and J2/1be the postprocessing interpolation operators: where ψi,i=1,2,…,8be the associated basis functions of the biquadratic element space Q2(τ), Q1(τ) is the bilinear function on the element τ. Then we have:
Theorem4. Suppose all condition of the thmm above are valid,|Π2huh-u|1+||J2hPh-p||o≤Ch2(||u||3+||p||2).
1. Finite volume element method for the linear elasticity problem.
Consider the pure displacement boundary value problem for the equations of plain strain linear isotropic elasticity: where u denotes the displacements, f the body forces and μ,λ are Lame parameters.
Let Th={K} which is called primal partition be a strictly convex and quadrilateral partition of Ω. The dual partition is nonoverlapping barycenter quadrilateral partition. Choose CNRQ1function space Uh as the trial function space and the piecewise constant space Vh as the test function space for the displacements. The FVEM for the linear elasticity problem is:
Suppose the primal partition is quasi-uniform and regular. Each quadrilateral in Th satisfies quasi-parallel quadrilateral condition.Then,we can obtain the following results to prove the FVEM is locking-free.
(1).Error estimate in energy norm. Theorem5.Suppose Uh∈Uh and u∈H01(Ь)∩H2(Ω) are solutions of finite uolume element scheme(19)and the equations(18),respectively,then||u-uh||h≤Ch||f||0Ω
(2).Superconvergence in the energy norm.
Theorem6.Suppose Uh∈Uh and u∈H01(Ω)∩H3(Ω) are solutions of finite uolume element scheme(19)and the equations(18),respectively,then||Πu-uh||h≤Ch2(||u||3,Ω+λ||divu||2,Ω).
对于求解Stokes问题的CNRQ1-P0元有限体积元格式,我们选取了原始剖分上的带约束的非协调旋转Q1元为速度的试探函数空间(Uh),对偶剖分上的分片常数空间为速度的检验函数空间(Vh),宏单元上的分片常数为压力的近似函数空间(Ph).在分析了此方法的正定性和稳定性之后,我们证明了速度的H1误差和L2误差估计,压力的L2误差以及速度的H1误差和L2误差的超收敛性.同时给出了支持理论结果的数值算例.
对于求解平面弹性问题的CNRQ1元有限体积元法,我们选取了原始剖分上的带约束的非协调旋转Q1元为位移的试探函数空间(Uh),对偶剖分上的分片常数空间为位移的检验函数空间(Vh).同时我们证明了位移向量的能量模误差,以及能量模误差的超收敛性.最后,我们针对不同的Lame参数λ给出了数值算例结果,验证了我们理论的可靠性.
Finite volume method (FVM) is an important numerical discretization technique for solving partial differential equations, especially for those arising from physical conserva-tion laws including mass, momentum and energy. In general, FVM has both simplic-ity in implementation and local conservation property, so it has enjoyed great popular-ity in many fields, such as computational fluid dynamics, computational aerodynamics, petroleum engineering and so on. About some recent development of FVM, readers can refer to the monographs [46,49,61,79,88,92,93,112,118] for details.
In2003, Park and Sheen [98] have introduced a P1-nonconforming quadrilateral element which has only three degrees of freedom on each quadrilateral. Later, Man and Shi [93] proposed the P1-nonconforming quadrilateral FVM for the elliptic problem by using a dual partition of overlapping type. Following the line of the finite element in [98], In2005, Hu and Shi [73] presented a constrained nonconforming rotated Q1(CNRQ1) element and applied it to the second order elliptic problem. In [73], the authors also point out that the CNRQ1element and the P1-nonconforming element are equivalent on a rectangle since the constraint and the continuity are the same, however, the two elements are different on a general quadrilateral. Afterwards, Hu, Man and Shi [72] and Mao and Chen [94] investigated and analyzed the CNRQ1-P0finite element method for the Stokes problem on rectangular meshes. The application of the CNRQ1element to the nearly incompressible planar elasticity problem can be referred to [72,95]. Meanwhile, Mao and Chen [94] and Liu and Yan [90] discussed the supconvergence of the finite element scheme for the Stokes problem on rectangular meshes.
The main work of this paper is as follows:
1. Finite volume element method for the Stokes problems.
Let Ω be a bounded, convex and open polygon of R2with boundary (?)Q. We consider the following Stokes equations with the homogeneous Dirichlet boundary condition-△u+▽p=f, in Ω (9) divu=0, in Ω,(10) u=0, on (?)Ω,(11) where u=(u1,u2) represents the velocity vector, p is the pressure, and f indicates a prescribed body force.
Let Th={K} which is called primal partition be a strictly convex and quadrilateral partition of Ω. The dual partition is nonoverlapping barycenter quadrilateral partition. Choose CNRQ1function space Uh as the trial function space and the piecewise constant space Vh as the test function space for the velocity. For the pressure, we select the piecewise constant space(Ph) on the macro-elements as discrete space. The FVEM for the Stokes problem (2.1)-(2.3) investigated in this paper is:find a pair (uh,ph)∈Uh×Ph, such that ah(uh,vh)+bh(vh,ph)=(f,vh),(?)vh∈Vh,(12) b'h(uh,qh)=0,(?)qh∈Ph.(13)
(1). Stability.
Suppose the primal partition is quasi-uniform and regular. Each quadrilateral in Th satisfies quasi-parallel quadrilateral condition. Then, we obtain the following results:
Lemma1. For sufficiently small h, there exists a constant C0>0, we have ah(uh,Γhuh)≥C0|uh|1,h2,(?)uh∈Uh.
Lemma2. The pair (Uh,Ph) satisfies an uniform inf-sup condition, i.e., there holds
(2). Error estimates.
We prove the errors in the H1norm and L2norm for the velocity, the error in the L2norm for the pressure, and the superconvergence of the velocity in the broken H1norm and the pressure in the L2norm..
Theorem1. Let the pair (u,p)∈(H01(Ω)∩H2(Ω))2×(L02(Ω)∩H1(Ω)) be the solution of (9)-(11); and (uh,Ph)∈Gh×be the solution of (12)-(13). There exists a positive constant C independent of h such that|u-uh|1,h+||p-Ph||0≤Ch(||u||2+||p||1).
Theorem2. Let (u,p)∈(H01(Ω)∩H2(Ω))2×(L02(Ω)∩H1(Ω)) and (uh,ph)∈Uh×P be the solutions of (9)-(11) and (12)-(13), respectively. If f∈H1(Ω)2, then||u-uh|0≤Ch2(||u||2+||p||1+||f||1).
Theorem3. Let (u,p)∈(H01(Ω)∩H3(Ω))2×(L02(Ω)∩H1(Ω)) and (uh,ph)∈Uh×Ph be the solutions of (9)-(11) and (12)-(13), respectively. There exists a positive constant C independent of h such that|Πhu-uh|1,h+||Jhp-Ph||0≤Ch2(||u||3+||p||2).(15)
Let Π2hvh∈Q2(T) and J2/1be the postprocessing interpolation operators: where ψi,i=1,2,…,8be the associated basis functions of the biquadratic element space Q2(τ), Q1(τ) is the bilinear function on the element τ. Then we have:
Theorem4. Suppose all condition of the thmm above are valid,|Π2huh-u|1+||J2hPh-p||o≤Ch2(||u||3+||p||2).
1. Finite volume element method for the linear elasticity problem.
Consider the pure displacement boundary value problem for the equations of plain strain linear isotropic elasticity: where u denotes the displacements, f the body forces and μ,λ are Lame parameters.
Let Th={K} which is called primal partition be a strictly convex and quadrilateral partition of Ω. The dual partition is nonoverlapping barycenter quadrilateral partition. Choose CNRQ1function space Uh as the trial function space and the piecewise constant space Vh as the test function space for the displacements. The FVEM for the linear elasticity problem is:
Suppose the primal partition is quasi-uniform and regular. Each quadrilateral in Th satisfies quasi-parallel quadrilateral condition.Then,we can obtain the following results to prove the FVEM is locking-free.
(1).Error estimate in energy norm. Theorem5.Suppose Uh∈Uh and u∈H01(Ь)∩H2(Ω) are solutions of finite uolume element scheme(19)and the equations(18),respectively,then||u-uh||h≤Ch||f||0Ω
(2).Superconvergence in the energy norm.
Theorem6.Suppose Uh∈Uh and u∈H01(Ω)∩H3(Ω) are solutions of finite uolume element scheme(19)and the equations(18),respectively,then||Πu-uh||h≤Ch2(||u||3,Ω+λ||divu||2,Ω).
引文
[1]于长华,李永海.解两点边值问题的基于应力佳点的二次有限体积元法[J].吉林大学学报(理学版),40(4)：639-648,2009.[2]于长华,王晓玲,李永海.解两点边值问题的一类修改的三次有限体积元法[J].计算数学,32(4)：358-398,2010.[3]李潜.二维抛物方程的广义Galerkin方法[J].山东大学学报(自然科学版),(1)：41-52,1982.[4]李荣华.两点边值问题的广义差分法[J].吉林大学自然科学学报,(1)：26-40,1982.[5]李荣华,祝丕琦.二阶椭圆偏微分方程的广义差分法(三角网情形)[J].高等学校计算数学学报,(2)：140-152,1982.[6]祝丕琦,李荣华.二阶椭圆偏微分方程的广义差分法(四边形网情形)[J].高等学校计算数学学报,(4)：360-375,1982.[7]吴微,李荣华.解一维二阶椭圆和抛物微分方程的广义差分法[J].数学年刊,5A(3)：303-312,1984.[8]李潜.一类非线性双曲型方程的广义Galerkin方法[J].计算数学,(2)：150-158,1986.[9]陈仲英.热传导方程的一种广义差分法[J].中山大学学报(自然科学版),(1)：6-13,1990.[10]王申林.Stokes(?)司题的非协调广义差分法[J].计算物理,10(2)：129-136,1993.[11]李荣华,陈仲英.微分方程广义差分法[M].吉林大学出版社,长春,101994.[12]明平兵.非协调元vs locking(?)司题[D].中国科学院数学与系统科学院计算数学所,2000.[13]齐禾.位移边界条件下弹性问题的locking-free有限元方法[D].中国科学院数学与系统科学研究院,2002.[14]何国良.二维不可压Navier-Stokes方程的有限体积法研究[D].西安交通大学,2007.[15]于长华,李永海.解possion方程的基于应力佳点的双二次有限体积元法[J].计算数学,32(1)：59-74,2010.[16]丁玉琼,左平.Lagrange三次有限体积元法的超收敛现象[J].吉林大学学报(理学版),49(2)：159-163,2011.[17]Douglas N. Arnold. Discretization by finite elements of a model parameter depen-dent problem[J]. Numer. Math.,37(3):405-421,1981.[18]J. P. Aubin. Approximation of Elliptic Boundary-Value Problems[M]. Wiley-Interscience., New York,1972.[19]Ivo Babuska and Manil Suri. Locking effects in the finite element approximation of elasticity problems[J]. Numer. Math.,62(4):439-463,1992.[20]Ivo Babuska and Manil Suri. On locking and robustness in the finite element method[J]. SIAM J. Numer. Anal,29(5):1261-1293,1992.[21]C. Baiocchi and F. Brezzi. Stabilization of unstable numerical methods[R]. Rome,1993. Current Problems of Analysis and Mathematical Physics,(Taormina,1992), Univ. Roma "La Sapienza,"[22]R. E. Bank and D. J. Rose. Some error estimates for the box method[J]. SIAM J. Numer. Anal,24:777-787,1987.[23]T. Barth, P. Bochev, M. Gunzburger, and J. Shadid. A taxonomy of consistently stabilized finite element methods for the Stokes problem [J]. SIAM J. Sci. Comput.,25:1585-1607,2004.[24]M. Behr, L. Franca, and T. Tezduyar. Stabilized finite element methods for the velocitypressure-stress formulation of incompressible flows [J]. Comput. Methods Appl. Mech. Engrg.,104:31-48,1993.[25]J. Blasco and R. Codina. Stabilized finite element method for the transient Navier-Stokes equations based on a pressure gradient projection[J]. Comput. Methods Appl. Mech. Engrg.,182:277-300,2000.
[26] J. Blasco and R. Codina. Space and time error estimates for a first-order, pressurestabilized finite element method for the incompressible Navier-Stokes equations[J].Appl. Numer. Math.,38:475–497,2001.
[27] P. Bochev and M. Gunzburger. Finite element methods of least-squares type[J].SIAM Rev.,40:789–837,1998.
[28] J. M. Boland and R. A. Nicolaides. On the stability of bilinear-constant velocity-pressure finite elements[J]. Numer. Math.,44(2):219–222,1984.
[29] S. Brenner and L. Sung. Linear finite element methods for planar elasticity[J]. Math.Comput.,59:321–338,1992.
[30] S. C. Brenner and L. R. Scott. The mathematical theory of finite element meth-ods[M], volume15of Texts in Applied Mathematics. Springer, New York, thirdedition,2008.
[31] Susanne C. Brenner. A nonconforming mixed multigrid method for the pure tractionproblem in planar linear elasticity[J]. Math. Comp.,63(208):435–460, S1–S5,1994.
[32] Susanne C. Brenner and Li-Yeng Sung. Linear finite element methods for planarlinear elasticity[J]. Math. Comp.,59(200):321–338,1992.
[33] F. Brezzi and M. Fortin. Mixed and hybrid finite element methods[M], volume15ofSpringer Series in Computational Mathematics. Springer-Verlag, New York,1991.
[34] Franco Brezzi, Klaus-Ju¨rgen Bathe, and Michel Fortin. Mixed-interpolated elementsfor Reissner-Mindlin plates[J]. Internat. J. Numer. Methods Engrg.,28(8):1787–1801,1989.
[35] Z. Cai. On the finite volume element method[J]. Numer. Math.,58:713–735,1991.
[36] Z. Cai, J. Mandel, and S. McCormick. The finite volume element for difusionequations on general triangulations[J]. SIAM J. Numer. Anal.,28:392–402,1991.
[37] Z. Cai and S. McCormick. On the accuracy of the finite volume element methodfor difusion equations on composite grids[J]. SIAM J. Numer. Anal.,27:636–655,1990.
[38] Z. Q. Cai, J. Douglas, Jr., and X. Ye. A stable nonconforming quadrilateral finiteelement method for stationary Stokes and Navier-Stokes equations[J]. Calcolo,36(4):215–232,1999.
[39] P. Chatzipantelidis. Finite volume methods for elliptic PDE’s: A new approach[J].M2AN,36:307–324,2002.
[40] Shao Chun Chen and Dong Yang Shi. Error estimates for double set parameterelements[J]. Gongcheng Shuxue Xuebao,22(4):599–605,2005.
[41] Shao Chun Chen and Zhong Ci Shi. Double set parameter method of constructinga stifness matrix[J]. Chinese J. Numer. Math. Appl.,13(4):55–69,1991.
[42] Z. X. Chen. Finite element methods and their applications[M]. Scientific Computa-tion. Springer-Verlag, Berlin,2005.
[43] Z. X. Chen. On the control volume finite element methods and their applicationsto multiphase flow[J]. Netw. Heterog. Media,1(4):689–706(electronic),2006.
[44] Z. Y. Chen. L2-estimates for one-dimensional schemes for generalized diferencemethods[J]. Acta Sci. Natur. Univ. Sunyatseni,33(4):22–28,1994.
[45] Z. Y. Chen, C. N. He, and B. Wu. High order finite volume methods for singularperturbation problems[J]. Sci. China Ser. A,51(8):1391–1400,2008.
[46] Z. Y. Chen, R. H. Li, and A. H. Zhou. A note on the optimal L2-estimate of thefinite volume element method[J]. Adv. Comput. Math.,16(4):291–303,2002.
[47] S.-H. Chou. Analysis and convergence of a covolume method for the generalizedStokes problem[J]. Math. Comp.,36:85–104,1997.
[48] S.-H. Chou and D. Y. Kwak. Analysis and convergence of a MAC-like scheme forthe generalized Stokes problem[J]. Numer. Methods Partial Diferential Equations,13(2):147–162,1997.
[49] S.-H. Chou and D. Y. Kwak. A covolume method based on rotated bilinears for thegeneralized Stokes problem[J]. SIAM J. Numer. Anal.,35(2):494–507(electronic),1998.
[50] S.-H. Chou and D. Y. Kwak. Multigrid algorithms for a vertex-centered covolumemethod for elliptic problems[J]. Numer. Math.,90:441–458,2002.
[51] S.-H. Chou, D. Y. Kwak, and K. Y. Kim. Mixed finite volume methods on nonstag-gered quadrilateral grids for elliptic problems[J]. Math. Comp.,72(242):525–539,2002.
[52] S.-H. Chou, D. Y. Kwak, and Q. Li. Lperror estimates and superconvergence forcovolume or finite volume element methods[J]. Numer. Methods Partial DiferentialEquations,19(4):463–486,2003.
[53] S.-H. Chou, D. Y. Kwak, and P. S. Vassilevski. Mixed covolume methods for el-liptic problems on triangular grids[J]. SIAM J. Numer. Anal.,35(5):1850–1861(electronic),1998.
[54] S.-H. Chou and Q. Li. Error estimates in L2, H1and L∞in covolume methods forelliptic and parabolic problems: a unified approach[J]. Math. Comp.,69(229):103–120,2000.
[55] S.-H. Chou and S. R. Tang. Comparing two approaches of analyzing mixed finitevolume methods[J]. J. KSIAM,1:55–78,2001.
[56] S.-H. Chou and P. S. Vassilevski. A general mixed covolume framework for con-structing conservative schemes for elliptic problems[J]. Math. Comp.,68(227):991–1011,1999.
[57] P. G. Ciarlet. The finite element method for elliptic problems[M]. North-HollandPublishing Co., Amsterdam,1978. Studies in Mathematics and its Applications,Vol.4.
[58] R. Codina and J. Blasco. Analysis of a pressure stabilized finite element approxima-tion of the stationary Navier-Stokes equations[J]. Numer. Math.,87:59–81,2000.
[59] J. P. Croisille. Finite volume box-schemes and mixed methods[J]. Math. Model.Numer. Anal.,34(5):1087–1106,2000.
[60] M. Crouzeix and P. A. Raviart. Conforming and nonconforming finite elementmethods for solving the stationary stokes equations[J]. R.A.I.R.O.Anal. Numer.,3(7):33–75,1973.
[61] M. Cui and X. Ye. Superconvergence of finite volume methods for the Stokes equa-tions[J]. Numerical Methods for Partial Diferential Equations,25(5):1212–1230,2009.
[62] Yuqiong Ding and Li Yonghai. Finite volume element method with lagrangian cubicfunctions[J]. Journal of Systems Science and Complexity., On line.
[63] J. Douglas, Jr. and J. P. Wang. An absolutely stabilized finite element method forthe Stokes problem[J]. Math. Comp.,52(186):495–508,1989.
[64] R. E. Ewing, T. Lin, and Y. P. Lin. On the accuracy of the finite volume ele-ment method based on piecewise linear polynomials[J]. SIAM J. Numer. Anal.,39(6):1865–1888(electronic),2002.
[65] I. Faille. A control volume method to solve an elliptic equation on a two-dimensionalirregular mesh[J]. Comput. Methods Appl. Mech. Eng.,100:275–290,1992.
[66] R. Falk. An analysis of the penalty method and extrapolation for the stationaryStokes equations[J]. Advances in computer methods for partial diferential equations,pages66–69,1975.
[67] Richard S. Falk. Nonconforming finite element methods for the equations of linearelasticity[J]. Math. Comp.,57(196):529–550,1991.
[68] Richard S. Falk and Mary E. Morley. Equivalence of finite element methods forproblems in elasticity[J]. SIAM J. Numer. Anal.,27(6):1486–1505,1990.
[69] M. Fortin and R. Glowinski. Augmented lagrangian methods: applications to nu-merical solution of boundary value problems[M]. North-Holland, Amsterdam,1983.Stud. Math. Appl. Vol.15.
[70] V. Girault and P.-A. Raviart. Finite element methods for Navier-Stokes equations:Theory and algorithms[M], volume5of Springer Series in Computational Mathe-matics. Springer-Verlag, Berlin,1986.
[71] W. Hackbusch. On first and second order box schemes[J]. Computing,41:277–296,1989.
[72] J. Hu, H. Y. Man, and Z. C. Shi. Constrained nonconforming rotated Q1element forStokes flow and planar elasticity[J]. Mathematica Numerica Sinica,27(3):311–324,2005.
[73] J. Hu and Z. C. Shi. Constrained quadrilateral nonconforming rotated Q1ele-ment[J]. J. Comp. Math.,23(5):561–586,2005.
[74] T. J. R. Hughes and L. P. Franca. A new finite element formulation for computa-tional fluid dynamics: VII. The Stokes problem with various well-posed boundaryconditions: Symmetric formulations that converge for all velocity pressure spaces[J].Comput. Methods Appl. Mech. Engrg.,65:85–96,1987.
[75] T. J. R. Hughes, L. P. Franca, and M. Balestra. A new finite element formulationfor computational fluid dynamics. V. Circumventing the Babuˇska-Brezzi condition:a stable Petrov-Galerkin formulation of the Stokes problem accommodating equal-order interpolations[J]. Comput. Methods Appl. Mech. Engrg.,59(1):85–99,1986.
[76] T. J. R. Hughes, W. Liu, and A. Brooks. Finite element analysis of incompressibleviscous flows by the penalty function formulation[J]. J. Comput. Phys.,30:1–60,1979.
[77] Thomas J. R. Hughes. The Finite Element Method: Linear Static and DynamicFinite Element Analysis [R]. Prentice Hall, Englewood Clifts, NJ,1987.
[78] C. Johnson and J. Pintkaranat. Analysis of mixed finite element methods relatedto reduced integration[J]. Math. Comp.,42:9–23,1984.
[79] K. S. Kang and D. Y. Kwak. Covolume method for new nonconforming rectangularelement for the Stokes problem[J]. Computers and Mathematics with Applications,43(8-9):1063–1078,2002.
[80] Reijo Kouhia and Rolf Stenberg. A linear nonconforming finite element method fornearly incompressible elasticity and Stokes flow[J]. Comput. Methods Appl. Mech.Engrg.,124(3):195–212,1995.
[81] Chang-Ock Lee, Jongwoo Lee, and Dongwoo Sheen. A locking-free nonconformingfinite element method for planar linear elasticity[J]. Adv. Comput. Math.,19(1-3):277–291,2003. Challenges in computational mathematics (Pohang,2001).
[82] J. Li and Z. X. Chen. A new stabilized finite volume method for the stationaryStokes equations[J]. Adv. Comput. Math.,30(2):141–152,2009.
[83] J. Li and Y. N. He. A stabilized finite element method based on two local Gaussintegrations for the Stokes equations[J]. J. Comput. Appl. Math.,214(1):58–65,2008.
[84] Li Kang Li. Discretization of the Timoshenko beam problem by the p and the h-pversions of the finite element method[J]. Numer. Math.,57(4):413–420,1990.
[85] R. H. Li. Generalized diference methods for a nonlinear Dirichlet problem[J]. SIAMJ. Numer. Anal.,24(1):77–88,1987.
[86] R. H. Li, Z. Y. Chen, and W. Wu. Generalized diference methods for diferentialequations (Numerical analysis of finite volume methods)[M], volume226of Mono-graphs and Textbooks in Pure and Applied Mathematics. Marcel Dekker Inc., NewYork,2000. Numerical analysis of finite volume methods.
[87] R. H. Li and J. H. Shen. On a class of high-accuracy upwind schemes[J]. North-eastern Math. J.,7:11–26,1991.
[88] Y. H. Li and R. H. Li. Generalized diference methods on arbitrary quadrilateralnetworks[J]. J. Comput. Math.,17:653–672,1999.
[89] F. Liebau. The finite volume element method with quadratic basis functions[J].Computing,57:281–299,1996.
[90] H. P. Liu and N. N. Yan. Superconvergence analysis of the nonconforming quadrilat-eral linear-constant scheme for Stokes equations[J]. Adv. Comput. Math.,29(4):375–392,2008.
[91] J. L. Lv. L2error estimates and superconvergence of the finite volume elementmethods on quadrilateral meshes[D]. Ph. D. Jilin University, Changchun,1999.
[92] J. L. Lv and Y. H. Li. L2error estimate of the finite volume element methods onquadrilateral meshes[J]. Adv. Comput. Math.,33:129–148,2010.
[93] H. Y. Man and Z. C. Shi. P1-nonconforming quadrilateral finite volume elementmethod and its casadic multigrid algorithm for elliptic problems[J]. J. Comp. Math.,24:59–80,2006.
[94] S. P. Mao and S. C. Chen. Convergence and superconvergence of a nonconformingfinite element method for the Stokes problem[J]. J. Numer. Math.,14(2):83–101,2006.
[95] S. P. Mao and S. C. Chen. A quadrilateral nonconforming finite element for linearelasticity problem[J]. Adv. Comput. Math.,28(1):81–100,2008.
[96] Shipeng Mao, Shaochun Chen, and Huixia Sun. A quadrilateral, anisotropic, super-convergent, nonconforming double set parameter element[J]. Appl. Numer. Math.,56(7):937–961,2006.
[97] T. J. Oden. RIP-methods for Stokesian flow[M]. John Wiley, New York,1982.Finite Elements in Fluids, Vol.4.
[98] C. Park and D. Sheen. P1-nonconforming quadrilateral finite element methods forsecond-order elliptic problems[J]. SIAM J.Numer.Anal,41(2):624–640,2003.
[99] J. Pitkaranta and R. Strnberg. Error bounds for the approximation of the Stokesproblem using bilinear/constant elements on irregular quadrilateral meshes[J]. inThe Mathematics of finite elements and applications V, ed. J. Whiteman, AcademicPress, London, pages325–334,1985.
[100] He Qi, Lie-heng Wang, and Wei-ying Zheng. On locking-free finite element schemesfor three-dimensional elasticity[J]. J. Comput. Math.,23(1):101–112,2005.
[101] R. Rannacher and S. Turek. Simple nonconforming quadrilateral stokes element[J].Numerical Methods for Partial Diferential Equations,8(2):97–111,1992.
[102] T. Schmidt. Box schemes on quadrilateral meshes[J]. Computing,51:271–292,1993.
[103] L. R. Scott and M. Vogelius. Norm estimates for a maximal right inverse of thedivergence operator in spaces of piecewise polynomials[J]. RAIRO Mod′el. Math.Anal. Num′er.,19(1):111–143,1985.
[104] E. Su¨li. Convergence of finite volume schemes for Poisson’s equation on nonuniformmeshes[J]. SIAM J. Numer. Anal.,28:1419–1430,1991.
[105] E. Su¨li. The accuracy of cell vertex finite volume methods on quadrilateralmeshes[J]. Math. Comp.,59:359–382,1992.
[106] Dongyang Shi, Shipeng Mao, and Shaochun Chen. A locking-free anisotropic non-conforming finite element for planar linear elasticity problem[J]. Acta Math. Sci.Ser. B Engl. Ed.,27(1):193–202,2007.
[107] Zhongci Shi. On the accuracy of the quasi-conforming and generalize conformingfinite elements[J]. China Ann. Math.(Series B),11:148–155,1990.
[108] Zhongci Shi. A rectangulat plate element with high accuracy [J]. Sci. inChina(Series A),30:504–515,2000.
[109] Zhongci Shi, Xuejun Xu, and Zhimin Zhang. The patch recovery for finite ele-ment approximation of elasticity problems under quadrilateral meshes[J]. DiscreteContin. Dyn. Syst. Ser. B,9(1):163–182(electronic),2008.
[110] R. Stenberg. Analysis of mixed finite element methods for the Stokes problem:aunified approach[J]. Math. of Comp.,42(165):9–23,1984.
[111] Hui Xia Sun, Shao Chun Chen, and Mi Luo Feng. Double set parameter trapezoidplate element with symmetry[J]. Math. Appl.(Wuhan),14(3):52–58,2001.
[112] W. F. Tian. Mixed Finite Volume Element Method[D]. Ph. D. Jilin University,Changchun,2010.
[113] Michael Vogelius. An analysis of the p-version of the finite element method fornearly incompressible materials. Uniformly valid, optimal error estimates[J]. Nu-mer. Math.,41(1):39–53,1983.
[114] Jian Jun Wan and Shao Chun Chen. Multigrid method of double set parameterMorley finite element[J]. Numer. Math. J. Chinese Univ.,27(3):239–249,2005.
[115] J. P. Wang. A superconvergence analysis for finite element solutions by the least-squares surface fitting on irregular meshes for smooth problems[J]. J. Math. Study.,33:229–243,2000.
[116] Lie Heng Wang and He Qi. On locking-free finite element schemes for the puredisplacement boundary value problem in planar elasticity[J]. Math. Numer. Sin.,24(2):243–256,2002.
[117] Lie-heng Wang and He Qi. A locking-free scheme of nonconforming rectangularfinite element for the planar elasticity[J]. J. Comput. Math.,22(5):641–650,2004.
[118] M. Yang. A second-order finite volume element method on quadrilateral meshes forelliptic equations[J]. ESAIM: M2AN.,40(6):1053–1067,2006.
[119] X. Ye. On the relationship between finite volume and finite element methods ap-plied to the Stokes equations[J]. Numer. Methods Partial Diferential Equations,17(5):440–453,2001.
[120] X. Ye and L. Zikatanov. Some observations on Babus ka and Brezzi theories[J].Numer.Math.,94(1):195–202,2003.
[121] Xiu Ye. A new discontinuous finite volume method for elliptic problems[J]. SIAMJ. Numer. Anal.,42(3):1062–1072(electronic),2004.
[122] Changhua Yu and Yonghai Li. Biquadratic finite volume element method basedon optimal stress points for second order hyperbolic equations[J]. J. Comp. App.Math.,236:1055–1068,2011.
[123] Zhimin Zhang. Analysis of some quadrilateral nonconforming elements for incom-pressible elasticity[J]. SIAM J. Numer. Anal.,34(2):640–663,1997.
[26] J. Blasco and R. Codina. Space and time error estimates for a first-order, pressurestabilized finite element method for the incompressible Navier-Stokes equations[J].Appl. Numer. Math.,38:475–497,2001.
[27] P. Bochev and M. Gunzburger. Finite element methods of least-squares type[J].SIAM Rev.,40:789–837,1998.
[28] J. M. Boland and R. A. Nicolaides. On the stability of bilinear-constant velocity-pressure finite elements[J]. Numer. Math.,44(2):219–222,1984.
[29] S. Brenner and L. Sung. Linear finite element methods for planar elasticity[J]. Math.Comput.,59:321–338,1992.
[30] S. C. Brenner and L. R. Scott. The mathematical theory of finite element meth-ods[M], volume15of Texts in Applied Mathematics. Springer, New York, thirdedition,2008.
[31] Susanne C. Brenner. A nonconforming mixed multigrid method for the pure tractionproblem in planar linear elasticity[J]. Math. Comp.,63(208):435–460, S1–S5,1994.
[32] Susanne C. Brenner and Li-Yeng Sung. Linear finite element methods for planarlinear elasticity[J]. Math. Comp.,59(200):321–338,1992.
[33] F. Brezzi and M. Fortin. Mixed and hybrid finite element methods[M], volume15ofSpringer Series in Computational Mathematics. Springer-Verlag, New York,1991.
[34] Franco Brezzi, Klaus-Ju¨rgen Bathe, and Michel Fortin. Mixed-interpolated elementsfor Reissner-Mindlin plates[J]. Internat. J. Numer. Methods Engrg.,28(8):1787–1801,1989.
[35] Z. Cai. On the finite volume element method[J]. Numer. Math.,58:713–735,1991.
[36] Z. Cai, J. Mandel, and S. McCormick. The finite volume element for difusionequations on general triangulations[J]. SIAM J. Numer. Anal.,28:392–402,1991.
[37] Z. Cai and S. McCormick. On the accuracy of the finite volume element methodfor difusion equations on composite grids[J]. SIAM J. Numer. Anal.,27:636–655,1990.
[38] Z. Q. Cai, J. Douglas, Jr., and X. Ye. A stable nonconforming quadrilateral finiteelement method for stationary Stokes and Navier-Stokes equations[J]. Calcolo,36(4):215–232,1999.
[39] P. Chatzipantelidis. Finite volume methods for elliptic PDE’s: A new approach[J].M2AN,36:307–324,2002.
[40] Shao Chun Chen and Dong Yang Shi. Error estimates for double set parameterelements[J]. Gongcheng Shuxue Xuebao,22(4):599–605,2005.
[41] Shao Chun Chen and Zhong Ci Shi. Double set parameter method of constructinga stifness matrix[J]. Chinese J. Numer. Math. Appl.,13(4):55–69,1991.
[42] Z. X. Chen. Finite element methods and their applications[M]. Scientific Computa-tion. Springer-Verlag, Berlin,2005.
[43] Z. X. Chen. On the control volume finite element methods and their applicationsto multiphase flow[J]. Netw. Heterog. Media,1(4):689–706(electronic),2006.
[44] Z. Y. Chen. L2-estimates for one-dimensional schemes for generalized diferencemethods[J]. Acta Sci. Natur. Univ. Sunyatseni,33(4):22–28,1994.
[45] Z. Y. Chen, C. N. He, and B. Wu. High order finite volume methods for singularperturbation problems[J]. Sci. China Ser. A,51(8):1391–1400,2008.
[46] Z. Y. Chen, R. H. Li, and A. H. Zhou. A note on the optimal L2-estimate of thefinite volume element method[J]. Adv. Comput. Math.,16(4):291–303,2002.
[47] S.-H. Chou. Analysis and convergence of a covolume method for the generalizedStokes problem[J]. Math. Comp.,36:85–104,1997.
[48] S.-H. Chou and D. Y. Kwak. Analysis and convergence of a MAC-like scheme forthe generalized Stokes problem[J]. Numer. Methods Partial Diferential Equations,13(2):147–162,1997.
[49] S.-H. Chou and D. Y. Kwak. A covolume method based on rotated bilinears for thegeneralized Stokes problem[J]. SIAM J. Numer. Anal.,35(2):494–507(electronic),1998.
[50] S.-H. Chou and D. Y. Kwak. Multigrid algorithms for a vertex-centered covolumemethod for elliptic problems[J]. Numer. Math.,90:441–458,2002.
[51] S.-H. Chou, D. Y. Kwak, and K. Y. Kim. Mixed finite volume methods on nonstag-gered quadrilateral grids for elliptic problems[J]. Math. Comp.,72(242):525–539,2002.
[52] S.-H. Chou, D. Y. Kwak, and Q. Li. Lperror estimates and superconvergence forcovolume or finite volume element methods[J]. Numer. Methods Partial DiferentialEquations,19(4):463–486,2003.
[53] S.-H. Chou, D. Y. Kwak, and P. S. Vassilevski. Mixed covolume methods for el-liptic problems on triangular grids[J]. SIAM J. Numer. Anal.,35(5):1850–1861(electronic),1998.
[54] S.-H. Chou and Q. Li. Error estimates in L2, H1and L∞in covolume methods forelliptic and parabolic problems: a unified approach[J]. Math. Comp.,69(229):103–120,2000.
[55] S.-H. Chou and S. R. Tang. Comparing two approaches of analyzing mixed finitevolume methods[J]. J. KSIAM,1:55–78,2001.
[56] S.-H. Chou and P. S. Vassilevski. A general mixed covolume framework for con-structing conservative schemes for elliptic problems[J]. Math. Comp.,68(227):991–1011,1999.
[57] P. G. Ciarlet. The finite element method for elliptic problems[M]. North-HollandPublishing Co., Amsterdam,1978. Studies in Mathematics and its Applications,Vol.4.
[58] R. Codina and J. Blasco. Analysis of a pressure stabilized finite element approxima-tion of the stationary Navier-Stokes equations[J]. Numer. Math.,87:59–81,2000.
[59] J. P. Croisille. Finite volume box-schemes and mixed methods[J]. Math. Model.Numer. Anal.,34(5):1087–1106,2000.
[60] M. Crouzeix and P. A. Raviart. Conforming and nonconforming finite elementmethods for solving the stationary stokes equations[J]. R.A.I.R.O.Anal. Numer.,3(7):33–75,1973.
[61] M. Cui and X. Ye. Superconvergence of finite volume methods for the Stokes equa-tions[J]. Numerical Methods for Partial Diferential Equations,25(5):1212–1230,2009.
[62] Yuqiong Ding and Li Yonghai. Finite volume element method with lagrangian cubicfunctions[J]. Journal of Systems Science and Complexity., On line.
[63] J. Douglas, Jr. and J. P. Wang. An absolutely stabilized finite element method forthe Stokes problem[J]. Math. Comp.,52(186):495–508,1989.
[64] R. E. Ewing, T. Lin, and Y. P. Lin. On the accuracy of the finite volume ele-ment method based on piecewise linear polynomials[J]. SIAM J. Numer. Anal.,39(6):1865–1888(electronic),2002.
[65] I. Faille. A control volume method to solve an elliptic equation on a two-dimensionalirregular mesh[J]. Comput. Methods Appl. Mech. Eng.,100:275–290,1992.
[66] R. Falk. An analysis of the penalty method and extrapolation for the stationaryStokes equations[J]. Advances in computer methods for partial diferential equations,pages66–69,1975.
[67] Richard S. Falk. Nonconforming finite element methods for the equations of linearelasticity[J]. Math. Comp.,57(196):529–550,1991.
[68] Richard S. Falk and Mary E. Morley. Equivalence of finite element methods forproblems in elasticity[J]. SIAM J. Numer. Anal.,27(6):1486–1505,1990.
[69] M. Fortin and R. Glowinski. Augmented lagrangian methods: applications to nu-merical solution of boundary value problems[M]. North-Holland, Amsterdam,1983.Stud. Math. Appl. Vol.15.
[70] V. Girault and P.-A. Raviart. Finite element methods for Navier-Stokes equations:Theory and algorithms[M], volume5of Springer Series in Computational Mathe-matics. Springer-Verlag, Berlin,1986.
[71] W. Hackbusch. On first and second order box schemes[J]. Computing,41:277–296,1989.
[72] J. Hu, H. Y. Man, and Z. C. Shi. Constrained nonconforming rotated Q1element forStokes flow and planar elasticity[J]. Mathematica Numerica Sinica,27(3):311–324,2005.
[73] J. Hu and Z. C. Shi. Constrained quadrilateral nonconforming rotated Q1ele-ment[J]. J. Comp. Math.,23(5):561–586,2005.
[74] T. J. R. Hughes and L. P. Franca. A new finite element formulation for computa-tional fluid dynamics: VII. The Stokes problem with various well-posed boundaryconditions: Symmetric formulations that converge for all velocity pressure spaces[J].Comput. Methods Appl. Mech. Engrg.,65:85–96,1987.
[75] T. J. R. Hughes, L. P. Franca, and M. Balestra. A new finite element formulationfor computational fluid dynamics. V. Circumventing the Babuˇska-Brezzi condition:a stable Petrov-Galerkin formulation of the Stokes problem accommodating equal-order interpolations[J]. Comput. Methods Appl. Mech. Engrg.,59(1):85–99,1986.
[76] T. J. R. Hughes, W. Liu, and A. Brooks. Finite element analysis of incompressibleviscous flows by the penalty function formulation[J]. J. Comput. Phys.,30:1–60,1979.
[77] Thomas J. R. Hughes. The Finite Element Method: Linear Static and DynamicFinite Element Analysis [R]. Prentice Hall, Englewood Clifts, NJ,1987.
[78] C. Johnson and J. Pintkaranat. Analysis of mixed finite element methods relatedto reduced integration[J]. Math. Comp.,42:9–23,1984.
[79] K. S. Kang and D. Y. Kwak. Covolume method for new nonconforming rectangularelement for the Stokes problem[J]. Computers and Mathematics with Applications,43(8-9):1063–1078,2002.
[80] Reijo Kouhia and Rolf Stenberg. A linear nonconforming finite element method fornearly incompressible elasticity and Stokes flow[J]. Comput. Methods Appl. Mech.Engrg.,124(3):195–212,1995.
[81] Chang-Ock Lee, Jongwoo Lee, and Dongwoo Sheen. A locking-free nonconformingfinite element method for planar linear elasticity[J]. Adv. Comput. Math.,19(1-3):277–291,2003. Challenges in computational mathematics (Pohang,2001).
[82] J. Li and Z. X. Chen. A new stabilized finite volume method for the stationaryStokes equations[J]. Adv. Comput. Math.,30(2):141–152,2009.
[83] J. Li and Y. N. He. A stabilized finite element method based on two local Gaussintegrations for the Stokes equations[J]. J. Comput. Appl. Math.,214(1):58–65,2008.
[84] Li Kang Li. Discretization of the Timoshenko beam problem by the p and the h-pversions of the finite element method[J]. Numer. Math.,57(4):413–420,1990.
[85] R. H. Li. Generalized diference methods for a nonlinear Dirichlet problem[J]. SIAMJ. Numer. Anal.,24(1):77–88,1987.
[86] R. H. Li, Z. Y. Chen, and W. Wu. Generalized diference methods for diferentialequations (Numerical analysis of finite volume methods)[M], volume226of Mono-graphs and Textbooks in Pure and Applied Mathematics. Marcel Dekker Inc., NewYork,2000. Numerical analysis of finite volume methods.
[87] R. H. Li and J. H. Shen. On a class of high-accuracy upwind schemes[J]. North-eastern Math. J.,7:11–26,1991.
[88] Y. H. Li and R. H. Li. Generalized diference methods on arbitrary quadrilateralnetworks[J]. J. Comput. Math.,17:653–672,1999.
[89] F. Liebau. The finite volume element method with quadratic basis functions[J].Computing,57:281–299,1996.
[90] H. P. Liu and N. N. Yan. Superconvergence analysis of the nonconforming quadrilat-eral linear-constant scheme for Stokes equations[J]. Adv. Comput. Math.,29(4):375–392,2008.
[91] J. L. Lv. L2error estimates and superconvergence of the finite volume elementmethods on quadrilateral meshes[D]. Ph. D. Jilin University, Changchun,1999.
[92] J. L. Lv and Y. H. Li. L2error estimate of the finite volume element methods onquadrilateral meshes[J]. Adv. Comput. Math.,33:129–148,2010.
[93] H. Y. Man and Z. C. Shi. P1-nonconforming quadrilateral finite volume elementmethod and its casadic multigrid algorithm for elliptic problems[J]. J. Comp. Math.,24:59–80,2006.
[94] S. P. Mao and S. C. Chen. Convergence and superconvergence of a nonconformingfinite element method for the Stokes problem[J]. J. Numer. Math.,14(2):83–101,2006.
[95] S. P. Mao and S. C. Chen. A quadrilateral nonconforming finite element for linearelasticity problem[J]. Adv. Comput. Math.,28(1):81–100,2008.
[96] Shipeng Mao, Shaochun Chen, and Huixia Sun. A quadrilateral, anisotropic, super-convergent, nonconforming double set parameter element[J]. Appl. Numer. Math.,56(7):937–961,2006.
[97] T. J. Oden. RIP-methods for Stokesian flow[M]. John Wiley, New York,1982.Finite Elements in Fluids, Vol.4.
[98] C. Park and D. Sheen. P1-nonconforming quadrilateral finite element methods forsecond-order elliptic problems[J]. SIAM J.Numer.Anal,41(2):624–640,2003.
[99] J. Pitkaranta and R. Strnberg. Error bounds for the approximation of the Stokesproblem using bilinear/constant elements on irregular quadrilateral meshes[J]. inThe Mathematics of finite elements and applications V, ed. J. Whiteman, AcademicPress, London, pages325–334,1985.
[100] He Qi, Lie-heng Wang, and Wei-ying Zheng. On locking-free finite element schemesfor three-dimensional elasticity[J]. J. Comput. Math.,23(1):101–112,2005.
[101] R. Rannacher and S. Turek. Simple nonconforming quadrilateral stokes element[J].Numerical Methods for Partial Diferential Equations,8(2):97–111,1992.
[102] T. Schmidt. Box schemes on quadrilateral meshes[J]. Computing,51:271–292,1993.
[103] L. R. Scott and M. Vogelius. Norm estimates for a maximal right inverse of thedivergence operator in spaces of piecewise polynomials[J]. RAIRO Mod′el. Math.Anal. Num′er.,19(1):111–143,1985.
[104] E. Su¨li. Convergence of finite volume schemes for Poisson’s equation on nonuniformmeshes[J]. SIAM J. Numer. Anal.,28:1419–1430,1991.
[105] E. Su¨li. The accuracy of cell vertex finite volume methods on quadrilateralmeshes[J]. Math. Comp.,59:359–382,1992.
[106] Dongyang Shi, Shipeng Mao, and Shaochun Chen. A locking-free anisotropic non-conforming finite element for planar linear elasticity problem[J]. Acta Math. Sci.Ser. B Engl. Ed.,27(1):193–202,2007.
[107] Zhongci Shi. On the accuracy of the quasi-conforming and generalize conformingfinite elements[J]. China Ann. Math.(Series B),11:148–155,1990.
[108] Zhongci Shi. A rectangulat plate element with high accuracy [J]. Sci. inChina(Series A),30:504–515,2000.
[109] Zhongci Shi, Xuejun Xu, and Zhimin Zhang. The patch recovery for finite ele-ment approximation of elasticity problems under quadrilateral meshes[J]. DiscreteContin. Dyn. Syst. Ser. B,9(1):163–182(electronic),2008.
[110] R. Stenberg. Analysis of mixed finite element methods for the Stokes problem:aunified approach[J]. Math. of Comp.,42(165):9–23,1984.
[111] Hui Xia Sun, Shao Chun Chen, and Mi Luo Feng. Double set parameter trapezoidplate element with symmetry[J]. Math. Appl.(Wuhan),14(3):52–58,2001.
[112] W. F. Tian. Mixed Finite Volume Element Method[D]. Ph. D. Jilin University,Changchun,2010.
[113] Michael Vogelius. An analysis of the p-version of the finite element method fornearly incompressible materials. Uniformly valid, optimal error estimates[J]. Nu-mer. Math.,41(1):39–53,1983.
[114] Jian Jun Wan and Shao Chun Chen. Multigrid method of double set parameterMorley finite element[J]. Numer. Math. J. Chinese Univ.,27(3):239–249,2005.
[115] J. P. Wang. A superconvergence analysis for finite element solutions by the least-squares surface fitting on irregular meshes for smooth problems[J]. J. Math. Study.,33:229–243,2000.
[116] Lie Heng Wang and He Qi. On locking-free finite element schemes for the puredisplacement boundary value problem in planar elasticity[J]. Math. Numer. Sin.,24(2):243–256,2002.
[117] Lie-heng Wang and He Qi. A locking-free scheme of nonconforming rectangularfinite element for the planar elasticity[J]. J. Comput. Math.,22(5):641–650,2004.
[118] M. Yang. A second-order finite volume element method on quadrilateral meshes forelliptic equations[J]. ESAIM: M2AN.,40(6):1053–1067,2006.
[119] X. Ye. On the relationship between finite volume and finite element methods ap-plied to the Stokes equations[J]. Numer. Methods Partial Diferential Equations,17(5):440–453,2001.
[120] X. Ye and L. Zikatanov. Some observations on Babus ka and Brezzi theories[J].Numer.Math.,94(1):195–202,2003.
[121] Xiu Ye. A new discontinuous finite volume method for elliptic problems[J]. SIAMJ. Numer. Anal.,42(3):1062–1072(electronic),2004.
[122] Changhua Yu and Yonghai Li. Biquadratic finite volume element method basedon optimal stress points for second order hyperbolic equations[J]. J. Comp. App.Math.,236:1055–1068,2011.
[123] Zhimin Zhang. Analysis of some quadrilateral nonconforming elements for incom-pressible elasticity[J]. SIAM J. Numer. Anal.,34(2):640–663,1997.