四阶椭圆方程的非协调有限元方法
|
摘要
在论文中,我们主要讨论了四阶椭圆问题的一些非协调有限元逼近。由于技术上的困难,我们通常采用非协调有限元来逼近四阶问题。但是,并不是所有的板元对四阶奇异摄动问题都关于摄动参数一致收敛,而且大多数的讨论都是在网格满足正则性条件或拟一致假设的前提下进行的。本文主要是在不同的网格剖分下讨论了一些非协调板元应用到四阶板弯曲问题和四阶奇异摄动问题时的收敛性。
第一章,我们给出了一些基本空间的定义,记号和有限元方法的一些基础知识以及其相关的性质。
第二章,我们给出了两个Morley型的非C~0非协调板元在各向异性网格下对四阶板弯曲问题的逼近,利用Poincare不等式得到了插值误差的一个显式估计,通过引入特殊的插值算子得到了相容误差的估计,从而证明了收敛性结果,得到了能量模和零模意义下和正则网格下相同的收敛速度,相应的数值例子再次验证了我们的理论分析。
第三章,我们首先通过一个反例说明了第二章中的第一个Morley型的非C~0非协调板元逼近像Poisson方程这样的二阶问题时并不收敛,这就意味着该单元在通常的有限元离散格式下逼近四阶奇异摄动问题时关于奇异摄动参数不是一致收敛的。接着,我们证明了它在一个修正的有限元离散格式下是收敛的,并且在各向异性网格下也得到了关于摄动参数一致收敛的估计结果,并进行了相关的数值试验。最后,我们证明了无论网格是否满足正则性条件或拟一致假设,第二章中的第二个Morley型非协调板元逼近四阶奇异摄动问题时关于奇异摄动参数都是一致收敛的。由于该单元也是非C~0的,这就表明对四阶奇异摄动问题的收敛性并不要求有限元空间一定是H~1的子空间。最后的数值例子再次表明了该单元的有效性。
第四章,在本章中,我们首先证明了一类C~0型非协调板元逼近四阶奇异摄动问题时的一个一般收敛性结果,并给出了边界层情形时的误差分析,得到了只依赖于右端项的误差估计。接着,我们分析了一个双参数三角形元的性质,利用前面的结论得到了相应的收敛性结果,最后,我们也进行了相关的数值试验。
In this thesis,several nonconforming finite element methods for plate problems and fourth order elliptic singular perturbation problems are discussed.For the reason of technique difficulties,nonconforming plate elements are always employed to solve plate bending problems.Many successful plate elements have been constructed,however, they are not always convergent for fourth order singular perturbation problems.On the other hand,most of the discussions are based on regular condition or quasi-uniform assumption of triangulations.Here,we focus on the convergence analysis of different kinds of nonconforming finite element approximations to fourth order elliptic problems with different mesh fashions.
In Chapter 1,we introduce some definitions and notations of basic spaces,present the fundamental knowledge and relational properties of finite element method,such as some important notions,inequalities and useful theorems.
In Chapter 2,the approximation of plate bending problem by employing two Morley-type non-C~0 nonconforming plate elements under anisotropic meshes are discussed, the optimal anisotropic interpolation error and consistency error estimates are obtained by using some novel approaches.Some numerical tests are given to confirm the theoretical analysis.
In Chapter 3,a counterexample is given to show that the first element in Chapter 2 diverges for the reduced second order equations,which means that it is not convergent for fourth order elliptic singular perturbation problems uniformly.As an alternative,the convergence in the energy norm uniformly with respect to the perturbation parameter is derived when a modified finite element approximation is employed.Numerical experiments are carried out to confirm our theoretical results.Moreover,by employing some new tricks,we obtain the convergence results of the second element in Chapter 2 even when the regular condition or quasi-uniform assumption on the triangulation is not satisfied.Since this element is also non-C~0,this means that the convergence does not require the finite element space being a subspace of H~1(Ω).Similar estimates are also presented for other two different approximation formulations.Numerical results are also given at the end of this chapter.
In Chapter 4,a general convergence theorem for C~0 nonconforming finite element to solve the elliptic fourth order singular perturbation problem is presented.The error estimates only related to the righthand term of the equation are also derived when there exist boundary layers.Then the properties of a nine parameter triangular element constructed by double set method is studied,the corresponding convergence results are obtained by the former theorem,numerical experiments are carried out to check our analysis results.
第一章,我们给出了一些基本空间的定义,记号和有限元方法的一些基础知识以及其相关的性质。
第二章,我们给出了两个Morley型的非C~0非协调板元在各向异性网格下对四阶板弯曲问题的逼近,利用Poincare不等式得到了插值误差的一个显式估计,通过引入特殊的插值算子得到了相容误差的估计,从而证明了收敛性结果,得到了能量模和零模意义下和正则网格下相同的收敛速度,相应的数值例子再次验证了我们的理论分析。
第三章,我们首先通过一个反例说明了第二章中的第一个Morley型的非C~0非协调板元逼近像Poisson方程这样的二阶问题时并不收敛,这就意味着该单元在通常的有限元离散格式下逼近四阶奇异摄动问题时关于奇异摄动参数不是一致收敛的。接着,我们证明了它在一个修正的有限元离散格式下是收敛的,并且在各向异性网格下也得到了关于摄动参数一致收敛的估计结果,并进行了相关的数值试验。最后,我们证明了无论网格是否满足正则性条件或拟一致假设,第二章中的第二个Morley型非协调板元逼近四阶奇异摄动问题时关于奇异摄动参数都是一致收敛的。由于该单元也是非C~0的,这就表明对四阶奇异摄动问题的收敛性并不要求有限元空间一定是H~1的子空间。最后的数值例子再次表明了该单元的有效性。
第四章,在本章中,我们首先证明了一类C~0型非协调板元逼近四阶奇异摄动问题时的一个一般收敛性结果,并给出了边界层情形时的误差分析,得到了只依赖于右端项的误差估计。接着,我们分析了一个双参数三角形元的性质,利用前面的结论得到了相应的收敛性结果,最后,我们也进行了相关的数值试验。
In this thesis,several nonconforming finite element methods for plate problems and fourth order elliptic singular perturbation problems are discussed.For the reason of technique difficulties,nonconforming plate elements are always employed to solve plate bending problems.Many successful plate elements have been constructed,however, they are not always convergent for fourth order singular perturbation problems.On the other hand,most of the discussions are based on regular condition or quasi-uniform assumption of triangulations.Here,we focus on the convergence analysis of different kinds of nonconforming finite element approximations to fourth order elliptic problems with different mesh fashions.
In Chapter 1,we introduce some definitions and notations of basic spaces,present the fundamental knowledge and relational properties of finite element method,such as some important notions,inequalities and useful theorems.
In Chapter 2,the approximation of plate bending problem by employing two Morley-type non-C~0 nonconforming plate elements under anisotropic meshes are discussed, the optimal anisotropic interpolation error and consistency error estimates are obtained by using some novel approaches.Some numerical tests are given to confirm the theoretical analysis.
In Chapter 3,a counterexample is given to show that the first element in Chapter 2 diverges for the reduced second order equations,which means that it is not convergent for fourth order elliptic singular perturbation problems uniformly.As an alternative,the convergence in the energy norm uniformly with respect to the perturbation parameter is derived when a modified finite element approximation is employed.Numerical experiments are carried out to confirm our theoretical results.Moreover,by employing some new tricks,we obtain the convergence results of the second element in Chapter 2 even when the regular condition or quasi-uniform assumption on the triangulation is not satisfied.Since this element is also non-C~0,this means that the convergence does not require the finite element space being a subspace of H~1(Ω).Similar estimates are also presented for other two different approximation formulations.Numerical results are also given at the end of this chapter.
In Chapter 4,a general convergence theorem for C~0 nonconforming finite element to solve the elliptic fourth order singular perturbation problem is presented.The error estimates only related to the righthand term of the equation are also derived when there exist boundary layers.Then the properties of a nine parameter triangular element constructed by double set method is studied,the corresponding convergence results are obtained by the former theorem,numerical experiments are carried out to check our analysis results.
引文
[1]G.Acosta,R.G.Duran,The maximum angle condition for mixed and nonconforming element:Application to Stokes equations[J],SIAM J.Numer.Anal.,1999,37(1):18-36.
[2]R.A.Adams,Sobolev spaces[M],New York:Academic Press,1975.
[3]T.Apel,Anisotropic finite element:Local estimates and approximations[M],B.G.Teubner Leipzig,1999.
[4]T.Apel,M.Dobrowolski,Anisotropic interpolation with applications to the finite element method [J],Computing.,1992,47:277-293.
[5]T.Apel,S.Nicaise,J.Sch(o|¨)berl,Crouzeix-Raviart Type finite elements on anisotropic meshes[J],Numer.Math.,2001,89:193-223.
[6]T.Apel,S.Nicaise,J.Sch(o|¨)berl,A nonconforming finite element method with anisotropic mesh grading for the stokes problem in domains with edges[J],IMA J.Numer.Anal.,2001,21:843-856.
[7]J.W.Barrett,J.F.Blowey,H.Garcke,Finite element approximation of a fourth order nonlinear degenerate parabolic equation[J],Numer.Math.,1998,80:525-556.
[8]G.P.Bazeley,Y.K.Cheung,B.M.Irons,O.C.Zienkiewicz,Triangular elements in bending-conforming and nonconforming solutions[C],in Proceedings of the Conference on Matrix Methods in Structural Mechanics,Wright Patterson A.F.B.,Ohio,1965.
[9]L.Beirao da Veiga,J.Niiranen,R.Stenberg,A posteriori error estimates for the Morley plate bending element[J],Numer.Math.,2007,106(2):165-179.
[10]S.C.Brenner,L.R.Scott,The Mathematical theory of finite element method[M],New York,Springer-Verlag,1994.
[11]X.M.Bu,双参数法的一些扩展及其在矩形板元中的应用[J],计算数学,J995,17(1):65-72.
[12]J.W.Cahn,C.M.Elliott,A.Novick-Cohen,The Cahn-Hilliard equation with a concentration dependent mobility:motion by minus the Laplacian of the mean curvature[J],European J.Appl.Math.,1996,7(3):287-301.
[13]S.C.Chen,Y.Li,S.P.Mao,An anisotropic,superconvergent nonconforming plate finite element [J],J.Comput.Appl.Math.,2008,220:65-72.
[14]S.C.Chen,D.Y.Shi,G.B.Ren,Three dimension quasi-Wilson element for plat hexahedron meshes[J],J.Comput.Math.,2004,22(2):178-187.
[15]S.C.Chen,D.Y.Shi,Y.C.Zhao,Anisotropic Interpolation and Quasi-Wilson Element for Narrow Quadrilateral Meshes[J],IMA.J.Numer.Anal.,2004,24:77-95.
[16]S.C.Chen,D.Y.Shi,H.Ichiro,Generalized estimate formulation of nonconforming finite elements [J],Numer.Math.Sinica,2000,22(3):295-300.
[17]S.C.Chen,Y.C.Zhao,D.Y.Shi,Anisotropic interpolation with application to nonconforming finite elements[J],Appl.Numer.Math.,2004,49(2):135-152.
[18]S.C.Chen,Y.C.Zhao and D.Y.Shi,Non C~0 nonconforming elements for elliptic fourth order singular perpurbation problem[J],J.Comptu.Math.,2005,23(2):185-198.
[19]S.C.Chen,L.C.Xiao,Interpolation theory of anisotropic finite elements and applications[J],Science in China,Series A,2008,51(8):1361-1375.
[20]陈绍春,石钟慈,构造单元刚度矩阵的双参数方法[J],计算数学,1991,13(3):286-296.
[21]P.G.Ciarlet,The finite element method for elliptic problems[M],Amsterdam,North-Holland,1978.
[22]R.G.Duran,Error estimates for anisotropic finite elements and applications[C],In:Proceedings of the International Congress of Mathematicians,Volume Ⅲ,Madrid:EuroMS,2006,1181-1200.
[23]R.G.Duran,A.L.Lombardi,Error estimates for the Raviart-Thomas interpolation under the maximum angle condition[J],SIAM J.Numer.Anat.,2008,46(3):1442-1453.
[24]冯康,石钟慈,弹性结构的数学理论[M],北京:科学出版社,1981.
[25]L.Formaggia,S.Perotto,New anisotropic a priori error estimates[J],Numer.Math.,2001,89(4):641-667.
[26]L.Formaggia,S.Perotto,Anisotropic error estimates for elliptic problems[J],Numer.Math.,2003,94(1):67-92.
[27]J.Hu,Z.C.Shi,A new a posteriori error estimate for the Morley element[J],Numer.Math.,2009,112(1):25-40.
[28]B.M.Irons,A.Razzaque,Experience with Patch Test for convergence of finite elements[C],Proc.Symps.on Mathematical Foundations of Finite Element Method with Applications to Partial Differential Operators,(Baltimore,1972),A.K.Aziz ed.,Academic Press,New York,1972,557-587.
[29]P.Lascaux and P.Lesaint,Some nonconforming finite element for the plate bending problem[J],RAIRO,Anal.Numer.,1975,R-I:9-53.
[30]李开泰,黄艾香,黄庆怀.有限元方法及其应用[M].北京:科学出版社,2006.
[31]林群,严宁宁,高效有限元与分析[M],保定:河北大学出版社,1996.
[32]Z.C.Lin,Singular pertubation of the fourth order elliptic equation when the limit equation is elliptic-parabolic[J],Appl.Math.Mech.,1991,12(1):77-84.
[33]S.T.Liu,Y.S.Xu,Galerkin Methods based on Hermite splines for singular perturbation problems [J],SIAM J.Numer.Anal.,2006,43(6):2607-2623.
[34]Y.Q.Long,K.G.Xin,Generalized conforming element for bending and buckling analysis of plates [J],Finite Element analysis & design,1989,5:15-30.
[35]龙驭球,辛克贵,广义协调元[J],土木工程学报,1987,20(1):1-14.
[36]S.P.Mao,S.C.Chen,Convergence analysis of Morley element on anisotropic meshes[J],J.Comput.Math.,2006,2(24):169-180.
[37]S.P.Mao,S.C.Chen,Accuracy analysis of Adini's nonconforming plate element on anisotropic meshes[J],Commun.Numer.Meth.Engng.,2006,22(5):433-440.
[38]S.P.Mao,S.C.Chen,H.X.Sun,A quadrilateral,anisotropic,superconvergent,nonconforming double set parameter element[J],Appl.Numer.Math.,2006,56(7):937-961.
[39]S.P.Mao,Z.C.Shi,High accuracy analysis of two nonconforming plate elements[J],Numer.Math.,2009,111(3):407-433.
[40]L.S.D.Morley,The triangular equilibrium element in the solution of plate bending problems[J],Aero.Quart.,1968,19:149-169.
[41]T.K.Nissen,X.C.Tai,R.Winther,A robust nonconforming H~2-element[J],Math.Comp.,2001,70(234):489-505.
[42]A.Novick-Cohen,On Cahn-Hilliard type equations[J],Nonlinear Anal.,1990,15(9):797-814.
[43]L.E.Payne,H.F.Weinberger,An optimal Poincare inequality for convex domain[J],Arch.Rational.Mech.Anal.,1960,5:286-292.
[44]I.Pitacco,A new plate bending element based on orthogonal polynomials expansion of the curvature field[J],Int.J.Numer.Meth.Engng.,2007,72:156-179.
[45]S.B.Pomeranz,A posteriori finite element method error estimates for fourth-order problems[J],Commun.Numer.Meth.Engng,1995,11(2):213-226.
[46]B.Semper,Conforming finite element approximations for a fourth order singular perturbation problem [J],SIAM J.Numer.Anal.,1992,29(4),1043-1058.
[47]D.Y.Shi,S.C.Chen,I.Hagiwara,Convergence analysis for a nonconforming membrane element on anisotropic meshes[J],J.Comput.Math.,2005,23(4):373-382.
[48]D.Y.Shi,S.P.Mao,S.C.Chen,On the anisotropic accuracy analysis of ACM's nonconforming finite element[J],J.Comput.Math.,2005,23(6):635-646.
[49]D.Y.Shi,S.P.Mao,S.C.Chen,An anisotropic nonconforming finite element with some superconvergence results[J],J.Comput.Math.,2005,23(2):261-274.
[50]石东洋,陈绍春,一类八自由度矩形板元[J],应用数学,1996,3:303-306.
[51]石东洋,陈绍春,构造8-参数非协调矩形板元的新方法[J],工程数学学报,2001,18(3):83-88.
[52]石东洋,宋士仓,荷载属于H~(-1)(Ω)时不完全双二次非协调板元的新的误差估计式[J],高校应用数学学报A辑,2003,18(1):39-44.
[53]Z.C.Shi,An explicit analysis of Stummel's patch test examples[J],Int.J.Numer.Meth.Engrg.,1984,20:1233-1246.
[54]Z.C.Shi,The generalized patch test for Zienkiewicz's triangle[J],J.Comput.Math.,1984,2:276-286.
[55]Z.C.Shi,The F-E-M Test for convergence of nonconforming finite elements[J],Math.Comp.,1987,49(180):391-405.
[56]Z.C.Shi,S.C.Chen,Analysis of a nine degree plate bending element of Specht[J],Chinese J.Numer.Math.Appl.,1989,4:73-79.
[57]Z.C.Shi,On the accuacy of quasi-conforming and generalized comforming finite elements[J],Chin.Ann.of Math.,1990,11B(2):148-155.
[58]石钟慈,陈绍春,九参数广义协调元的收敛性[J],计算数学,1991,13(2):193-203.
[59]石钟慈.关于不完全二次板元的收敛性[J],计算数学,1986,8(1):53-62.
[60]石钟慈.关于Morley元的误差估计[J],计算数学,1990,2:113-118.
[61]石钟慈,陈千勇,一个高精度矩形板元[J],中国科学(A辑),2000,30(6):504-515.
[62]G.Strang,Variational crimes in the finite element method[C],in the Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations,A.R.Aziz,ed.,Academic Press,New York,1972,689-710.
[63]F.Stummel,The generalized patch test[J],SIAM J.Numer.Anal.,1979,16:449-471.
[64]F.Stummel,The limitations of the Patch Test[J],Int.J.Numer.Meth.Engrg.,1980,15:177-188.
[65]G.F.Sun,M.Stynes,Finite-element methods for singularly perturbed high-order elliptic two-point boundary problems,Ⅰ:Reaction-diffusion-type problems[J],IMA J.Numer.Anal.,1995,15(1):117-139.
[66]G.F.Sun,M.Stynes,Finite-element methods for singularly perturbed high-order elliptic two-point boundary problems,Ⅱ:Convection-diffusion-type problems[J],IMA J.Numer.Anal.,1995,15(2):197-219.
[67]王烈衡,许学军,有限元方法的数学基础[M],北京:科学出版社,2004.
[68]M.Wang,J.C.Xu,Y.C.Hu,Modified Morley element method for a fourth order elliptic singular perturbation problem[J],J.Comput.Math.,2006,24(2):113-120.
[69]M.Wang,J.C.Xu,The Morley element for fourth order elliptic equations in any dimensions[J],Numer.Math.,2006,103:155-169.
[70]M.Wang,J.C.Xu,Some tetrahedron nonconforming elements for fourth order elliptic equations [J],Math.Comp.,2007,76:1-18.
[71]M.Wang,Z.C.Shi,J.C.Xu,Some n-rectangle nonconforming elements for fourth order elliptic equations[J],J.Comput.Math.,2007,25(4):408-420.
[72]M.Wang,On the necessity and sufficiency of the patch test for convergence of nonconforming finite elements[J],SIAM J.Numer.Anal.,2001,39(2):363-384.
[73]M.Wang,Error estimates of nonconforming finite elements for the biharmonic equation[J],J.Comput.Math.,1993,11:276-288.
[74]M.Wang,X.R.Meng,A robust finite element method for a 3-D elliptic singular perturbation problem[J],J.Comput.Math.,2007,25:631-633.
[75]吴茂庆,不完全双二次非协调板元,苏州大学学报[J],1983,1:20-29.
[76]P.L.Xie,D.Y.Shi,H.Li,A new robust C~0 nonconforming triangular element for fourth order elliptic singular perturbation problems[J],submitted to Applied Mathematics and Computation.
[77]张鸿庆,王鸣,有限元的数学理论[M],北京:科学出版社,1991.
[78]S.Zhang,M.Wang,A posteriori estimator of nonconforming finite element method for fourth order perturbation problems[J],J.Comput.Math.,2008,26(4):554-577.
[79]A.Zenisek,M.Vanmaele,The interpolation theorem for narrow quadrilateral isoparametric finite elements[J],Numer.Math.,1995,72:123-141.
[80]A.Zenisek,M.Vanmaele,Applicability of the Bramble-Hilbert Lemma in interpolation problems of narrow quadrilateral isoparametric finite elements IJ],J.Comp.Appl.Math.,1995,63:109-122.
[81]O.C.Zienkiewicz,R.L.Taylor.The finite element method IMI,Fourth Ed.,London:McGraw-Hill Book Company,1989.
[2]R.A.Adams,Sobolev spaces[M],New York:Academic Press,1975.
[3]T.Apel,Anisotropic finite element:Local estimates and approximations[M],B.G.Teubner Leipzig,1999.
[4]T.Apel,M.Dobrowolski,Anisotropic interpolation with applications to the finite element method [J],Computing.,1992,47:277-293.
[5]T.Apel,S.Nicaise,J.Sch(o|¨)berl,Crouzeix-Raviart Type finite elements on anisotropic meshes[J],Numer.Math.,2001,89:193-223.
[6]T.Apel,S.Nicaise,J.Sch(o|¨)berl,A nonconforming finite element method with anisotropic mesh grading for the stokes problem in domains with edges[J],IMA J.Numer.Anal.,2001,21:843-856.
[7]J.W.Barrett,J.F.Blowey,H.Garcke,Finite element approximation of a fourth order nonlinear degenerate parabolic equation[J],Numer.Math.,1998,80:525-556.
[8]G.P.Bazeley,Y.K.Cheung,B.M.Irons,O.C.Zienkiewicz,Triangular elements in bending-conforming and nonconforming solutions[C],in Proceedings of the Conference on Matrix Methods in Structural Mechanics,Wright Patterson A.F.B.,Ohio,1965.
[9]L.Beirao da Veiga,J.Niiranen,R.Stenberg,A posteriori error estimates for the Morley plate bending element[J],Numer.Math.,2007,106(2):165-179.
[10]S.C.Brenner,L.R.Scott,The Mathematical theory of finite element method[M],New York,Springer-Verlag,1994.
[11]X.M.Bu,双参数法的一些扩展及其在矩形板元中的应用[J],计算数学,J995,17(1):65-72.
[12]J.W.Cahn,C.M.Elliott,A.Novick-Cohen,The Cahn-Hilliard equation with a concentration dependent mobility:motion by minus the Laplacian of the mean curvature[J],European J.Appl.Math.,1996,7(3):287-301.
[13]S.C.Chen,Y.Li,S.P.Mao,An anisotropic,superconvergent nonconforming plate finite element [J],J.Comput.Appl.Math.,2008,220:65-72.
[14]S.C.Chen,D.Y.Shi,G.B.Ren,Three dimension quasi-Wilson element for plat hexahedron meshes[J],J.Comput.Math.,2004,22(2):178-187.
[15]S.C.Chen,D.Y.Shi,Y.C.Zhao,Anisotropic Interpolation and Quasi-Wilson Element for Narrow Quadrilateral Meshes[J],IMA.J.Numer.Anal.,2004,24:77-95.
[16]S.C.Chen,D.Y.Shi,H.Ichiro,Generalized estimate formulation of nonconforming finite elements [J],Numer.Math.Sinica,2000,22(3):295-300.
[17]S.C.Chen,Y.C.Zhao,D.Y.Shi,Anisotropic interpolation with application to nonconforming finite elements[J],Appl.Numer.Math.,2004,49(2):135-152.
[18]S.C.Chen,Y.C.Zhao and D.Y.Shi,Non C~0 nonconforming elements for elliptic fourth order singular perpurbation problem[J],J.Comptu.Math.,2005,23(2):185-198.
[19]S.C.Chen,L.C.Xiao,Interpolation theory of anisotropic finite elements and applications[J],Science in China,Series A,2008,51(8):1361-1375.
[20]陈绍春,石钟慈,构造单元刚度矩阵的双参数方法[J],计算数学,1991,13(3):286-296.
[21]P.G.Ciarlet,The finite element method for elliptic problems[M],Amsterdam,North-Holland,1978.
[22]R.G.Duran,Error estimates for anisotropic finite elements and applications[C],In:Proceedings of the International Congress of Mathematicians,Volume Ⅲ,Madrid:EuroMS,2006,1181-1200.
[23]R.G.Duran,A.L.Lombardi,Error estimates for the Raviart-Thomas interpolation under the maximum angle condition[J],SIAM J.Numer.Anat.,2008,46(3):1442-1453.
[24]冯康,石钟慈,弹性结构的数学理论[M],北京:科学出版社,1981.
[25]L.Formaggia,S.Perotto,New anisotropic a priori error estimates[J],Numer.Math.,2001,89(4):641-667.
[26]L.Formaggia,S.Perotto,Anisotropic error estimates for elliptic problems[J],Numer.Math.,2003,94(1):67-92.
[27]J.Hu,Z.C.Shi,A new a posteriori error estimate for the Morley element[J],Numer.Math.,2009,112(1):25-40.
[28]B.M.Irons,A.Razzaque,Experience with Patch Test for convergence of finite elements[C],Proc.Symps.on Mathematical Foundations of Finite Element Method with Applications to Partial Differential Operators,(Baltimore,1972),A.K.Aziz ed.,Academic Press,New York,1972,557-587.
[29]P.Lascaux and P.Lesaint,Some nonconforming finite element for the plate bending problem[J],RAIRO,Anal.Numer.,1975,R-I:9-53.
[30]李开泰,黄艾香,黄庆怀.有限元方法及其应用[M].北京:科学出版社,2006.
[31]林群,严宁宁,高效有限元与分析[M],保定:河北大学出版社,1996.
[32]Z.C.Lin,Singular pertubation of the fourth order elliptic equation when the limit equation is elliptic-parabolic[J],Appl.Math.Mech.,1991,12(1):77-84.
[33]S.T.Liu,Y.S.Xu,Galerkin Methods based on Hermite splines for singular perturbation problems [J],SIAM J.Numer.Anal.,2006,43(6):2607-2623.
[34]Y.Q.Long,K.G.Xin,Generalized conforming element for bending and buckling analysis of plates [J],Finite Element analysis & design,1989,5:15-30.
[35]龙驭球,辛克贵,广义协调元[J],土木工程学报,1987,20(1):1-14.
[36]S.P.Mao,S.C.Chen,Convergence analysis of Morley element on anisotropic meshes[J],J.Comput.Math.,2006,2(24):169-180.
[37]S.P.Mao,S.C.Chen,Accuracy analysis of Adini's nonconforming plate element on anisotropic meshes[J],Commun.Numer.Meth.Engng.,2006,22(5):433-440.
[38]S.P.Mao,S.C.Chen,H.X.Sun,A quadrilateral,anisotropic,superconvergent,nonconforming double set parameter element[J],Appl.Numer.Math.,2006,56(7):937-961.
[39]S.P.Mao,Z.C.Shi,High accuracy analysis of two nonconforming plate elements[J],Numer.Math.,2009,111(3):407-433.
[40]L.S.D.Morley,The triangular equilibrium element in the solution of plate bending problems[J],Aero.Quart.,1968,19:149-169.
[41]T.K.Nissen,X.C.Tai,R.Winther,A robust nonconforming H~2-element[J],Math.Comp.,2001,70(234):489-505.
[42]A.Novick-Cohen,On Cahn-Hilliard type equations[J],Nonlinear Anal.,1990,15(9):797-814.
[43]L.E.Payne,H.F.Weinberger,An optimal Poincare inequality for convex domain[J],Arch.Rational.Mech.Anal.,1960,5:286-292.
[44]I.Pitacco,A new plate bending element based on orthogonal polynomials expansion of the curvature field[J],Int.J.Numer.Meth.Engng.,2007,72:156-179.
[45]S.B.Pomeranz,A posteriori finite element method error estimates for fourth-order problems[J],Commun.Numer.Meth.Engng,1995,11(2):213-226.
[46]B.Semper,Conforming finite element approximations for a fourth order singular perturbation problem [J],SIAM J.Numer.Anal.,1992,29(4),1043-1058.
[47]D.Y.Shi,S.C.Chen,I.Hagiwara,Convergence analysis for a nonconforming membrane element on anisotropic meshes[J],J.Comput.Math.,2005,23(4):373-382.
[48]D.Y.Shi,S.P.Mao,S.C.Chen,On the anisotropic accuracy analysis of ACM's nonconforming finite element[J],J.Comput.Math.,2005,23(6):635-646.
[49]D.Y.Shi,S.P.Mao,S.C.Chen,An anisotropic nonconforming finite element with some superconvergence results[J],J.Comput.Math.,2005,23(2):261-274.
[50]石东洋,陈绍春,一类八自由度矩形板元[J],应用数学,1996,3:303-306.
[51]石东洋,陈绍春,构造8-参数非协调矩形板元的新方法[J],工程数学学报,2001,18(3):83-88.
[52]石东洋,宋士仓,荷载属于H~(-1)(Ω)时不完全双二次非协调板元的新的误差估计式[J],高校应用数学学报A辑,2003,18(1):39-44.
[53]Z.C.Shi,An explicit analysis of Stummel's patch test examples[J],Int.J.Numer.Meth.Engrg.,1984,20:1233-1246.
[54]Z.C.Shi,The generalized patch test for Zienkiewicz's triangle[J],J.Comput.Math.,1984,2:276-286.
[55]Z.C.Shi,The F-E-M Test for convergence of nonconforming finite elements[J],Math.Comp.,1987,49(180):391-405.
[56]Z.C.Shi,S.C.Chen,Analysis of a nine degree plate bending element of Specht[J],Chinese J.Numer.Math.Appl.,1989,4:73-79.
[57]Z.C.Shi,On the accuacy of quasi-conforming and generalized comforming finite elements[J],Chin.Ann.of Math.,1990,11B(2):148-155.
[58]石钟慈,陈绍春,九参数广义协调元的收敛性[J],计算数学,1991,13(2):193-203.
[59]石钟慈.关于不完全二次板元的收敛性[J],计算数学,1986,8(1):53-62.
[60]石钟慈.关于Morley元的误差估计[J],计算数学,1990,2:113-118.
[61]石钟慈,陈千勇,一个高精度矩形板元[J],中国科学(A辑),2000,30(6):504-515.
[62]G.Strang,Variational crimes in the finite element method[C],in the Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations,A.R.Aziz,ed.,Academic Press,New York,1972,689-710.
[63]F.Stummel,The generalized patch test[J],SIAM J.Numer.Anal.,1979,16:449-471.
[64]F.Stummel,The limitations of the Patch Test[J],Int.J.Numer.Meth.Engrg.,1980,15:177-188.
[65]G.F.Sun,M.Stynes,Finite-element methods for singularly perturbed high-order elliptic two-point boundary problems,Ⅰ:Reaction-diffusion-type problems[J],IMA J.Numer.Anal.,1995,15(1):117-139.
[66]G.F.Sun,M.Stynes,Finite-element methods for singularly perturbed high-order elliptic two-point boundary problems,Ⅱ:Convection-diffusion-type problems[J],IMA J.Numer.Anal.,1995,15(2):197-219.
[67]王烈衡,许学军,有限元方法的数学基础[M],北京:科学出版社,2004.
[68]M.Wang,J.C.Xu,Y.C.Hu,Modified Morley element method for a fourth order elliptic singular perturbation problem[J],J.Comput.Math.,2006,24(2):113-120.
[69]M.Wang,J.C.Xu,The Morley element for fourth order elliptic equations in any dimensions[J],Numer.Math.,2006,103:155-169.
[70]M.Wang,J.C.Xu,Some tetrahedron nonconforming elements for fourth order elliptic equations [J],Math.Comp.,2007,76:1-18.
[71]M.Wang,Z.C.Shi,J.C.Xu,Some n-rectangle nonconforming elements for fourth order elliptic equations[J],J.Comput.Math.,2007,25(4):408-420.
[72]M.Wang,On the necessity and sufficiency of the patch test for convergence of nonconforming finite elements[J],SIAM J.Numer.Anal.,2001,39(2):363-384.
[73]M.Wang,Error estimates of nonconforming finite elements for the biharmonic equation[J],J.Comput.Math.,1993,11:276-288.
[74]M.Wang,X.R.Meng,A robust finite element method for a 3-D elliptic singular perturbation problem[J],J.Comput.Math.,2007,25:631-633.
[75]吴茂庆,不完全双二次非协调板元,苏州大学学报[J],1983,1:20-29.
[76]P.L.Xie,D.Y.Shi,H.Li,A new robust C~0 nonconforming triangular element for fourth order elliptic singular perturbation problems[J],submitted to Applied Mathematics and Computation.
[77]张鸿庆,王鸣,有限元的数学理论[M],北京:科学出版社,1991.
[78]S.Zhang,M.Wang,A posteriori estimator of nonconforming finite element method for fourth order perturbation problems[J],J.Comput.Math.,2008,26(4):554-577.
[79]A.Zenisek,M.Vanmaele,The interpolation theorem for narrow quadrilateral isoparametric finite elements[J],Numer.Math.,1995,72:123-141.
[80]A.Zenisek,M.Vanmaele,Applicability of the Bramble-Hilbert Lemma in interpolation problems of narrow quadrilateral isoparametric finite elements IJ],J.Comp.Appl.Math.,1995,63:109-122.
[81]O.C.Zienkiewicz,R.L.Taylor.The finite element method IMI,Fourth Ed.,London:McGraw-Hill Book Company,1989.