三维四阶问题及不可压缩流的有限元分析
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摘要
本文针对三维四阶椭圆方程本文构造了两个新的非协调四面体单元,一个是十四参数单元,单元自由度个数较少；另一个为十六参数单元,具有能量正交的形函数空间。证明了两个新单元对三维四阶椭圆方程收敛。
文章的第二部分,关于三维四阶椭圆方程我们构造出了一个20参数三棱柱单元,并分析了该单元的收敛性。
对于定常Stokes问题,我们在压力空间取Q={q∈H1(Ω)；∫Ωqdx=0}情形下得到了新的变分形式,关于新的变分问题我们构造出了基于泡函数的任意阶混合元格式,给出了误差估计。同时,还给出了压力空间取二次元时基于泡函数新的一阶、二阶和三阶格式,所得结果中对压力的误差估计比已有的相应阶格式收敛阶分别高出一阶。
关于Darcy-Stokes问题我们运用宏元技巧探讨了P2-P1元在不规则十字网格、扭曲十字网格以及重心剖分网格上的一致稳定性和逼近性。
In this paper, two new nonconforming tetrahedral elements are presented for the fourth order elliptic partial differential operators in three spatial dimensions. Of the two newly constructed elements one is the 14-parameter tetrahedral element which has lower degree, the other is the 16-parameter energy-orthogonal tetrahedral element which has an energy-orthogonal shape function space. These two elements are proved to be convergent for a model biharmonic equation in three dimensions.
In the second part of this paper, we constructed a 20-parameter triangular prism element for the fourth order elliptic equation and analysed the approximation of this element.
For the stationnary Stokes problem, when the pressure space is taken as Q = {q∈H1 (Ω);∫Ωqdx= 0}, we get the new variational form. The mixed element formats of any order based on bubble func-tions are presented for the new variational problem in triangular and tetrahedral meshes. and the convergence of any order mixed element formats are proved. Furthermore, when the pressure finite element space consists piecewise polynomials of degree two, we propose the new mixed element formats of one order, two order and three order based on bubble functions for the new variational problem, the error estimates of the pressure in L2-norm for the new formats are one degree higher than that of the existing formats.
At the end of this paper, by the macroelement partition theorem, we study the uniform stability and approximation properties of P2-P1 element for Darcy-Stokes problem on irregular crisscross meshes, on distorted crisscross meshes and on barycentric trisected meshes respectively.
文章的第二部分,关于三维四阶椭圆方程我们构造出了一个20参数三棱柱单元,并分析了该单元的收敛性。
对于定常Stokes问题,我们在压力空间取Q={q∈H1(Ω)；∫Ωqdx=0}情形下得到了新的变分形式,关于新的变分问题我们构造出了基于泡函数的任意阶混合元格式,给出了误差估计。同时,还给出了压力空间取二次元时基于泡函数新的一阶、二阶和三阶格式,所得结果中对压力的误差估计比已有的相应阶格式收敛阶分别高出一阶。
关于Darcy-Stokes问题我们运用宏元技巧探讨了P2-P1元在不规则十字网格、扭曲十字网格以及重心剖分网格上的一致稳定性和逼近性。
In this paper, two new nonconforming tetrahedral elements are presented for the fourth order elliptic partial differential operators in three spatial dimensions. Of the two newly constructed elements one is the 14-parameter tetrahedral element which has lower degree, the other is the 16-parameter energy-orthogonal tetrahedral element which has an energy-orthogonal shape function space. These two elements are proved to be convergent for a model biharmonic equation in three dimensions.
In the second part of this paper, we constructed a 20-parameter triangular prism element for the fourth order elliptic equation and analysed the approximation of this element.
For the stationnary Stokes problem, when the pressure space is taken as Q = {q∈H1 (Ω);∫Ωqdx= 0}, we get the new variational form. The mixed element formats of any order based on bubble func-tions are presented for the new variational problem in triangular and tetrahedral meshes. and the convergence of any order mixed element formats are proved. Furthermore, when the pressure finite element space consists piecewise polynomials of degree two, we propose the new mixed element formats of one order, two order and three order based on bubble functions for the new variational problem, the error estimates of the pressure in L2-norm for the new formats are one degree higher than that of the existing formats.
At the end of this paper, by the macroelement partition theorem, we study the uniform stability and approximation properties of P2-P1 element for Darcy-Stokes problem on irregular crisscross meshes, on distorted crisscross meshes and on barycentric trisected meshes respectively.
引文
[1]冯康.数值计算方法[M].北京：国防工业出版社,1978.
[2]P. G. Ciarlet. The Finite Element Method for Elliptic Problems[M]. North-Holland, Amsterdam,1978.
[3]S. C. Brenner,L. R. Scott. The Mathematical Theory of Finite Element Methods[M]. New York:Springer-Verlag,1998.
[4]姜礼尚,庞之垣.有限元方法及其理论基础[M].北京：人民教育出版社,1979.
[5]李开泰,黄艾香.黄庆怀.有限元方法及其应用[M].西安：西安交通大学出版社,1984.
[6]Zenicek A, A general theorem on triangular finite Cm-elements. RAIRO Anal. Numer.,1974, R-2, 119127.
[7]Zenicek A,Polynomial approximation on tetrahedrons in the finite element method. J. App-rox. Theory,1973,7,334-351.
[8]Landau L and Lifchitz E,Theory of Elasticity. Pergamon Press,London,1959.
[9]Cahn J. W.,Hilliard J. E.,Free enegy of a nonuniform system Ⅰ. Interfacial free enegy, J. Chem. Phys., 1958,28,258-267.
[10]Gatti S,et al.,Hyperbolic relaxation of the viscous Cahn-Hilliard equation in 3-D. M3 AS,15,2(2005),165-198.MR2119676(2006C:35257).
[11]D. N. Arnold, F. Brezzi and M. Fortin. A stable finite element method for the Stokes equations. Calcolo, 1984,21:337-344.
[12]I. Babuska, Error-bound for finite element method. Numer. Math.,1971,16:322-333.
[13]V. Girault and P. A. Raviart. Finite element methods for Navier-Stokes equations. Theory and algo-rithms, Volume 5 of Springer Series in Computational Mathematics. Springer, Berlin,1986.
[14]F. Brezzi. On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers. SIAM J. Anal. Numer.,1974,13:185-197.
[15]C. Bernardi and G. Raugel. Analysis of some mixed finite element methods for the Stokes problem. Math. Comput.,1985, Vol.44 No.169:71-79.
[16]P. Hood, C. Taylor. A numerical solution of the Navier-Stokes equations using the finite element technique. Comp. and Fluids,1973,1:73-100.
[17]M. Crouzeix and P. A. Raviart. Conforming and nonconforming finite element methods for solving the stationary Stokes equations. RAIRO Math. Modeling Numer. Anal.,1973,3:33-75.
[18]F. Brezzi and M. Fortin. Mixed and hybrid finite element methods. Spring-Verlag,1991.
[19]J. Qin. On the convergence of some low order mixed finite elements for incompressible fluids, ph. D. Thesis, Penn State University, Department of Mathematics,1994.
[20]王烈衡Stokes问题的混合有限元分析.计算数学,1987,9卷1期：70-81.
[21]罗振东.三维空间的Stokes问题的混合有限元法.贵州师范大学学报(自然科学版),1988,9卷2期：70-74.
[22]罗振东.Stokes问题的三阶混合元法.贵州科学,1990,8卷2期：282-286.
[23]V. Temam. Navier-Stokes Equation, Theory and Numerical Analysis (Third Edition). North-Holland, Amsterdam, New York Oxford,1984.
[24]K.A. Mardal, X.C. Tai and R. Winther, A robust finite element method for Darcy-Stokes flow. SIAM J. Numer. Anal.2002,40:1605-1631.
[25]M. Crouzeix and P. A. Raviart. Conforming and nonconforming finite element methods for solving the stationary Stokes equations. RAIRO Math. Modeling Numer. Anal.,1973,3:33-75.
[26]P. A. Raviart, J. M. Thomas. A mixed finite element method for second order elliptic problems. Math-ematical Aspects of the Finite Element Method (Galligani I, Magences E, eds). Lecture Notes in Mathematics 606, Springer-Verlag,1977,292-315.
[27]D. N. Arnold, F. Brezzi and M. Fortin. A stable finite element method for the Stokes equations. Calcolo, 1984,21:337-344.
[28]Xue-Cheng Tai and R. Winther. A discrete Rham complex with enhanced smoothness.CALCOLO, 2006,43:287-306.
[29]E. Burman and P. Hansbo. A unified stablized method for stokes'and darcy's equations. J. Comput. Appl. Math.,2007,198(1):35-51.
[30]Chen Long, Sun Pengtao, Xu Jinchao. Optimal anisotropic meshes for minimizing interpolation errors in Lp-norm. Math. Comp.,2007,76(257):179-204.
[31]B. Riviere. Analysis of a discontinuous finite element method for the coupled Stokes and Darcy prob-lems. Journal of Scientific Computing,2005,22:479-500.
[32]B. Riviere and I. Yotov. Locally conservative coupling of Stokes and Darcy flows. SIAM J. Numer. Anal.2005,42(5):1959-1977.
[33]X. P. Xie J.C. Xu and G. R. Xue. Uniformly-stable finite element methods for Darcy-Stokes-Brinkman modes. Math. Comput.,2008, Vol.26, No.3,437-455.
[34]F. Brezzi, Jr. J. Douglas R. Duran and M. Fortin. Mixed finite elements for second order elliptic problems in three variables. Numer. Math.,1987,51:237-250.
[35]F. Brezzi, Jr. J. Douglas M. Fortin and L. D. Marini. Efficient rectangular mixed finite elements in two and three space variables. Math. Model. Numer. Anal.,1987,21:581-604.
[36]F. Brezzi, Jr. J. Douglas and L. D. Marini. Two families of mixed finite element methods for second order elliptic problems. Numer.Math.,1985,47:217-235.
[37]P. A. Raviart, J. M. Thomas. A mixed finite element method for second order elliptic problems. Math-ematical Aspects of the Finite Element Method (Galligani I, Magences E, eds), Lecture Notes in Math-ematics 606, Springer-Verlag,1977,292-315.
[38]P. A. Raviart and J. M. Thomas. Primal hybrid finite element methods for 2nd order elliptic problems. Math.Comp.,1977,31:391-413.
[39]P. G. Ciarlet and P. A. Raviart. A mixed finiteelement method for the biharmonic equation, Symposium on Mathematical Aspects of Finite Element in Partial Differential Equation. C de Boor, Ed., New York: Academic Press,1974,125-143.
[40]R. S. Falk and J. E. Osborn. Error estimates for mixed methods. RAIRO. Numer. Anal.,1980,14(3): 246-277.
[41]C. Johnson. On the convergence of a mixed finite element method for plate bending problem. Numer. Math.,1973,21:43-62.
[42]T. Miyoshi. A finite element method for the solution of fourth order partial differential equations. Kunamoto J.Sc. (Math.) 1974,9:133-149.
[43]D. N. Arnold and R. S. Falk. A new mixed formulation for elasticity. Numer. Math.,1988,53:13-30.
[44]D.N. Arnold and R. Winther. Mixed finite elements for elasticity. Numer. Math.,2002,92:401-419.
[45]D.N. Arnold and R. Winther. Nonconforming mixed finite elements for elasticity. Mathematical Mod-els and Mathods in Applied Science,2003,13(3):295-307.
[46]F. Brezzi. On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers. RAIRO Anal. Numer.,1974,8:129-151.
[47]石钟慈,关于Meley元的误差估计[J].计算数学,1990,12(2),113-118.
[48]Stummel F, Basic compactness properties of nonconforming and hybrid finite element spaces. RAIRO,Anal. Numer.,1980,4(1):81-115.MR0566091 (81h:65058)
[49]Wang Ming, Jinchao Xu,Some tetrahedral Nonconforming finite elements for fourth order elliptic equa-tions. Research Report,79(2004),School of Mathematical Sciences and Institute of Mathematics,Peking University.
[50]Wang Ming, On the necessity and sufficiency of the patch test for convergence of nonconforming finite elements. SIAM J.Numer.Anal.,2002,39(2):363-384.MR1860273 (2002h:65197).
[51]Wang Ming,Jinchao Xu, The Morley element for fourth order elliptic equations in any dimensions. Resear Report,89(2004), School of Mathematical Sciences and Institute of Mathematics,Peking Uni-versity. Numer. Math.,2006,103:155-169.
[52]Wang Ming, Shi Zhong-ci and Jinchao Xu, Some n-rectangle nonconfoming elements for fourth order elliptic equations. Resear Report,50(2005), School of Mathematical Sciences and Institute of Mathe-matics,Peking University.
[53]D. A. Nield and A. Bejan. Convection in porous media. Spring-Verlag,2006.
[54]Scott, Conforming finite element methods for incompressible and nearly incompressible continua. Lec-tures in Applied Mathematics,22(1985), pp.221-244.
[55]Y-S. Lee and P. R. Dawson, Obtaining residual stresses in metal forming after neglecting elasticity on loading. J. of Applied Mechanics,56(1989), pp.318-327.
[56]M. Kitahama and P. R. Dawson, A prectical formulation for Lagrangian viscoplastic analyses with adaptive remeshing. Preprint, Sibley School of Mechanical and Aerospace Engineering, Cornell Uni-versity, August,1990.
[57]C. de Boor and R. Devore, Approximation by smooth multivariate splines. Trans. Amer. Math. Soc., 276(1983), pp.775-788.
[58]C. de Boor and K. Hollig, Approximation order from bivariate C1-cubics:a counterexample. Proc. Amer. Math. Soc.,87(1983), pp.649-655.
[59]B. Mercier, A conforming finite element method for two-dimensinal incompressible elasticity. Internat. J. Numer. Methods Engrg.,14(1979), pp.942-945.
[60]J. F. Ciavaldini and J. C. Nedelec, Sur l'element de Fraeijs de Veubeke et Sander. RAIRO Anal. Numer., 1974.
[61]J. Morgan and L. R. Scott, The dimension of the space of C1 piecewise polynomials. Preprint, August, 1990.
[2]P. G. Ciarlet. The Finite Element Method for Elliptic Problems[M]. North-Holland, Amsterdam,1978.
[3]S. C. Brenner,L. R. Scott. The Mathematical Theory of Finite Element Methods[M]. New York:Springer-Verlag,1998.
[4]姜礼尚,庞之垣.有限元方法及其理论基础[M].北京：人民教育出版社,1979.
[5]李开泰,黄艾香.黄庆怀.有限元方法及其应用[M].西安：西安交通大学出版社,1984.
[6]Zenicek A, A general theorem on triangular finite Cm-elements. RAIRO Anal. Numer.,1974, R-2, 119127.
[7]Zenicek A,Polynomial approximation on tetrahedrons in the finite element method. J. App-rox. Theory,1973,7,334-351.
[8]Landau L and Lifchitz E,Theory of Elasticity. Pergamon Press,London,1959.
[9]Cahn J. W.,Hilliard J. E.,Free enegy of a nonuniform system Ⅰ. Interfacial free enegy, J. Chem. Phys., 1958,28,258-267.
[10]Gatti S,et al.,Hyperbolic relaxation of the viscous Cahn-Hilliard equation in 3-D. M3 AS,15,2(2005),165-198.MR2119676(2006C:35257).
[11]D. N. Arnold, F. Brezzi and M. Fortin. A stable finite element method for the Stokes equations. Calcolo, 1984,21:337-344.
[12]I. Babuska, Error-bound for finite element method. Numer. Math.,1971,16:322-333.
[13]V. Girault and P. A. Raviart. Finite element methods for Navier-Stokes equations. Theory and algo-rithms, Volume 5 of Springer Series in Computational Mathematics. Springer, Berlin,1986.
[14]F. Brezzi. On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers. SIAM J. Anal. Numer.,1974,13:185-197.
[15]C. Bernardi and G. Raugel. Analysis of some mixed finite element methods for the Stokes problem. Math. Comput.,1985, Vol.44 No.169:71-79.
[16]P. Hood, C. Taylor. A numerical solution of the Navier-Stokes equations using the finite element technique. Comp. and Fluids,1973,1:73-100.
[17]M. Crouzeix and P. A. Raviart. Conforming and nonconforming finite element methods for solving the stationary Stokes equations. RAIRO Math. Modeling Numer. Anal.,1973,3:33-75.
[18]F. Brezzi and M. Fortin. Mixed and hybrid finite element methods. Spring-Verlag,1991.
[19]J. Qin. On the convergence of some low order mixed finite elements for incompressible fluids, ph. D. Thesis, Penn State University, Department of Mathematics,1994.
[20]王烈衡Stokes问题的混合有限元分析.计算数学,1987,9卷1期：70-81.
[21]罗振东.三维空间的Stokes问题的混合有限元法.贵州师范大学学报(自然科学版),1988,9卷2期：70-74.
[22]罗振东.Stokes问题的三阶混合元法.贵州科学,1990,8卷2期：282-286.
[23]V. Temam. Navier-Stokes Equation, Theory and Numerical Analysis (Third Edition). North-Holland, Amsterdam, New York Oxford,1984.
[24]K.A. Mardal, X.C. Tai and R. Winther, A robust finite element method for Darcy-Stokes flow. SIAM J. Numer. Anal.2002,40:1605-1631.
[25]M. Crouzeix and P. A. Raviart. Conforming and nonconforming finite element methods for solving the stationary Stokes equations. RAIRO Math. Modeling Numer. Anal.,1973,3:33-75.
[26]P. A. Raviart, J. M. Thomas. A mixed finite element method for second order elliptic problems. Math-ematical Aspects of the Finite Element Method (Galligani I, Magences E, eds). Lecture Notes in Mathematics 606, Springer-Verlag,1977,292-315.
[27]D. N. Arnold, F. Brezzi and M. Fortin. A stable finite element method for the Stokes equations. Calcolo, 1984,21:337-344.
[28]Xue-Cheng Tai and R. Winther. A discrete Rham complex with enhanced smoothness.CALCOLO, 2006,43:287-306.
[29]E. Burman and P. Hansbo. A unified stablized method for stokes'and darcy's equations. J. Comput. Appl. Math.,2007,198(1):35-51.
[30]Chen Long, Sun Pengtao, Xu Jinchao. Optimal anisotropic meshes for minimizing interpolation errors in Lp-norm. Math. Comp.,2007,76(257):179-204.
[31]B. Riviere. Analysis of a discontinuous finite element method for the coupled Stokes and Darcy prob-lems. Journal of Scientific Computing,2005,22:479-500.
[32]B. Riviere and I. Yotov. Locally conservative coupling of Stokes and Darcy flows. SIAM J. Numer. Anal.2005,42(5):1959-1977.
[33]X. P. Xie J.C. Xu and G. R. Xue. Uniformly-stable finite element methods for Darcy-Stokes-Brinkman modes. Math. Comput.,2008, Vol.26, No.3,437-455.
[34]F. Brezzi, Jr. J. Douglas R. Duran and M. Fortin. Mixed finite elements for second order elliptic problems in three variables. Numer. Math.,1987,51:237-250.
[35]F. Brezzi, Jr. J. Douglas M. Fortin and L. D. Marini. Efficient rectangular mixed finite elements in two and three space variables. Math. Model. Numer. Anal.,1987,21:581-604.
[36]F. Brezzi, Jr. J. Douglas and L. D. Marini. Two families of mixed finite element methods for second order elliptic problems. Numer.Math.,1985,47:217-235.
[37]P. A. Raviart, J. M. Thomas. A mixed finite element method for second order elliptic problems. Math-ematical Aspects of the Finite Element Method (Galligani I, Magences E, eds), Lecture Notes in Math-ematics 606, Springer-Verlag,1977,292-315.
[38]P. A. Raviart and J. M. Thomas. Primal hybrid finite element methods for 2nd order elliptic problems. Math.Comp.,1977,31:391-413.
[39]P. G. Ciarlet and P. A. Raviart. A mixed finiteelement method for the biharmonic equation, Symposium on Mathematical Aspects of Finite Element in Partial Differential Equation. C de Boor, Ed., New York: Academic Press,1974,125-143.
[40]R. S. Falk and J. E. Osborn. Error estimates for mixed methods. RAIRO. Numer. Anal.,1980,14(3): 246-277.
[41]C. Johnson. On the convergence of a mixed finite element method for plate bending problem. Numer. Math.,1973,21:43-62.
[42]T. Miyoshi. A finite element method for the solution of fourth order partial differential equations. Kunamoto J.Sc. (Math.) 1974,9:133-149.
[43]D. N. Arnold and R. S. Falk. A new mixed formulation for elasticity. Numer. Math.,1988,53:13-30.
[44]D.N. Arnold and R. Winther. Mixed finite elements for elasticity. Numer. Math.,2002,92:401-419.
[45]D.N. Arnold and R. Winther. Nonconforming mixed finite elements for elasticity. Mathematical Mod-els and Mathods in Applied Science,2003,13(3):295-307.
[46]F. Brezzi. On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers. RAIRO Anal. Numer.,1974,8:129-151.
[47]石钟慈,关于Meley元的误差估计[J].计算数学,1990,12(2),113-118.
[48]Stummel F, Basic compactness properties of nonconforming and hybrid finite element spaces. RAIRO,Anal. Numer.,1980,4(1):81-115.MR0566091 (81h:65058)
[49]Wang Ming, Jinchao Xu,Some tetrahedral Nonconforming finite elements for fourth order elliptic equa-tions. Research Report,79(2004),School of Mathematical Sciences and Institute of Mathematics,Peking University.
[50]Wang Ming, On the necessity and sufficiency of the patch test for convergence of nonconforming finite elements. SIAM J.Numer.Anal.,2002,39(2):363-384.MR1860273 (2002h:65197).
[51]Wang Ming,Jinchao Xu, The Morley element for fourth order elliptic equations in any dimensions. Resear Report,89(2004), School of Mathematical Sciences and Institute of Mathematics,Peking Uni-versity. Numer. Math.,2006,103:155-169.
[52]Wang Ming, Shi Zhong-ci and Jinchao Xu, Some n-rectangle nonconfoming elements for fourth order elliptic equations. Resear Report,50(2005), School of Mathematical Sciences and Institute of Mathe-matics,Peking University.
[53]D. A. Nield and A. Bejan. Convection in porous media. Spring-Verlag,2006.
[54]Scott, Conforming finite element methods for incompressible and nearly incompressible continua. Lec-tures in Applied Mathematics,22(1985), pp.221-244.
[55]Y-S. Lee and P. R. Dawson, Obtaining residual stresses in metal forming after neglecting elasticity on loading. J. of Applied Mechanics,56(1989), pp.318-327.
[56]M. Kitahama and P. R. Dawson, A prectical formulation for Lagrangian viscoplastic analyses with adaptive remeshing. Preprint, Sibley School of Mechanical and Aerospace Engineering, Cornell Uni-versity, August,1990.
[57]C. de Boor and R. Devore, Approximation by smooth multivariate splines. Trans. Amer. Math. Soc., 276(1983), pp.775-788.
[58]C. de Boor and K. Hollig, Approximation order from bivariate C1-cubics:a counterexample. Proc. Amer. Math. Soc.,87(1983), pp.649-655.
[59]B. Mercier, A conforming finite element method for two-dimensinal incompressible elasticity. Internat. J. Numer. Methods Engrg.,14(1979), pp.942-945.
[60]J. F. Ciavaldini and J. C. Nedelec, Sur l'element de Fraeijs de Veubeke et Sander. RAIRO Anal. Numer., 1974.
[61]J. Morgan and L. R. Scott, The dimension of the space of C1 piecewise polynomials. Preprint, August, 1990.