Minimal finite element spaces for 2𝑚-th-order partial differential equations in 𝑅ⁿ

Wang, Ming; Xu, Jinchao

doi:10.1090/S0025-5718-2012-02611-1

Minimal finite element spaces for $2m$-th-order partial differential equations in $R^n$
HTML articles powered by AMS MathViewer

by Ming Wang and Jinchao Xu;

Math. Comp. 82 (2013), 25-43

DOI: https://doi.org/10.1090/S0025-5718-2012-02611-1

Published electronically: June 8, 2012

PDF | Request permission

Abstract:

This paper is devoted to a canonical construction of a family of piecewise polynomials with the minimal degree capable of providing a consistent approximation of Sobolev spaces $H^m$ in $R^n$ (with $n\ge m\ge 1$) and also a convergent (nonconforming) finite element space for $2m$-th-order elliptic boundary value problems in $R^n$. For this class of finite element spaces, the geometric shape is $n$-simplex, the shape function space consists of all polynomials with a degree not greater than $m$, and the degrees of freedom are given in terms of the integral averages of the normal derivatives of order $m-k$ on all subsimplexes with the dimension $n-k$ for $1\le k\le m$. This sequence of spaces has some natural inclusion properties as in the corresponding Sobolev spaces in the continuous cases.

The finite element spaces constructed in this paper constitute the only class of finite element spaces, whether conforming or nonconforming, that are known and proven to be convergent for the approximation of any $2m$-th-order elliptic problems in any $R^n$, such that $n\ge m\ge 1$. Finite element spaces in this class recover the nonconforming linear elements for Poisson equations ($m=1$) and the well-known Morley element for biharmonic equations ($m=2$).

References

Robert A. Adams, Sobolev spaces, Pure and Applied Mathematics, Vol. 65, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1975. MR 450957
Adini, A. and Glough, R. W., Analysis of plate bending by the finite element method, NSF report G, 7337, 1961.
Douglas N. Arnold, Franco Brezzi, Bernardo Cockburn, and L. Donatella Marini, Unified analysis of discontinuous Galerkin methods for elliptic problems, SIAM J. Numer. Anal. 39 (2001/02), no. 5, 1749–1779. MR 1885715, DOI 10.1137/S0036142901384162
Garth A. Baker, Finite element methods for elliptic equations using nonconforming elements, Math. Comp. 31 (1977), no. 137, 45–59. MR 431742, DOI 10.1090/S0025-5718-1977-0431742-5
John W. Barrett, Stephen Langdon, and Robert Nürnberg, Finite element approximation of a sixth order nonlinear degenerate parabolic equation, Numer. Math. 96 (2004), no. 3, 401–434. MR 2028722, DOI 10.1007/s00211-003-0479-4
Bazeley, G. P., Cheung, Y. K., Irons, B. M., and Zienkiewicz, O. C., Triangular elements in bending–conforming and nonconforming solutions, in Proc. Conf. Matrix Methods in Structural Mechanics, Air Force Ins. Tech., Wright-Patterson A. F. Base, Ohio, 1965.
Cahn, J. W., On spinodal decomposition, Acta Metallurgica, 9 (1961), 795–801.
Cahn, J. W. and Hilliard, J. E., Free energy of a nonuniform system I. interfacial free energy, J Chem. Phys., 28 (1958), 258–267.
Sun-Yung Alice Chang and Wenxiong Chen, A note on a class of higher order conformally covariant equations, Discrete Contin. Dynam. Systems 7 (2001), no. 2, 275–281. MR 1808400, DOI 10.3934/dcds.2001.7.275
Philippe G. Ciarlet, The finite element method for elliptic problems, Studies in Mathematics and its Applications, Vol. 4, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978. MR 520174
M. Crouzeix and P.-A. Raviart, Conforming and nonconforming finite element methods for solving the stationary Stokes equations. I, Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge 7 (1973), no. R-3, 33–75. MR 343661
Charles M. Elliott and Zheng Songmu, On the Cahn-Hilliard equation, Arch. Rational Mech. Anal. 96 (1986), no. 4, 339–357. MR 855754, DOI 10.1007/BF00251803
Richard S. Falk, Nonconforming finite element methods for the equations of linear elasticity, Math. Comp. 57 (1991), no. 196, 529–550. MR 1094947, DOI 10.1090/S0025-5718-1991-1094947-6
Kang Feng, On the theory of discontinuous finite elements, Math. Numer. Sinica 1 (1979), no. 4, 378–385 (Chinese, with English summary). MR 657317
Paul C. Fife, Models for phase separation and their mathematics, Electron. J. Differential Equations (2000), No. 48, 26. MR 1772733
Bruce M. Irons and Abdur Razzaque, Experience with the patch test for convergence of finite elements, The mathematical foundations of the finite element method with applications to partial differential equations (Proc. Sympos., Univ. Maryland, Baltimore, Md., 1972) Academic Press, New York-London, 1972, pp. 557–587. MR 423839
P. Lascaux and P. Lesaint, Some nonconforming finite elements for the plate bending problem, Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge Anal. Numér. 9 (1975), no. R-1, 9–53 (English, with French summary). MR 423968
Pierre Lesaint, On the convergence of Wilson’s nonconforming element for solving the elastic problems, Comput. Methods Appl. Mech. Engrg. 7 (1976), no. 1, 1–16. MR 455479, DOI 10.1016/0045-7825(76)90002-5
P. Lesaint and M. Zlámal, Convergence of the nonconforming Wilson element for arbitrary quadrilateral meshes, Numer. Math. 36 (1980/81), no. 1, 33–52. MR 595805, DOI 10.1007/BF01395987
Melosh, R. J., Basis of derivation of matrices for direct stiffness method, AIAA J., 1 (1963), 1631–1637.
Morley, L. S. D., The triangular equilibrium element in the solution of plate bending problems, Aero. Quart., 19 (1968), 149–169.
Rolf Rannacher, On nonconforming and mixed finite element method for plate bending problems. The linear case, RAIRO Anal. Numér. 13 (1979), no. 4, 369–387 (English, with French summary). MR 555385, DOI 10.1051/m2an/1979130403691
Vitoriano Ruas, A quadratic finite element method for solving biharmonic problems in $\textbf {R}^n$, Numer. Math. 52 (1988), no. 1, 33–43. MR 918315, DOI 10.1007/BF01401020
Zhong Ci Shi, An explicit analysis of Stummel’s patch test examples, Internat. J. Numer. Methods Engrg. 20 (1984), no. 7, 1233–1246. MR 751335, DOI 10.1002/nme.1620200705
Zhong Ci Shi, A convergence condition for the quadrilateral Wilson element, Numer. Math. 44 (1984), no. 3, 349–361. MR 757491, DOI 10.1007/BF01405567
Zhong Ci Shi, On the convergence properties of the quadrilateral elements of Sander and Beckers, Math. Comp. 42 (1984), no. 166, 493–504. MR 736448, DOI 10.1090/S0025-5718-1984-0736448-4
Zhong Ci Shi, The generalized patch test for Zienkiewicz’s triangles, J. Comput. Math. 2 (1984), no. 3, 279–286. MR 815422
Zhong Ci Shi, Convergence properties of two nonconforming finite elements, Comput. Methods Appl. Mech. Engrg. 48 (1985), no. 2, 123–137. MR 784479, DOI 10.1016/0045-7825(85)90100-8
Zhong Ci Shi, The F-E-M test for convergence of nonconforming finite elements, Math. Comp. 49 (1987), no. 180, 391–405. MR 906178, DOI 10.1090/S0025-5718-1987-0906178-7
Zhong Ci Shi, Error estimates for the Morley element, Math. Numer. Sinica 12 (1990), no. 2, 113–118 (Chinese, with English summary); English transl., Chinese J. Numer. Math. Appl. 12 (1990), no. 3, 102–108. MR 1070298
Gilbert Strang and George J. Fix, An analysis of the finite element method, Prentice-Hall Series in Automatic Computation, Prentice-Hall, Inc., Englewood Cliffs, NJ, 1973. MR 443377
Friedrich Stummel, The generalized patch test, SIAM J. Numer. Anal. 16 (1979), no. 3, 449–471. MR 530481, DOI 10.1137/0716037
Stummel, F., The limitation of the patch test, Int. J. Numer. Meth. Eng., 15 (1980), 177-188.
Friedrich Stummel, Basic compactness properties of nonconforming and hybrid finite element spaces, RAIRO Anal. Numér. 14 (1980), no. 1, 81–115 (English, with French summary). MR 566091
Roger Temam, Infinite-dimensional dynamical systems in mechanics and physics, Applied Mathematical Sciences, vol. 68, Springer-Verlag, New York, 1988. MR 953967, DOI 10.1007/978-1-4684-0313-8
B. Fraeijs de Veubeke, Variational principles and the patch test, Internat. J. Numer. Methods Engrg. 8 (1974), 783–801. MR 375911, DOI 10.1002/nme.1620080408
Ming Wang, On the necessity and sufficiency of the patch test for convergence of nonconforming finite elements, SIAM J. Numer. Anal. 39 (2001), no. 2, 363–384. MR 1860273, DOI 10.1137/S003614299936473X
Ming Wang and Jinchao Xu, The Morley element for fourth order elliptic equations in any dimensions, Numer. Math. 103 (2006), no. 1, 155–169. MR 2207619, DOI 10.1007/s00211-005-0662-x
Wang Ming and Jinchao Xu, Nonconforming tetrahedral finite elements for fourth order elliptic equations, Math. Comp. 76 (2007), no. 257, 1–18. MR 2261009, DOI 10.1090/S0025-5718-06-01889-8
Ming Wang, Zhong-ci Shi, and Jinchao Xu, A new class of Zienkiewicz-type non-conforming element in any dimensions, Numer. Math. 106 (2007), no. 2, 335–347. MR 2291941, DOI 10.1007/s00211-007-0063-4
Ming Wang, Zhong-Ci Shi, and Jinchao Xu, Some $n$-rectangle nonconforming elements for fourth order elliptic equations, J. Comput. Math. 25 (2007), no. 4, 408–420. MR 2337403
Wise, S. M., Lowengrub, J. S., Kim, J. S., et al., Efficient phase-field simulation of quantum dot formation on a trained heteroepitaxial film, Superlattices and Microstructure, 36 (2004), 293–304.
Wise, S. M., Lowengrub, J. S., Kim, J. S., et al., Quantum dot formation on a train-patterned epitaxial thin film, Appl. Phys. Lett., 87, 133102 (2005).
Jinchao Xu, Iterative methods by space decomposition and subspace correction, SIAM Rev. 34 (1992), no. 4, 581–613. MR 1193013, DOI 10.1137/1034116
Alexander Ženíšek, Polynomial approximation on tetrahedrons in the finite element method, J. Approximation Theory 7 (1973), 334–351. MR 350260, DOI 10.1016/0021-9045(73)90036-1
Hong Qing Zhang and Ming Wang, On the compactness of quasiconforming element spaces and the convergence of quasiconforming element method, Appl. Math. Mech. 7 (1986), no. 5, 409–423 (Chinese, with English summary); English transl., Appl. Math. Mech. (English Ed.) 7 (1986), no. 5, 443–459. MR 860997, DOI 10.1007/BF01895764
Zhang, H. Q. and Wang, M., The Mathematical Theory of Finite Elements, Science Press, Beijing, 1991.