## 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 PDF
- Math. Comp.
**82**(2013), 25-43 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**0450957** - 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**0520174** - 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, 1972, pp. 557–587. MR**0423839** - 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, N.J., 1973. MR**0443377** - 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.

## Additional Information

**Ming Wang**- Affiliation: Laboratory of Pure and Applied Mathematics, School of Mathematical Sciences, Peking University, Beijing, China
- Email: mwang@math.pku.edu.cn
**Jinchao Xu**- Affiliation: Laboratory of Pure and Applied Mathematics, School of Mathematical Sciences, Peking University, Beijing China –and– Department of Mathematics, Pennsylvania State University
- MR Author ID: 228866
- Email: xu@math.psu.edu
- Received by editor(s): September 30, 2008
- Received by editor(s) in revised form: August 30, 2011
- Published electronically: June 8, 2012
- Additional Notes: This work was supported by the National Natural Science Foundation of China (10871011)

This work was supported by the National Science Foundation, DMS 0749202 and DMS 0915153 - © Copyright 2012
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Math. Comp.
**82**(2013), 25-43 - MSC (2010): Primary 65N30
- DOI: https://doi.org/10.1090/S0025-5718-2012-02611-1
- MathSciNet review: 2983014