Skip to Main Content

Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.78.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.


Nonconforming tetrahedral finite elements for fourth order elliptic equations
HTML articles powered by AMS MathViewer

by Wang Ming and Jinchao Xu PDF
Math. Comp. 76 (2007), 1-18 Request permission


This paper is devoted to the construction of nonconforming finite elements for the discretization of fourth order elliptic partial differential operators in three spatial dimensions. The newly constructed elements include two nonconforming tetrahedral finite elements and one quasi-conforming tetrahedral element. These elements are proved to be convergent for a model biharmonic equation in three dimensions. In particular, the quasi-conforming tetrahedron element is a modified Zienkiewicz element, while the nonmodified Zienkiewicz element (a tetrahedral element of Hermite type) is proved to be divergent on a special grid.
  • Alfeld P, A trivariate Clough-Tocher scheme for tetrahedral data, Comput. Aided Geom. Design, 1 (1984), 169-181.
  • John W. Barrett and James F. Blowey, Finite element approximation of the Cahn-Hilliard equation with concentration dependent mobility, Math. Comp. 68 (1999), no. 226, 487–517. MR 1609678, DOI 10.1090/S0025-5718-99-01015-7
  • John W. Barrett, James F. Blowey, and Harald Garcke, Finite element approximation of the Cahn-Hilliard equation with degenerate mobility, SIAM J. Numer. Anal. 37 (1999), no. 1, 286–318. MR 1742748, DOI 10.1137/S0036142997331669
  • Bazeley G P, Cheung Y K, Irons B M and Zienkiewicz O C, Triangular elements in plate bending — conforming and nonconforming solutions, in Proceedings of the Conference on Matrix Methods in Structural Mechanics, Wright Patterson A. F. Base, Ohio, 1965, 547-576.
  • Boettinger W J, Warren J A, Beckermann C and Karma A, Phase-field simulation of solidification, Annu. Rev. Mater. Res., 32, 1(2002), 63-94.
  • Cahn J W, Hilliard J E, Free energy of a nonuniform system I. Interfacial free energy, J. Chem. Phys., 28 (1958), 258-267.
  • Chen Wanji, Liu Yingxi and Tang Limin, The formulation of quasi-conforming elements, Journal of Dalian Institute of Technology, 19, 2(1980), 37-49.
  • Philippe G. Ciarlet, The finite element method for elliptic problems, Classics in Applied Mathematics, vol. 40, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2002. Reprint of the 1978 original [North-Holland, Amsterdam; MR0520174 (58 #25001)]. MR 1930132, DOI 10.1137/1.9780898719208
  • Qiang Du and R. A. Nicolaides, Numerical analysis of a continuum model of phase transition, SIAM J. Numer. Anal. 28 (1991), no. 5, 1310–1322. MR 1119272, DOI 10.1137/0728069
  • Charles M. Elliott and Donald A. French, A nonconforming finite-element method for the two-dimensional Cahn-Hilliard equation, SIAM J. Numer. Anal. 26 (1989), no. 4, 884–903. MR 1005515, DOI 10.1137/0726049
  • Charles M. Elliott and Stig Larsson, Error estimates with smooth and nonsmooth data for a finite element method for the Cahn-Hilliard equation, Math. Comp. 58 (1992), no. 198, 603–630, S33–S36. MR 1122067, DOI 10.1090/S0025-5718-1992-1122067-1
  • Elliott C M, and Zhang S M, On the Cahn-Hilliard equation, Arch. Rational Mech. Anal., 96 (1986), 339-357.
  • Stefania Gatti, Maurizio Grasselli, Vittorino Pata, and Alain Miranville, Hyperbolic relaxation of the viscous Cahn-Hilliard equation in 3-D, Math. Models Methods Appl. Sci. 15 (2005), no. 2, 165–198. MR 2119676, DOI 10.1142/S0218202505000327
  • Ming-Jun Lai and A. LeMéhauté, A new kind of trivariate $C^1$ macro-element, Adv. Comput. Math. 21 (2004), no. 3-4, 273–292. MR 2073143, DOI 10.1023/B:ACOM.0000032047.05052.27
  • Landau L and Lifchitz E, Theory of Elasticity, Pergamon Press, London, 1959.
  • 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
  • LeMehaute A, Interpolation et approximation par des fonctions polynomiales par morceaux dans $R^n$, Ph.D. Thesis, Univ. Rennes, France, 1984.
  • Morley L S D, The triangular equilibrium element in the solution of plate bending problems, Aero. Quart., 19 (1968), 149-169.
  • Seol D J, Hu S Y, Li Y L, Shen J, Oh K H and Chen L Q, Computer simulation of spinodal decomposition in constrained films, Acta Materialia, 51 (2003), 5173-5185.
  • Zhong Ci Shi, The generalized patch test for Zienkiewicz’s triangles, J. Comput. Math. 2 (1984), no. 3, 279–286. MR 815422
  • Shi Zhong-ci, On the error estimates of Morley element, Numerica Mathematica Sinica 12, 2(1990), 113-118.
  • 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
  • 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
  • Tang Limin, Chen Wanji and Liu Yingxi, Quasi-conforming elements in finite element analysis, J. Dalian Inst. of Technology, 19, 2(1980), 19-35.
  • 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
  • Wang Ming, Shi Zhong-ci and Jinchao Xu, Some $n$-rectangle nonconforming elements for fourth order elliptic equations, Research Report, 50(2005), School of Mathematical Sciences and Institute of Mathematics, Peking University; J. Comput. Math., to appear.
  • Wang Ming and Jinchao Xu, The Morley element for fourth order elliptic equations in any dimensions, Research Report, 89 (2004), School of Mathematical Sciences and Institute of Mathematics, Peking University. Numer. Math., 103 (2006), 155–169.
  • A. J. Worsey and G. Farin, An $n$-dimensional Clough-Tocher interpolant, Constr. Approx. 3 (1987), no. 2, 99–110. MR 889547, DOI 10.1007/BF01890556
  • Zenicek A, Polynomial approximation on tetrahedrons in the finite element method, J. Approx. Theory, 7 (1973), 334-351.
  • Zenicek A, A general theorem on triangular finite $C^{(m)}$-elements, RAIRO Anal. Numer., R-2 (1974), 119-127.
  • Zhang Hongqing and Wang Ming, On the compactness of quasi-conforming element spaces and the convergence of quasi-conforming element method, Appl. Math. Mech. (English edition), 7 (1986), 443-459.
  • Zhang Hongqing and Wang Ming, The Mathematical Theory of Finite Elements, Science Press, Beijing, 1991.
Similar Articles
  • Retrieve articles in Mathematics of Computation with MSC (2000): 65N30
  • Retrieve articles in all journals with MSC (2000): 65N30
Additional Information
  • Wang Ming
  • Affiliation: LMAM, School of Mathematical Sciences, Peking University, Beijing, People’s Republic of China
  • Email:
  • Jinchao Xu
  • Affiliation: The School of Mathematical Sciences, Peking University; Beijing, People’s Republic of China; and Department of Mathematics, Pennsylvania State University, University Park, Pennsylvania 16802
  • MR Author ID: 228866
  • Email:
  • Received by editor(s): October 8, 2004
  • Received by editor(s) in revised form: September 16, 2005
  • Published electronically: August 1, 2006
  • Additional Notes: The work of the first author was supported by the National Natural Science Foundation of China (10571006).
    The work of the second author was supported by National Science Foundation DMS-0209479 and DMS-0215392 and the Changjiang Professorship through Peking University
  • © Copyright 2006 American Mathematical Society
  • Journal: Math. Comp. 76 (2007), 1-18
  • MSC (2000): Primary 65N30
  • DOI:
  • MathSciNet review: 2261009