Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

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



Superconvergence of quadratic finite elements on mildly structured grids

Authors: Yunqing Huang and Jinchao Xu
Journal: Math. Comp. 77 (2008), 1253-1268
MSC (2000): Primary 65N50, 65N30
Published electronically: March 4, 2008
MathSciNet review: 2398767
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Superconvergence estimates are studied in this paper on quadratic finite element discretizations for second order elliptic boundary value problems on mildly structured triangular meshes. For a large class of practically useful grids, the finite element solution $ u_h$ is proven to be superclose to the interpolant $ u_I$ and as a result a postprocessing gradient recovery scheme for $ u_h$ can be devised. The analysis is based on a number of carefully derived identities. In addition to its own theoretical interests, the result in this paper can be used for deriving asymptotically exact a posteriori error estimators for quadratic finite element methods.

References [Enhancements On Off] (What's this?)

  • 1. A. Andreev and R. Lazarov.
    Error estimate of type superconvergence of the gradient for quadratic triangular elements.
    C.R. Acad. Bulgare Sci., 36:1179-1182, 1984. MR 779566 (86f:65186)
  • 2. A. Andreev and R. Lazarov.
    Superconvergence of the gradient for quadratic triangular finite elements.
    Numer. Methods Partial Differential Equations, 4:15-32, 1988. MR 1012472 (90m:65190)
  • 3. I. Babuska and W. Rheinboldt.
    A posteriori error estimates for the finite element method.
    Internat. J. Numer. Methods Engrg., 12, 1978.
  • 4. I. Babuska and T. Strouboulis.
    The finite element method and its reliability.
    Numerical Mathematics and Scientific Computation, 2001. MR 1857191 (2002k:65001)
  • 5. R. Bank.
    A software package for solving elliptic partial differential equations.
    Software, Environments and Tools, 5, 1998.
  • 6. R. Bank and J. Xu.
    Asymptotically exact a posteriori error estimators. I. grids with superconvergence.
    SIAM J. Numerical Analysis, 41, No. 6, 2003. MR 2034616 (2004k:65194)
  • 7. J. Brandts.
    Superconvergence for triangular order $ k=1$ Raviart-Thomas mixed finite elements and for triangular standard quadratic finite element methods.
    Appl. Numer. Anal., 34:39-58, 2000. MR 1755693 (2001c:65142)
  • 8. C. M. Chen and Y. Huang.
    High accuracy theory of finite element methods.
    Hunan, Science Press, Hunan, China (in Chinese), 1995.
  • 9. G. Goodsell and J. Whiteman.
    Superconvergence of recovered gradients of piecewise quadratic finite element approximations.
    Numer. Methods Partial Differential Equations, 7:61-8, 1991. MR 1088856 (92e:65151a)
  • 10. R. Hiptmair.
    Canonical construction of finite elements.
    Math. Comp., 68:1325-1346, 1999.
    unreadable. MR 1665954 (2000b:65214)
  • 11. B. Li.
    Superconvergence for higher-order triangular finite elements.
    Chinese J. Numer. Math. Appl. (English), 12:75-79, 1990. MR 1118707 (92d:65196)
  • 12. L. B. Wahlbin.
    Superconvergence in Galkerkin finite element methods.
    Springer-Verlag, Berlin, 1995. MR 1439050 (98j:65083)
  • 13. Jinchao Xu and Z. M. Zhang.
    Analysis of recovery type a posteriori error estimators for mildly structured grids.
    Math. Comp., pages 781-801, 2003.
  • 14. J. Z. Zhu and O. C. Zienkiewicz.
    Superconvergence recovery technique and a posteriori error estimators.
    Internat. J. Numer. Methods Engrg., 30(7):1321-1339, 1990. MR 1078744 (91i:65141)
  • 15. Q. Zhu.
    The derivative good points for the finite element method with 2-degree triangular element (in chinese)
    Natural Science Journal of Xiangtan University, 4:36-45, 1981.
  • 16. Q. Zhu.
    Natural inner superconvergence for the finite element method (in chinese).
    Proc. China-France Symposium on the FEM, Beijing, pages 935-960, 1982. MR 754041 (85h:65253)
  • 17. Q. Zhu and Q. Lin.
    Finite element superconvergence theory.
    Hunan Science Press, 1989.

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2000): 65N50, 65N30

Retrieve articles in all journals with MSC (2000): 65N50, 65N30

Additional Information

Yunqing Huang
Affiliation: Institute for Computational and Applied Mathematics and Hunan Key Laboratory for Computation & Simulation in Science & Engineering, Xiangtan University, People’s Republic of China, 411105

Jinchao Xu
Affiliation: Institute for Computational and Applied Mathematics, Xiangtan University, People’s Republic of China and Center for Computational Mathematics and Applications, Pennsylvania State University, USA

Keywords: Superconvergence, gradient recovery, a posteriori error estimates
Received by editor(s): February 2, 2006
Received by editor(s) in revised form: March 2, 2007
Published electronically: March 4, 2008
Additional Notes: The work of the first author was supported in part by the NSFC for Distinguished Young Scholars (10625106) and the National Basic Research Program of China under the grant 2005CB321701
The second author was supported in part by the Furong Scholar Program of Hunan Province through Xiangtan University
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society