• Received : 2016.03.24
  • Accepted : 2016.06.15
  • Published : 2016.06.25


In this paper we analyze an a posteriori error estimator based on flux recovery for lowest-order finite element discretizations of elliptic interface problems. The flux recovery considered here is based on averaging the discrete normal fluxes and/or tangential derivatives at midpoints of edges with weight factors adapted to discontinuous coefficients. It is shown that the error estimator based on this flux recovery is equivalent to the error estimator of Bernardi and $Verf{\ddot{u}}rth$ based on the standard edge residuals uniformly with respect to jumps of the coefficient between subdomains. Moreover, as a byproduct, we obtain slightly modified weight factors in the edge residual estimator which are expected to produce more accurate results.


Supported by : Kangwon National University


  1. C. Bernardi and R. Verfurth, Adaptive finite element methods for elliptic equations with non-smooth coefficients, Numer. Math., 85 (2000), 579-608.
  2. M. Petzoldt, A posteriori error estimators for elliptic equations with discontinuous coefficients, Adv. Comput. Math., 16 (2002), 47-75.
  3. C. Carstensen and R. Verfurth, Edge residuals dominate a posteriori error estimates for low order finite element methods, SIAM J. Numer. Anal., 36 (1999), 1571-1587.
  4. Z. Cai and S. Zhang, Recovery-based error estimator for interface problems: conforming linear elements, SIAM J. Numer. Anal., 47 (2009), 2132-2156.
  5. Z. Cai and S. Zhang, Recovery-based error estimators for interface problems: mixed and nonconforming finite elements, SIAM J. Numer. Anal., 48 (2010), 30-52.
  6. O. C. Zienkiewicz and J. Z. Zhu, A simple error estimator and adaptive procedure for practical engineering analysis, Internat. J. Numer. Methods Engrg., 24 (1987), 337-357.
  7. C. Carstensen and S. Bartels, Each averaging technique yields reliable a posteriori error control in FEM on unstructured grids. Part I: Low order conforming, nonconforming, and mixed FEM, Math. Comp., 71 (2002), 945-969.
  8. G. Goodsell and J. R. Whiteman, A unified treatment of superconvergent recovered gradient functions for piecewise linear finite element approximations, Internat. J. Numer. Methods Engrg., 27 (1989), 469-481.
  9. N. Levine, Superconvergent recovery of the gradient from piecewise linear finite-element approximations, IMA J. Numer. Anal., 5 (1985), 407-427.
  10. J. H. Brandts, Superconvergence and a posteriori error estimation for triangular mixed finite elements, Numer. Math., 68 (1994), 311-324.
  11. J. Hu and R. Ma, Superconvergence of both the Crouzeix-Raviart and Morley elements, Numer. Math., 132 (2016), 491-509.
  12. K. Y. Kim, A posteriori error analysis for locally conservative mixed methods, Math. Comp., 76 (2007), 43-66.
  13. B. I. Wohlmuth and R. H. W. Hoppe, A comparison of a posteriori error estimators for mixed finite element discretizations by Raviart-Thomas elements, Math. Comp., 68 (1999), 1347-1378.
  14. R. B. Kellogg, On the Poisson equation with intersecting interfaces, Appl. Anal., 4 (1974), 101-129.