• Kim, Kwang-Yeon (Department of Mathematics Kangwon National University) ;
  • Park, Ju-Seong (Department of Mathematics Kangwon National University)
  • Received : 2019.03.14
  • Accepted : 2019.12.18
  • Published : 2020.03.31


We analyze a posteriori error estimator for the conforming P2 finite element on triangular meshes which is based on the solution of local Neumann problems. This error estimator extends the one for the conforming P1 finite element proposed in [4]. We prove that it is asymptotically exact for the Poisson equation when the underlying triangulations are mildly structured and the solution is smooth enough.


Supported by : Kangwon National University


  1. M. Ainsworth, The influence and selection of subspaces for a posteriori error estimators, Numer. Math. 73 (1996), no. 4, 399-418.
  2. M. Ainsworth and J. T. Oden, A posteriori error estimation in finite element analysis, Pure and Applied Mathematics (New York), Wiley-Interscience, New York, 2000.
  3. A. Alonso, Error estimators for a mixed method, Numer. Math. 74 (1996), no. 4, 385-395.
  4. R. E. Bank and A. Weiser, Some a posteriori error estimators for elliptic partial differential equations, Math. Comp. 44 (1985), no. 170, 283-301. 2007953
  5. R. E. Bank and B. D. Welfert, A posteriori error estimates for the Stokes problem, SIAM J. Numer. Anal. 28 (1991), no. 3, 591-623.
  6. R. E. Bank and J. Xu, Asymptotically exact a posteriori error estimators. I. Grids with superconvergence, SIAM J. Numer. Anal. 41 (2003), no. 6, 2294-2312.
  7. S. C. Brenner and L. R. Scott, The mathematical theory of finite element methods, third edition, Texts in Applied Mathematics, 15, Springer, New York, 2008.
  8. R. Duran and R. Rodrguez, On the asymptotic exactness of Bank-Weiser's estimator, Numer. Math. 62 (1992), no. 3, 297-303.
  9. Y. Huang and J. Xu, Superconvergence of quadratic finite elements on mildly structured grids, Math. Comp. 77 (2008), no. 263, 1253-1268.
  10. D. Kay and D. Silvester, A posteriori error estimation for stabilized mixed approximations of the Stokes equations, SIAM J. Sci. Comput. 21 (1999/00), no. 4, 1321-1336.
  11. Q. Liao, Error estimation and stabilization for low order nite elements, Ph.D thesis, The University of Manchester, 2010.
  12. Q. Liao and D. Silvester, A simple yet effective a posteriori estimator for classical mixed approximation of Stokes equations, Appl. Numer. Math. 62 (2012), no. 9, 1242-1256.
  13. A. Maxim, Asymptotic exactness of an a posteriori error estimator based on the equilibrated residual method, Numer. Math. 106 (2007), no. 2, 225-253.
  14. A. Naga and Z. Zhang, A posteriori error estimates based on the polynomial preserving recovery, SIAM J. Numer. Anal. 42 (2004), no. 4, 1780-1800.
  15. J. S. Ovall, Function, gradient, and Hessian recovery using quadratic edge-bump functions, SIAM J. Numer. Anal. 45 (2007), no. 3, 1064-1080.
  16. R. Verfurth, A posteriori error estimators for the Stokes equations, Numer. Math. 55 (1989), no. 3, 309-325.
  17. R. Verfurth, A posteriori error estimation and adaptive mesh-refinement techniques, J. Comput. Appl. Math. 50 (1994), no. 1-3, 67-83.
  18. H. Wu and Z. Zhang, Can we have superconvergent gradient recovery under adaptive meshes?, SIAM J. Numer. Anal. 45 (2007), no. 4, 1701-1722.
  19. J. Xu and Z. Zhang, Analysis of recovery type a posteriori error estimators for mildly structured grids, Math. Comp. 73 (2004), no. 247, 1139-1152.
  20. O. C. Zienkiewicz and J. Z. Zhu, The superconvergent patch recovery and a posteriori error estimates. I. The recovery technique, Internat. J. Numer. Methods Engrg. 33 (1992), no. 7, 1331-1364.