Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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ASYMPTOTIC EXACTNESS OF SOME BANK-WEISER ERROR ESTIMATOR FOR QUADRATIC TRIANGULAR FINITE ELEMENT

  • 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
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Abstract

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.

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References

  1. M. Ainsworth, The influence and selection of subspaces for a posteriori error estimators, Numer. Math. 73 (1996), no. 4, 399-418. https://doi.org/10.1007/s002110050198
  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. https://doi.org/10.1002/9781118032824
  3. A. Alonso, Error estimators for a mixed method, Numer. Math. 74 (1996), no. 4, 385-395. https://doi.org/10.1007/s002110050222
  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. https://doi.org/10.2307/ 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. https://doi.org/10.1137/0728033
  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. https://doi.org/10.1137/S003614290139874X
  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. https://doi.org/10.1007/978-0-387-75934-0
  8. R. Duran and R. Rodrguez, On the asymptotic exactness of Bank-Weiser's estimator, Numer. Math. 62 (1992), no. 3, 297-303. https://doi.org/10.1007/BF01396231
  9. Y. Huang and J. Xu, Superconvergence of quadratic finite elements on mildly structured grids, Math. Comp. 77 (2008), no. 263, 1253-1268. https://doi.org/10.1090/S0025-5718-08-02051-6
  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. https://doi.org/10.1137/S1064827598333715
  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. https://doi.org/10.1016/j.apnum.2010.05.003
  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. https://doi.org/10.1007/s00211-007-0064-3
  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. https://doi.org/10.1137/S0036142903413002
  15. J. S. Ovall, Function, gradient, and Hessian recovery using quadratic edge-bump functions, SIAM J. Numer. Anal. 45 (2007), no. 3, 1064-1080. https://doi.org/10.1137/060648908
  16. R. Verfurth, A posteriori error estimators for the Stokes equations, Numer. Math. 55 (1989), no. 3, 309-325. https://doi.org/10.1007/BF01390056
  17. R. Verfurth, A posteriori error estimation and adaptive mesh-refinement techniques, J. Comput. Appl. Math. 50 (1994), no. 1-3, 67-83. https://doi.org/10.1016/0377-0427(94)90290-9
  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. https://doi.org/10.1137/060661430
  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. https://doi.org/10.1090/S0025-5718-03-01600-4
  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. https://doi.org/10.1002/nme.1620330702