Korean Journal of Mathematics

The Kangwon-Kyungki Mathematical Society (강원경기수학회)
권호 표지
DOI QR코드

DOI QR Code

ON THE ASYMPTOTIC EXACTNESS OF AN ERROR ESTIMATOR FOR THE LOWEST-ORDER RAVIART-THOMAS MIXED FINITE ELEMENT

  • Received : 2013.07.11
  • Accepted : 2013.08.13
  • Published : 2013.09.30
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper we analyze an error estimator for the lowest-order triangular Raviart-Thomas mixed finite element which is based on solution of local problems for the error. This estimator was proposed in [Alonso, Error estimators for a mixed method, Numer. Math. 74 (1996), 385{395] and has a similar concept to that of Bank and Weiser. We show that it is asymptotically exact for the Poisson equation if the underlying triangulations are uniform and the exact solution is regular enough.

Keywords

Acknowledgement

Supported by : National Research Foundation of Korea (NRF)

References

  1. M. Ainsworth, A posteriori error estimation for lowest order Raviart-Thomas mixed finite elements, SIAM J. Sci. Comput., 30 (2007), pp. 189-204.
  2. A. Alonso, Error estimators for a mixed method, Numer. Math., 74 (1996), pp. 385-395. https://doi.org/10.1007/s002110050222
  3. R. E. Bank and A. Weiser, Some a posteriori error estimators for elliptic partial differential equations, Math. Comp., 44 (1985), pp. 283-301. https://doi.org/10.1090/S0025-5718-1985-0777265-X
  4. D. Braess and R. Verfurth, A posteriori error estimators for the Raviart-Thomas element, SIAM J. Numer. Anal., 33 (1996), pp. 2431-2444. https://doi.org/10.1137/S0036142994264079
  5. J. H. Brandts, Superconvergence and a posteriori error estimation for triangular mixed finite elements, Numer. Math., 68 (1994), pp. 311-324. https://doi.org/10.1007/s002110050064
  6. F. Brezzi and M. Fortin, Mixed and hybrid finite element methods, Springer-Verlag, New York, 1991.
  7. C. Carstensen, A posteriori error estimate for the mixed finite element method, Math. Comp., 66 (1997), pp. 465-476. https://doi.org/10.1090/S0025-5718-97-00837-5
  8. T. F. Dupont and P. T. Keenan, Superconvergence and postprocessing of fluxes from lowest-order mixed methods on triangles and tetrahedra, SIAM J. Sci. Comput., 19 (1998), pp. 1322-1332. https://doi.org/10.1137/S1064827595280417
  9. R. Duran and R. Rodriguez, On the asymptotic exactness of Bank-Weiser's estimator, Numer. Math., 62 (1992), pp. 297-303. https://doi.org/10.1007/BF01396231
  10. K.-Y. Kim, Guaranteed a posteriori error estimator for mixed finite element methods of elliptic problems, Appl. Math. Comp., 218 (2012), pp. 11820-11831. https://doi.org/10.1016/j.amc.2012.04.084
  11. M. G. Larson and A. Malqvist, A posteriori error estimates for mixed finite element approximations of elliptic problems, Numer. Math., 108 (2008), pp. 487-500. https://doi.org/10.1007/s00211-007-0121-y
  12. C. Lovadina and R. Stenberg, Energy norm a posteriori error estimates for mixed finite element methods, Math. Comp., 75 (2006), pp. 1659-1674. https://doi.org/10.1090/S0025-5718-06-01872-2
  13. A. Maxim, Asymptotic exactness of an a posteriori error estimator based on the equilibrated residual method, Numer. Math., 106 (2007), pp. 225-253. https://doi.org/10.1007/s00211-007-0064-3
  14. P. A. Raviart and J. M. Thomas, A mixed finite element method for 2nd order elliptic problems, in Proc. Conf. on Mathematical Aspects of Finite Element Methods, Lecture Notes in Math., Vol. 606, Springer-Verlag, Berlin, 1977, pp. 292-315.
  15. M. Vohralik, A posteriori error estimates for lowest-order mixed finite element discretizations of convection-diffusion-reaction equations, SIAM J. Numer. Anal., 45 (2007), pp. 1570-1599. https://doi.org/10.1137/060653184
  16. 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), pp. 1347-1378. https://doi.org/10.1090/S0025-5718-99-01125-4
  17. J. Xu and Z. Zhang, Analysis of recovery type a posteriori error estimators for mildly structured grids, Math. Comp., 73 (2004), pp. 1139-1152.
  18. N. Yan and A. Zhou, Gradient recovery type a posteriori error estimates for finite element approximations on irregular meshes, Comput. Methods Appl. Mech. Engrg., 190 (2001), pp. 4289-4299. https://doi.org/10.1016/S0045-7825(00)00319-4

Cited by

  1. Guaranteed and asymptotically exact a posteriori error estimator for lowest-order Raviart–Thomas mixed finite element method vol.165, pp.None, 2013, https://doi.org/10.1016/j.apnum.2021.03.002