Korean J. Math. Vol. 22 No. 3 (2014) pp.429-441
DOI: https://doi.org/10.11568/kjm.2014.22.3.429

# Hierarchical error estimators for lowest-order mixed finite element methods

## Abstract

In this work we study two a posteriori error estimators of hierarchical type for lowest-order mixed finite element methods.One estimator is computed by solving a global defect problem based on the splitting of the lowest-order Brezzi--Douglas--Marini space, and the other estimator is locally computable by applying the standard localization to the first estimator. We establish the reliability and efficiency of both estimators by comparing them with the standard residual estimator. In addition, it is shown that the error estimator based on the global defect problem is asymptotically exact under suitable conditions.

## Supporting Agencies

This study was supported by 2013 Research Grant from Kangwon National Uni- versity (No. C1009911-01-01).

## References

 M. Ainsworth, A posteriori error estimation for lowest order Raviart–Thomas mixed finite elements, SIAM J. Sci. Comput. 30 (2007), 189–204. Google Scholar

 A. Alonso, Error estimators for a mixed method, Numer. Math. 74 (1996), 385– 395. Google Scholar

 D. Boffi, F. Brezzi, and M. Fortin, Mixed finite element methods and applications. Springer Series in Computational Mathematics, 44, Springer, Heidelberg, 2013.  D. Braess and R. Verfu ̈rth, A posteriori error estimators for the Raviart–Thomas element, SIAM J. Numer. Anal. 33 (1996), 2431–2444. Google Scholar

 J. H. Brandts, Superconvergence and a posteriori error estimation for triangular mixed finite elements, Numer. Math. 68 (1994), 311–324. Google Scholar

 C. Carstensen, A posteriori error estimate for the mixed finite element method, Math. Comp. 66 (1997), 465–476. Google Scholar

 K.-Y. Kim, A posteriori error analysis for locally conservative mixed methods, Math. Comp. 76 (2007), 43–66. Google Scholar

 K.-Y. Kim, Guaranteed a posteriori error estimator for mixed finite element methods of elliptic problems, Appl. Math. Comp. 218 (2012), 11820–11831. Google Scholar

 K.-Y. Kim, Asymptotically exact error estimator based on equilibrated fluxes, Int. J. Numer. Anal. Model. (2014), submitted. Google Scholar

 M. G. Larson and A. Ma ̇lqvist, A posteriori error estimates for mixed finite element approximations of elliptic problems, Numer. Math. 108 (2008), 487–500. Google Scholar

 C. Lovadina and R. Stenberg, Energy norm a posteriori error estimates for mixed finite element methods, Math. Comp. 75 (2006), 1659–1674. Google Scholar

 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, 292–315. Google Scholar

 M. Vohral ik, A posteriori error estimates for lowest-order mixed finite element Google Scholar

 discretizations of convection-diffusion-reaction equations, SIAM J. Numer. Anal. Google Scholar