An a posteriori error estimate for the variable-degree Raviart-Thomas method

Cockburn, Bernardo; Zhang, Wujun

doi:10.1090/S0025-5718-2013-02789-5

An a posteriori error estimate for the variable-degree Raviart-Thomas method
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by Bernardo Cockburn and Wujun Zhang;

Math. Comp. 83 (2014), 1063-1082

DOI: https://doi.org/10.1090/S0025-5718-2013-02789-5

Published electronically: October 31, 2013

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Abstract:

We propose a new a posteriori error analysis of the variable-degree, hybridized version of the Raviart-Thomas method for second-order elliptic problems on conforming meshes made of simplexes. We establish both the reliability and efficiency of the estimator for the $L_2$-norm of the error of the flux. We also find the explicit dependence of the estimator on the order of the local spaces $k\ge 0$; the only constants that are not explicitly computed are those depending on the shape-regularity of the simplexes. In particular, the constant of the local efficiency inequality is proven to behave like $(k+{2})^{3/2}$. However, we present numerical experiments suggesting that such a constant is actually independent of $k$.

References

Mark Ainsworth, A posteriori error estimation for lowest order Raviart-Thomas mixed finite elements, SIAM J. Sci. Comput. 30 (2007/08), no. 1, 189–204. MR 2377438, DOI 10.1137/06067331X
Mark Ainsworth and Xinhui Ma, Non-uniform order mixed FEM approximation: implementation, post-processing, computable error bound and adaptivity, J. Comput. Phys. 231 (2012), no. 2, 436–453. MR 2872084, DOI 10.1016/j.jcp.2011.09.011
A. Alonso, Error estimators for a mixed method, Numer. Math. 74 (1996), no. 4, 385–395. MR 1414415, DOI 10.1007/s002110050222
D. N. Arnold and F. Brezzi, Mixed and nonconforming finite element methods: implementation, postprocessing and error estimates, RAIRO Modél. Math. Anal. Numér. 19 (1985), no. 1, 7–32 (English, with French summary). MR 813687, DOI 10.1051/m2an/1985190100071
D. Braess and R. Verfürth, A posteriori error estimators for the Raviart-Thomas element, SIAM J. Numer. Anal. 33 (1996), no. 6, 2431–2444. MR 1427472, DOI 10.1137/S0036142994264079
James H. Bramble and Jinchao Xu, A local post-processing technique for improving the accuracy in mixed finite-element approximations, SIAM J. Numer. Anal. 26 (1989), no. 6, 1267–1275. MR 1025087, DOI 10.1137/0726073
Franco Brezzi, Jim Douglas Jr., Ricardo Durán, and Michel Fortin, Mixed finite elements for second order elliptic problems in three variables, Numer. Math. 51 (1987), no. 2, 237–250. MR 890035, DOI 10.1007/BF01396752
Franco Brezzi and Michel Fortin, Mixed and hybrid finite element methods, Springer Series in Computational Mathematics, vol. 15, Springer-Verlag, New York, 1991. MR 1115205, DOI 10.1007/978-1-4612-3172-1
Carsten Carstensen, A posteriori error estimate for the mixed finite element method, Math. Comp. 66 (1997), no. 218, 465–476. MR 1408371, DOI 10.1090/S0025-5718-97-00837-5
Bernardo Cockburn, Jayadeep Gopalakrishnan, and Raytcho Lazarov, Unified hybridization of discontinuous Galerkin, mixed, and continuous Galerkin methods for second order elliptic problems, SIAM J. Numer. Anal. 47 (2009), no. 2, 1319–1365. MR 2485455, DOI 10.1137/070706616
B. Cockburn and F.-J. Sayas, Divergence-conforming HDG methods for Stokes flows, Math. Comp., to appear.
Willy Dörfler, A convergent adaptive algorithm for Poisson’s equation, SIAM J. Numer. Anal. 33 (1996), no. 3, 1106–1124. MR 1393904, DOI 10.1137/0733054
Lucia Gastaldi and Ricardo H. Nochetto, Sharp maximum norm error estimates for general mixed finite element approximations to second order elliptic equations, RAIRO Modél. Math. Anal. Numér. 23 (1989), no. 1, 103–128 (English, with French summary). MR 1015921, DOI 10.1051/m2an/1989230101031
Ohannes A. Karakashian and Frederic Pascal, A posteriori error estimates for a discontinuous Galerkin approximation of second-order elliptic problems, SIAM J. Numer. Anal. 41 (2003), no. 6, 2374–2399. MR 2034620, DOI 10.1137/S0036142902405217
Ohannes A. Karakashian and Frederic Pascal, Convergence of adaptive discontinuous Galerkin approximations of second-order elliptic problems, SIAM J. Numer. Anal. 45 (2007), no. 2, 641–665. MR 2300291, DOI 10.1137/05063979X
Mats G. Larson and Axel Målqvist, A posteriori error estimates for mixed finite element approximations of elliptic problems, Numer. Math. 108 (2008), no. 3, 487–500. MR 2365826, DOI 10.1007/s00211-007-0121-y
Carlo Lovadina and Rolf Stenberg, Energy norm a posteriori error estimates for mixed finite element methods, Math. Comp. 75 (2006), no. 256, 1659–1674. MR 2240629, DOI 10.1090/S0025-5718-06-01872-2
Rolf Stenberg, A family of mixed finite elements for the elasticity problem, Numer. Math. 53 (1988), no. 5, 513–538. MR 954768, DOI 10.1007/BF01397550
Rolf Stenberg, Postprocessing schemes for some mixed finite elements, RAIRO Modél. Math. Anal. Numér. 25 (1991), no. 1, 151–167 (English, with French summary). MR 1086845, DOI 10.1051/m2an/1991250101511
Martin Vohralík, A posteriori error estimates for lowest-order mixed finite element discretizations of convection-diffusion-reaction equations, SIAM J. Numer. Anal. 45 (2007), no. 4, 1570–1599. MR 2338400, DOI 10.1137/060653184
T. Warburton and J. S. Hesthaven, On the constants in $hp$-finite element trace inverse inequalities, Comput. Methods Appl. Mech. Engrg. 192 (2003), no. 25, 2765–2773. MR 1986022, DOI 10.1016/S0045-7825(03)00294-9
Liang Zhu, Stefano Giani, Paul Houston, and Dominik Schötzau, Energy norm a posteriori error estimation for $hp$-adaptive discontinuous Galerkin methods for elliptic problems in three dimensions, Math. Models Methods Appl. Sci. 21 (2011), no. 2, 267–306. MR 2776669, DOI 10.1142/S0218202511005052