Abstract
This paper introduces and analyzes an equilibrated a posteriori error estimator for mixed finite element approximations to the diffusion problem in two dimensions. The estimator, which is a generalization of those in Braess and Schöberl (Math Comput 77:651–672, 2008) and Cai and Zhang (SIAM J Numer Anal 50(1):151–170, 2012), is based on the Prager–Synge identity and on a local recovery of a gradient in the curl free subspace of the \(H(\text {curl})\)-confirming finite element spaces. The resulting estimator admits guaranteed reliability, and its robust local efficiency is proved under the quasi-monotonicity condition of the diffusion coefficient. Numerical experiments are given to confirm the theoretical results.
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References
Ainsworth, M.: Robust a posteriori error estimation for nonconforming finite element approximation. SIAM J. Numer. Anal. 42(6), 2320–2341 (2005)
Ainsworth, M.: A posteriori error estimation for lowest order Raviart–Thomas mixed finite elements. SIAM J. Sci. Comput. 30, 189–204 (2007)
Babuška, I., Osborn, J., Pitkäranta, J.: Analysis of mixed methods using mesh dependent norms. Math. Comput. 35, 1039–1062 (1980)
Bernardi, C., Verfürth, R.: Adaptive finite element methods for elliptic equations with non-smooth coefficients. Numer. Math. 85(4), 579–608 (2000)
Boffi, D., Brezzi, F., Fortin, M.: Mixed Finite Element Methods and Applications. Springer, Berlin (2013)
Braess, D.: Finite Elements: Theory, Fast Solvers and Applications in Solid Mechanics, 3rd edn. Cambridge University Press, Cambridge (2007)
Braess, D., Fraunholz, T., Hoppe, R.H.W.: An equilibrated a posteriori error estimator for the Interior Penalty Discontinuous Galerkin method. SIAM J. Numer. Anal. 52, 2121–2136 (2014)
Braess, D., Schöberl, J.: Equilibrated residual error estimator for edge elements. Math. Comput. 77, 651–672 (2008)
Braess, D., Pillwein, V., Schöberl, J.: Equilibrated residual error estimates are p-robust. Comput. Methods Appl. Mech. Eng. 198, 1189–1197 (2009)
Braess, D., Verfürth, R.: A posteriori error estimators for the Raviart–Thomas element. SIAM J. Numer. Anal. 33, 2431–2444 (1996)
Cai, Z., He, C., Zhang, S.: Discontinuous finite element Methods for interface problems: robust a priori and a posteriori error estimates. SIAM J. Numer. Anal. 55(1), 400–418 (2017)
Cai, Z., Ye, X., Zhang, S.: Discontinuous Galerkin finite element methods for interface problems: a priori and a posteriori error estimations. SIAM J. Numer. Anal. 49(5), 1761–1787 (2011)
Cai, Z., Zhang, S.: Recovery-based error estimator for interface problems: conforming linear elements. SIAM J. Numer. Anal. 47(3), 2132–2156 (2009)
Cai, Z., Zhang, S.: Recovery-based error estimator for interface problems: mixed and nonconforming elements. SIAM J. Numer. Anal. 48(1), 30–52 (2010)
Cai, Z., Zhang, S.: Robust equilibrated residual error estimator for diffusion problems: conforming elements. SIAM J. Numer. Anal. 50(1), 151–170 (2012)
Cockburn, B., Gopalakrishnan, J.: Incompressible finite elements via hybridization. Part II: the Stokes system in three space dimensions. SIAM J. Numer Anal. 43, 1651–1672 (2005)
Destuynder, P., Métivet, B.: Explicit error bounds for a nonconforming finite element method. SIAM J. Numer. Anal. 35(5), 2099–2115 (1998)
Destuynder, P., Métivet, B.: Explicit error bounds in a conforming finite element method. Math. Comput. 68, 1379–1396 (1999)
Ern, A., Vohralik, M.: Polynomial-degree-robust a posteriori estimates in a unified setting for conforming, nonconforming, discontinuous Galerkin, and mixed discretizations. SIAM J. Numer. Anal. 53, 1058–1081 (2015)
Kim, K.-Y.: A posteriori error analysis for locally conservative mixed methods. Math. Comput. 76, 43–66 (2007)
Luce, R., Wohlmuth, B.I.: A local a posteriori error estimator based on equilibrated fluxes. SIAM J. Numer. Anal. 42(4), 1394–1414 (2004)
Nédélec, J.C.: Mixed finite elements in \(\Re ^3\). Numer. Math. 35, 315–341 (1980)
Nédélec, J.C.: A new family of mixed finite elements in \(\Re ^3\). Numer. Math. 50, 57–81 (1986)
Neittaanmäki, P., Repin, S.: Reliable Methods for Computer Simulation: Error Control and A Posteriori Estimates. Elsevier, Amsterdam (2004)
Petzoldt, M.: A posteriori error estimators for elliptic equations with discontinuous coefficients. Adv. Comput. Math. 16(1), 47–75 (2002)
Prager, W., Synge, J.L.: Approximations in elasticity based on the concept of function space. Q. Appl. Math. 5, 286–292 (1947)
Repin, S.: A Posteriori Error Estimation Methods for Partial Differential Equations. Walter de Gruyter, Berlin (2008)
Vejchodský, T.: Local a posteriori error estimator based on the hypercircle method. In: Proceedings of of the European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS 2004), Jyvaskylä, Finland (2004)
Vejchodský, T.: Guaranteed and locally computable a posteriori error estimate. IMA J. Numer. Anal. 26, 525–540 (2006)
Verfürth, R.: A note on constant-free a posteriori error estimates. SIAM J. Numer. Anal. 47, 3180–3194 (2009)
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Z. Cai: This work was supported in part by the National Science Foundation under Grant DMS-1522707. S. Zhang: This work was supported in part by Research Grants Council of the Hong Kong SAR, China under the GRF Grant Project No. 11305319, CityU.
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Cai, D., Cai, Z. & Zhang, S. Robust Equilibrated Error Estimator for Diffusion Problems: Mixed Finite Elements in Two Dimensions. J Sci Comput 83, 22 (2020). https://doi.org/10.1007/s10915-020-01199-9
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DOI: https://doi.org/10.1007/s10915-020-01199-9