Abstract
Back in 1992, Monk (Numer Math 63: 243–261, 1992) found that the numerical solution of time-harmonic Maxwell equations shows a superconvergence result \(O(h^2)\) in the discrete maximum norm when the problem is solved by Nédé1ec’s first type linear tetrahedral elements (i.e., the so-called edge elements). However, Monk did not provide any theoretical investigation of this superconvergence phenomenon. Since then, superconvergence analysis has been carried out for edge elements on hexahedral grids and triangular grids. Until now, the theoretical justification of the superconvergence phenomenon for linear edge elements on tetrahedral grids is still open (Monk in Finite element methods for Maxwell’s equations. Oxford University Press, Oxford, 2003, p.188) . The paper is motivated by this open issue. Our major goal of this paper is to fill this gap by providing a delicate theoretical analysis of this superconvergence phenomenon. We further provide some numerical results to demonstrate this phenomenon.
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References
Adjerid, S., Baccouch, M.: The discontinuous Galerkin method for two-dimensional hyperbolic problems. Part I: superconvergence error analysis. J. Sci. Comput. 33, 75–113 (2007)
Ainsworth, M., Oden, J.T.: A Posteriori Error Estimation in Finite Element Analysis. Wiley Interscience, New York (2000)
Bank, R.E., Xu, J.: Asymptotically exact a posteriori error estimators, Part II: General unstructured grids. SIAM J. Numer. Anal. 41, 2313–2332 (2004)
Celiker, F., Zhang, Z., Zhu, H.: Nodal superconvergence of SDFEM for singularly perturbed problems. J. Sci. Comput. 50, 405–433 (2012)
Chen, C.M., Huang, Y.: High Accuracy Theory of Finite Element Methods (in Chinese). Hunan Science Press, China (1995)
Cheng, Y., Shu, C.-W.: Superconvergence and time evolution of discontinuous Galerkin finite element solutions. J. Comput. Phys. 227, 9612–9627 (2008)
Chung, E.T., Ciarlet Jr, P., Yu, T.F.: Convergence and superconvergence of staggered discontinuous Galerkin methods for the three-dimensional Maxwell’s equations on Cartesian grids. J. Comput. Phys. 235, 14–31 (2013)
Ciarlet, P.G.: The Finite Element Method for Elliptic Problems, 2nd edn. Society for Industrial and Applied Mathematics (2002)
Cockburn, B., Dong, B., Guzmán, J., Restelli, M., Sacco, R.: Superconvergent and optimally convergent LDG-hybridizable discontinuous Galerkin methods for convection-diffusion-reaction problems. SIAM J. Sci. Comput. 31, 3827–3846 (2009)
Ewing, R.E., Lin, Y., Sun, T., Wang, J., Zhang, S.: Sharp L2 -error estimates and superconvergence of mixed finite element methods for non-Fickian flows in porous media. SIAM J. Numer. Anal. 40, 1538–1560 (2002)
Huang, Y., Li, J., Lin, Q.: Superconvergence analysis for time-dependent Maxwell’s equations in metamaterials. Numer. Methods Partial Differ. Equ. 28, 1794–1816 (2012)
Huang, Y., Li, J., Wu, C.: Averaging for superconvergence: verification and application of 2D edge elements to Maxwell’s equations in metamaterials. Comput. Methods Appl. Mech. Eng. 255, 121–132 (2013)
Huang, Y., Li, J., Yang, W., Sun, S.: Superconvergence of mixed finite element approximations to 3-D Maxwell’s equations in metamaterials. J. Comput. Phys. 230, 8275–8289 (2011)
Krizek, M., Neittaanamäki, P., Stenberg, R. (eds.): Finite Element Methods: Superconvergence. Postprocessing and A Posteriori Estimates. Marcel Dekker, New York (1998)
Li, J., Huang, Y., Yang, W.: An adaptive edge finite element method for electromagnatic cloaking simulation. J. Comput. Phys. 249, 216–232 (2013)
Lin, Q., Li, J.: Superconvergence analysis for Maxwell’s equations in dispersive media. Math. Comput. 77, 757–771 (2008)
Lin, Q., Yan, N.: Global superconvergence for Maxwell’s equations. Math. Comput. 69, 159–176 (1999)
Lin, Q., Yan, N.: The Construction and Analysis of High Accurate Finite Element Methods (in Chinese). Hebei University Press, Hebei, China (1996)
Monk, P.: A finite element method for approximating the time-harmonic Maxwell equations. Numer. Math. 63, 243–261 (1992)
Monk, P.: Superconvergence of finite element approximations to Maxwells equations. Numer. Methods Partial Differ. Equ. 10, 793–812 (1994)
Monk, P.: Finite Element Methods for Maxwell’s Equations. Oxford University Press, Oxford (2003)
Qiao, Z., Yao, C., Jia, S.: Superconvergence and extrapolation analysis of a nonconforming mixed finite element approximation for time-harmonic Maxwell’s equations. J. Sci. Comput. 46, 1–19 (2011)
Wahlbin, L.B.: Superconvergence in Galerkin Finite Element Methods. Springer, Berlin (1995)
Yang, Y., Shu, C.-W.: Analysis of optimal superconvergence of discontinuous Galerkin method for linear hyperbolic equations. SIAM J. Numer. Anal. 50, 3110–3133 (2012)
Zhang, Z.: (ed.): Special issues of Superconvergence and a posteriori error estimates in finite element methods. Int. J. Numer. Anal. Model 2(1), 1–126 (2005). vol. 3, no.3, pp. 255–376 (2006)
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The authors like to thank anonymous referees for their many insightful comments that improved the paper.
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Work partially supported by NSFC Project 11271310, NSFC Key Project 11031006, NSFC Project 11301446, IRT1179 of PCSIRT and MOST 2010DFR00700, Hunan Provincial Innovation Foundation for Postgraduate (CX2013B254), and a Grant from the Simons Foundation (#281296 to Jichun Li).
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Huang, Y., Li, J., Wu, C. et al. Superconvergence Analysis for Linear Tetrahedral Edge Elements. J Sci Comput 62, 122–145 (2015). https://doi.org/10.1007/s10915-014-9848-7
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DOI: https://doi.org/10.1007/s10915-014-9848-7