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Adaptive Hybridized Interior Penalty Discontinuous Galerkin Methods for H(curl)-Elliptic Problems

Published online by Cambridge University Press:  28 May 2015

C. Carstensen,
R. H. W. Hoppe,
N. Sharma and
T. Warburton
C. Carstensen
Affiliation:
Department of Mathematics, Humboldt Universität zu Berlin, D-10099 Berlin, GermanyDepartment of Computer Science Engineering, Yonsei University, Seoul 120-749, Korea
R. H. W. Hoppe
Affiliation:
Department of Mathematics, University of Houston, Houston TX 77204-3008, USAInstitute of Mathematics, University of Augsburg, D-86159 Augsburg, Germany
N. Sharma
Affiliation:
Department of Mathematics, University of Houston, Houston TX 77204-3008, USA
T. Warburton
Affiliation:
CAAM, Rice University, Houston, TX 77005-1892, USA
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Abstract

We develop and analyze an adaptive hybridized Interior Penalty Discontinuous Galerkin (IPDG-H) method for H(curl)-elliptic boundary value problems in 2D or 3D arising from a semi-discretization of the eddy currents equations. The method can be derived from a mixed formulation of the given boundary value problem and involves a Lagrange multiplier that is an approximation of the tangential traces of the primal variable on the interfaces of the underlying triangulation of the computational domain. It is shown that the IPDG-H technique can be equivalently formulated and thus implemented as a mortar method. The mesh adaptation is based on a residual-type a posteriori error estimator consisting of element and face residuals. Within a unified framework for adaptive finite element methods, we prove the reliability of the estimator up to a consistency error. The performance of the adaptive symmetric IPDG-H method is documented by numerical results for representative test examples in 2D.

Keywords

65N3065N5078M10Adaptive hybridized Interior Penalty Discontinuous Galerkin methoda posteriori error analysisH(curl)-elliptic boundary value problemssemi-discrete eddy currents equations

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Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 4 , Issue 1 , February 2011, pp. 13 - 37
Copyright
Copyright © Global Science Press Limited 2011

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References

[1] Ainsworth, M., A posteriori error estimation f or Discontinuous Galerkin finite element approximation, SIAM J. Numer. Anal., 45 (2007), pp. 1777–1798.Google Scholar
[2] Ainsworth, M. and Oden, T., A Posteriori Error Estimation in Finite Element Analysis, Wiley, Chichester, 2000.Google Scholar
[3] Arnold, D.N., An interior penalty finite element method with discontinuous elements, SIAM J. Numer. Anal., 19 (1982), pp. 742–760.Google Scholar
[4] Arnold, D.N. and Brezzi, F., Mixed and nonconforming finite element methods: implementation postprocessing and error estimates, RAIRO, Modél. Math. Anal. Numér, 19 (1985), pp. 7–32.Google Scholar
[5] Arnold, D.N, Brezzi, F., Cockburn, B., and Marini, D., Unified analysis of discontinuous Galerkin methods for elliptic problems, SIAM J. Numer. Anal., 39 (2002), pp. 1749–1779.Google Scholar
[6] Babuska, I. and Strouboulis, T., The Finite Element Method and its Reliability, Clarendon Press, Oxford, 2001.Google Scholar
[7] Bangerth, W. and Rannacher, R., Adaptive Finite Element Methods for Differential Equations, Lectures in Mathematics. ETH-Zürich. Birkhäuser, Basel, 2003.Google Scholar
[8] Beck, R., Deuflhard, P., Hiptmair, R., R.Hoppe, H.W, and Wohlmuth, B., Adaptive multilevel methods for edge element discretizations of Maxwell’s equations, Surveys of Math. in Industry, 8 (1999), pp. 271–312.Google Scholar
[9] Beck, R., Hiptmair, R., Hoppe, R.H.W., and Wohlmuth, B., Residual based a posteriori error estimators for eddy current computation, M2 AN Math. Modeling and Numer. Anal., 34 (2000), pp. 159–182.Google Scholar
[10] Beck, R., Hiptmair, R., and Wohlmuth, B., Hierarchical error estimator for eddy current computation, In: Proc. 2nd European Conf. on Advanced Numer. Meth. (ENUMATH99), Jyväskylä, Finland, July 26-30, 1999 (Neittaanmäki, P. et al.; eds.), pp. 111–120, World Scientific, Singapore, 2000.Google Scholar
[11] Becker, R., Hansbo, P., and Larson, M.G., Energy norm a posteriori error estimation f or discontinuous Galerkin methods, Comput. Methods Appl. Mech. Engrg., 192 (2003), pp. 723–733.Google Scholar
[12] Bonito, A. and Nochetto, R., Quasi-optimal convergence rate of an adaptive Discontinuous Galerkin method, SIAM J. Numer. Anal., 48 (2010), pp. 734–771.Google Scholar
[13] Bramble, J.H. and Xu, J., A local post-processing technique for improving the accuracy in mixed finite element approximations, SIAM J. Numer. Anal., 26 (1089), pp. 1267–1275.Google Scholar
[14] Brandts, J., Superconvergence and a posteriori error estimation f or triangular mixed finite element methods, Numer. Math., 68 (1994), pp. 311–324.Google Scholar
[15] Brezzi, F., Douglas, J. Jr., and Marini, L.D., Two families of mixed finite elements for second order elliptic problems, Numer. Math., 47 (1985), pp. 217–235.Google Scholar
[16] Buffa, A., Costabel, M., and Sheen, D., On traces for H(curl,ū) in Lipschitz domains, J. Math. Anal. Appl., 276 (2002), pp. 845–867.Google Scholar
[17] Carstensen, C., A unifying theory of a posteriori finite element error control, Numer. Math., 100 (2005), pp. 617–637.Google Scholar
[18] Carstensen, C., Gudi, T., and Jensen, M., A unifying theory of a posteriori error control for discontinuous Galerkin FEM, Numer. Math., 112 (2009), pp. 363–379.Google Scholar
[19] Carstensen, C. and Hoppe, R.H.W., Convergence analysis of an adaptive edge finite element method for the 2d eddy current equations, J. Numer. Math., 13 (2005), pp. 19–32.Google Scholar
[20] Carstensen, C. and R.H.Hoppe, W., Unified framework for an a posteriori error analysis of non-standard finite element approximations of H(curl)-elliptic problems, J. Numer. Math., 17 (2009), pp. 27–44.Google Scholar
[21] Carstensen, C. and Hu, J., A unifying theory of a posteriori error control for nonconforming finite element methods, Numer. Math., 107 (2007), pp. 473–502.Google Scholar
[22] Carstensen, C., Hu, J., and Orlando, A., Framework for the a posteriori error analysis of non-conforming finite elements, SIAM J. Numer. Anal., 45 (2007), pp. 68–82.Google Scholar
[23] Castillo, P., Cockburn, B., Perugia, I., and SchÖtzau, D., An a priori error estimate of the local discontinuous Galerkin method for elliptic problems, SIAM J. Numer. Anal., 38 (2000), pp. 1676–1706.CrossRefGoogle Scholar
[24] Cockburn, B., Discontinuous Galerkin methods, Z. Angew. Math. Mech., 83 (2003), pp. 731–754.Google Scholar
[25] Cockburn, B. and Gopalakrishnan, J., A characterization of hybridized mixed methods for second order elliptic problems, SIAM J. Numer. Anal., 42 (2004), pp. 283–301.Google Scholar
[26] Cockburn, B. and Gopalakrishnan, J., Error analysis of variable degree mixed methods for elliptic problems via hybridization, Math. Comp., 74 (2005), pp. 1653–1677.Google Scholar
[27] Cockburn, B. and Gopalakrishnan, J., New hybridization techniques, GAMM-Mitteilungen, 2 (2005), pp. 154–183.Google Scholar
[28] Cockburn, B., Gopalakrishnan, J., and Lazarov, R., Unified hybridization of discontinuous Galerkin, Mixed and continuous Galerkin methods for second order elliptic problems, SIAM J. Numer. Anal., 47 (2009), pp. 1319–1365.Google Scholar
[29] Dörfler, W., A convergent adaptive algorithm for Poisson’s equation, SIAM J. Numer. Anal., 33 (1996), pp. 1106–1124.Google Scholar
[30] Eriksson, K., Estep, D., Hansbo, P., and Johnson, C., Computational Differential Equations, Cambridge University Press, Cambridge, 1996.Google Scholar
[31] Veubeke, B.M. Fraejis de, Displacement and equilibrium methods in the finite element method, In: Stress Analysis (Zienkiewicz, O. and Hollister, G.; eds.), pp. 145–197, Wiley, New York, 1977.Google Scholar
[32]Girault, V. and P.Raviart, A., Finite Element Approximation of the Navier-Stokes Equations, Lecture Notes in Mathematics 749, Springer, Berlin-Heidelberg-New York, 1979.Google Scholar
[33] Hesthaven, J.S. and Warburton, T., Nodal Discontinuous Galerkin Methods, Springer, Berlin-Heidelberg-New York, 2008.Google Scholar
[34] R.Hoppe, H.W., Kanschat, G., and Warburton, T., Convergence analysis of an adaptive interior penalty discontinuous Galerkin method, SIAM J. Numer. Anal., 47 (2009), pp. 534–550.Google Scholar
[35] R.H.Hoppe, W. and Schöberl, J., Convergence of adaptive edge element methods for the 3D eddy currents equations, J. Comp. Math., 27 (2009), pp. 657–676.Google Scholar
[36] Houston, P., Perugia, I., and Schötzau, D., Mixed discontinuous Galerkin approximation of the Maxwell operator, SIAM J. Numer. Anal., 42 (2004), pp. 434–459.Google Scholar
[37] Houston, P., Perugia, I., and Schötzau, D., A posteriori error estimation for discontinuous Galerkin discretizations of H(curl)-elliptic partial differential equations, IMA Journal of Numerical Analysis, 27 (2007), pp. 122–150.Google Scholar
[38] Houston, P., Schötzau, D., and T.Wihler, P., Energy norm a posteriori error estimation ofhp-adaptive discontinuous Galerkin methods for elliptic problems, Math. Models Methods Appl. Sci., 17 (2007), pp. 33–62.Google Scholar
[39] Kanschat, G. and Rannacher, R., Local error analysis of the interior penalty discontinuous Galerkin method for second order problems, J. Numer. Math., 10 (2002), pp. 249–274.Google Scholar
[40] Karakashian, O. and Pascal, F., A posteriori error estimates for a discontinuous Galerkin approximation of second-order elliptic problems, SIAM J. Numer. Anal., (41), pp. 2003–2374.Google Scholar
[41] Karakashian, O. and Pascal, F., Convergence of adaptive Doiscontinuous Galerkin approximations of second-order elliptic problems, SIAM J. Numer. Anal., 45 (2007), pp. 641–665.Google Scholar
[42] Monk, P., Finite Element Methods for Maxwell’s Equations, Clarendon Press, Oxford, 2003.Google Scholar
[43] Nédélec, J.C., Mixedfinite elements in ℝ3 , Numer. Math., 35 (1980), pp. 315–341.Google Scholar
[44] Neittaanmäki, P. and Repin, S., Reliable Methods for Mathematical Modelling. Error Control and a Posteriori Estimates, Elsevier, New York, 2004.Google Scholar
[45] Nicaise, S. and Creuse, E., Isotropic and anisotropic a posteriori error estimation of the mixed finite element method for second order operators in divergence form, ETNA, 23 (2006), pp. 38–62.Google Scholar
[46] Perugia, I., Schötzau, D., and Monk, P., Stabilized interior penalty methods for the time-harmonic Maxwell equations, Comp. Meth. Appl. Mech. Engrg., 191 (2002), pp. 4675–4697.Google Scholar
[47] Riviere, B. and Wheeler, M.F., A posteriori error estimates and mesh adaptation strategy for discontinuous Galerkin methods applied to diffusion problems, Computers & Mathematics with Applications, 46 (2003), pp. 141–163.Google Scholar
[48] Schöberl, J., A posteriori error estimates for Maxwell equations, Math. Comp., 77 (2008), pp. 633–649.Google Scholar
[49] Verfürth, R., A Review of A Posteriori Estimation and Adaptive Mesh-Refinement Techniques, Wiley-Teubner, New York, Stuttgart, 1996.Google Scholar
[50] Wohlmuth, B. and Hoppe, R.H.W., A comparison of a posteriori error estimators for mixed finite element discretizations, Math. Comp., 82 (1999), pp. 253–279.Google Scholar
[51] Xu, X. and Hoppe, R.H.W., On the convergence of mortar edge element methods in R3 , SIAM J. Numer. Anal., 43 (2005), pp. 1276–1294.Google Scholar