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Adaptive hp-Finite Element Computations for Time-Harmonic Maxwell’s Equations

Published online by Cambridge University Press:  03 June 2015

Xue Jiang ,
Linbo Zhang  and
Weiying Zheng
Xue Jiang*
Affiliation:
LSEC, Institute of Computational Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China
Linbo Zhang*
Affiliation:
LSEC, Institute of Computational Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China
Weiying Zheng*
Affiliation:
LSEC, Institute of Computational Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China
*
Email:jxue@lsec.cc.ac.cn
Email:zlb@lsec.cc.ac.cn
Corresponding author.Email:zwy@lsec.cc.ac.cn
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Abstract

In this paper, hp-adaptive finite element methods are studied for time-harmonic Maxwell’s equations. We propose the parallel hp-adaptive algorithms on conforming unstructured tetrahedral meshes based on residual-based a posteriori error estimates. Extensive numerical experiments are reported to investigate the efficiency of the hp-adaptive methods for point singularities, edge singularities, and an engineering benchmark problem of Maxwell’s equations. The hp-adaptive methods show much better performance than the h-adaptive method.

Keywords

65M3078A45hp-adaptive finite element methodMaxwell’s equationseddy current problema posteriori error estimate
Type
Research Article
Information
Communications in Computational Physics , Volume 13 , Issue 2 , February 2013 , pp. 559 - 582
Copyright
Copyright © Global Science Press Limited 2013

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References

[1]Ainsworth, M. and Coyle, J., Hierarchic finite element bases on unstructured tetrahedral meshes, Int. J. Numr. Meth. Engng., 58 (2003), 2103–2130.Google Scholar
[2]Ainsworth, M. and Senior, B., An adaptive refinement strategy for hp-finite element computation, Appl. Numer. Math., 26 (1998), 165–178.Google Scholar
[3]Ammari, H., Buffa, A., and Nédélec, J.C., A justification of eddy current model for the maxwell equations, SIAM J. Appl. Math., 60 (2000), 1805–1823.Google Scholar
[4]Arnold, D.N., Mukherjee, A., and Pouly, L., Locally Adapted Tetrahedral Meshes Using Bisection, SIAM J. Sci. Comput., 22 (2000), 431–448.Google Scholar
[5]Babuška, I. and Aziz, A., Survey Lectures on Mathematical Foundations of the Finite Element Method, in The Mathematical Foundations of the Finite Element Method with Application to the Partial Differential Equations, ed. by Aziz, A., Academic Press, New York, 1973.Google Scholar
[6]Babuška, I., Andersson, B., Guo, B., Melenk, J.M., and Oh, H.S., Finite element method for solving problems with singular solutions, J. Comput. Appl. Math., 74 (1996), 51–70.Google Scholar
[7]Babuška, I. and Dorr, M.R., Error estimates for combined h and p versions of the finite element method, Numer. Math., 37 (1981), 257–277.Google Scholar
[8]Babuška, I. and Rheinboldt, C., Error estimates for adaptive finite element computations, SIAM J. Numer. Anal., 15 (1978), 736–754.Google Scholar
[9]Babuška, I., Szabo, B.A., and Katz, I.N., The p-version of the finite element method, SIAM J. Numer. Anal., 18 (1981), 515–545.Google Scholar
[10]Bachinger, F., Langer, U., and Schöberl, J., Numerical analysis of nonlinear multiharmonic eddy current problems, Numer. Math., 100 (2005), pp. 594–616.Google Scholar
[11]Beck, R., Hiptmar, R., Hoppe, R.H.W., and Wohlmoth, B., Residual based a posteriori error estimations for eddy current computation, M2AN Math. Modeling and Numer. Anal., 34 (2000), 159–182.Google Scholar
[12]Birman, M.Sh. and Solomyak, M.Z., L2-Theory of the Maxwell operator in arbitary domains, Uspekhi Mat. Nauk 42 (1987), pp. 61–76 (in Russian); Russian Math. Surveys 43 (1987), pp. 75–96 (in English).Google Scholar
[13]Braess, D., Pillwein, V., and Schöberl, J., Equilibrated residual error estimators are p-robust, Comput. Methods Appl. Mech. Eng., 198 (2009), 1189–1197.Google Scholar
[14]Braess, D. and Schöberl, J., Equilibrated residual error estimator for edge elements, Math. Comp., 77 (2008), 651–672.Google Scholar
[15]Chen, J., Chen, Z., Cui, T. and Zhang, L., An adaptive finite element method for the eddy current model with circuit/field couplings, SIAM J. Sci. Comput., 32 (2010), 1020–1042.Google Scholar
[16]Chen, Z., Wang, L., and Zheng, W., An adaptive multilevel method for time-harmonic Maxwell equations with singularities, SIAM J. Sci. Comput., 29 (2007), 118–138.Google Scholar
[17]Chen, Z., Guo, B., and Xiao, Y., An hp adaptive uniaxial perfectly matched layer method for Helmholtz scattering problems, Commun. in Comput. Phys., 5 (2009), 546–564.Google Scholar
[18]Cheng, Z., Takahashi, N., and Forghani, B., TEAM Problem 21 Family (V. 2009), approved by the International Compumag Society Board at Compumag-2009, Florianopolis, Brazil, http://www.compumag.org/jsite/team.html.Google Scholar
[19]Cherubini, C., Gizzi, A., Bertolaso, M., Tambone, V., and Filippi, S., A bistable field model of cancer dynamics, Commun. Comput. Phys., 11 (2012), 1–18.Google Scholar
[20]Demkowicz, L., Kurtz, J., Pardo, D., Paszyinski, M., Rachowicz, W., and Zdunek, A., Computing with hp-Adaptive Finite Elements, Vol. 2, Frontiers: Three Dimensional Elliptic and Maxwell Problems with Applications, Chapman & Hall/CRC, Boca Raton, London, 2008.Google Scholar
[21]Dhia, A.B., Hazard, C., and Lohrengel, S., A singular field method for the solution of Maxwell’s equations in polyhedral domains, SIAM J. Appl. Math., 59 (1999), pp. 2028–2044.CrossRefGoogle Scholar
[22]Dörfler, W., A convergent adaptive algorithm for Poissons equation, SIAM J. Numer. Anal., 33 (1996), 1106–1124.Google Scholar
[23]Guo, W. and Babŭska, I., The h-p version of the finite element method. Part 2. General results and applications, Comput. Mech., 1 (1986), 203–226.Google Scholar
[24]Heuveline, V. and Rannacher, R., Duality-based adaptivity in the hp-finite element method, J. Numer. Math., 11 (2003), 1–18.Google Scholar
[25]Holst, M., McCammon, J. A., Yu, Z., Zhou, Y. C., and Zhu, Y., Adaptive finite element modeling techniques for the Poisson-Boltzmann equation, Commun. Comput. Phys., 11 (2012), 179214.Google Scholar
[26]Hoppe, R. and Schöberl, J., Convergence of adaptive edge element methods for the 3D eddy currents equation, J. Comp. Math., 27 (2009), 657–676.Google Scholar
[27]Kossaczký, I., A recursive approach to local mesh refinement in two and three dimensions, J. Comput. Appl. Math., 55 (1994), 275–288.CrossRefGoogle Scholar
[28]Liu, H., Researches on Dynamic Load Balancing Algorithms and hp Adaptivity in 3-D Parallel Adaptive Finite Element Computations, Ph.D thesis, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, 2010 (accessible by sending email to Liu Hui, Dr.: liuhui@lsec.cc.ac.cn).Google Scholar
[29]Melenk, J.M., hp-interpolation of nonsmooth functions and an application to hp-a posteriori error estimation, SIAM J. Numer. Anal., 43 (2005), 127–155.Google Scholar
[30]Melenk, J.M. and Wohlmuth, B., On residual-based a posteriori error estimation in hp-FEM, Adv. Comp. Math., 15 (2001), 311–331.Google Scholar
[31]Mitchell, W.F. and McClain, M.A., A survey of hp-adaptive strategies for elliptic partial differential equations, Recent Advances in Computational and Applied Mathematics, Simos, T.E. (Ed.), Springer Dordrecht, 2011.Google Scholar
[32]Mitchell, W., Optimal multilevel iterative methods for adaptive grids, SIAM J. Sci. Stat. Comput., 13 (1992), pp. 146–167.Google Scholar
[33]Monk, P., Finite Element Methods for Maxwell’s Equations, Clarendon Press, Oxford, 2003.Google Scholar
[34]Morin, P., Nochetto, R.H., and Siebert, K., Convergence of adaptive finite element methods, SIAM Review, 44 (2002), 631–658.Google Scholar
[35] MUMPS: a MUltifrontal Massively Parallel sparse direct Solver. http://mumps.enseeiht.fr.Google Scholar
[36]Pardo, D., Demkowicz, L., Torres-Verdln, C., and Paszynski, M., Two-dimensional high-accuracy simulation of resistivity logging-while-drilling (lwd) measurements using a self-adaptive goal-oriented hp finite element method, SIAM J. Appl. Math., 66 (2006), 2085–2106.Google Scholar
[37]Pardo, D., Demkowicz, L., Torres-Verdln, C., and Paszynski, M., A self-adaptive goal-oriented hp-finite element method with electromagnetic applications. Part II: Electrodynamics, Computer Methods in Applied Mechanics and Engineering, 196 (2007), 37–40.Google Scholar
[38]Schmidt, A. and Siebert, K.G., Design of Adaptive Finite Element Software, The Finite Element Toolbox ALBERTA, Lecture Notes in Computational Science and Engineering, Vol. 42, Springer, Berlin, 2005.Google Scholar
[39]Schöberl, J., A posteriori error estimates for Maxwell equations, Math. Comp., 77 (2008), 633–649.Google Scholar
[40]Verfürth, R., A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Rechniques, Wiley-Teubner, Chichester, Stuttgart, 1996.Google Scholar
[41]Zhang, L., A Parallel Algorithm for Adaptive Local Refinement of Tetrahedral Meshes Using Bisection, Numer. Math.: Theor. Method Appl., 2 (2009), 65–89.Google Scholar
[42]Zheng, W., Chen, Z., and Wang, L., An adaptive finite element method for the H-ty formulation of time-dependent eddy current problems, Numer. Math., 103 (2006), 667–689.Google Scholar