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

  • Xue Jiang (a1), Linbo Zhang (a1) and Weiying Zheng (a1)

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.

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Corresponding author.Email:zwy@lsec.cc.ac.cn

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[1]Ainsworth, M. and Coyle, J., Hierarchic finite element bases on unstructured tetrahedral meshes, Int. J. Numr. Meth. Engng., 58 (2003), 21032130.
[2]Ainsworth, M. and Senior, B., An adaptive refinement strategy for hp-finite element computation, Appl. Numer. Math., 26 (1998), 165178.
[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), 18051823.
[4]Arnold, D.N., Mukherjee, A., and Pouly, L., Locally Adapted Tetrahedral Meshes Using Bisection, SIAM J. Sci. Comput., 22 (2000), 431448.
[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.
[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), 5170.
[7]Babuška, I. and Dorr, M.R., Error estimates for combined h and p versions of the finite element method, Numer. Math., 37 (1981), 257277.
[8]Babuška, I. and Rheinboldt, C., Error estimates for adaptive finite element computations, SIAM J. Numer. Anal., 15 (1978), 736754.
[9]Babuška, I., Szabo, B.A., and Katz, I.N., The p-version of the finite element method, SIAM J. Numer. Anal., 18 (1981), 515545.
[10]Bachinger, F., Langer, U., and Schöberl, J., Numerical analysis of nonlinear multiharmonic eddy current problems, Numer. Math., 100 (2005), pp. 594616.
[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), 159182.
[12]Birman, M.Sh. and Solomyak, M.Z., L2-Theory of the Maxwell operator in arbitary domains, Uspekhi Mat. Nauk 42 (1987), pp. 6176 (in Russian); Russian Math. Surveys 43 (1987), pp. 7596 (in English).
[13]Braess, D., Pillwein, V., and Schöberl, J., Equilibrated residual error estimators are p-robust, Comput. Methods Appl. Mech. Eng., 198 (2009), 11891197.
[14]Braess, D. and Schöberl, J., Equilibrated residual error estimator for edge elements, Math. Comp., 77 (2008), 651672.
[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), 10201042.
[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), 118138.
[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), 546564.
[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.
[19]Cherubini, C., Gizzi, A., Bertolaso, M., Tambone, V., and Filippi, S., A bistable field model of cancer dynamics, Commun. Comput. Phys., 11 (2012), 118.
[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.
[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. 20282044.
[22]Dörfler, W., A convergent adaptive algorithm for Poissons equation, SIAM J. Numer. Anal., 33 (1996), 11061124.
[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), 203226.
[24]Heuveline, V. and Rannacher, R., Duality-based adaptivity in the hp-finite element method, J. Numer. Math., 11 (2003), 118.
[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.
[26]Hoppe, R. and Schöberl, J., Convergence of adaptive edge element methods for the 3D eddy currents equation, J. Comp. Math., 27 (2009), 657676.
[27]Kossaczký, I., A recursive approach to local mesh refinement in two and three dimensions, J. Comput. Appl. Math., 55 (1994), 275288.
[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).
[29]Melenk, J.M., hp-interpolation of nonsmooth functions and an application to hp-a posteriori error estimation, SIAM J. Numer. Anal., 43 (2005), 127155.
[30]Melenk, J.M. and Wohlmuth, B., On residual-based a posteriori error estimation in hp-FEM, Adv. Comp. Math., 15 (2001), 311331.
[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.
[32]Mitchell, W., Optimal multilevel iterative methods for adaptive grids, SIAM J. Sci. Stat. Comput., 13 (1992), pp. 146167.
[33]Monk, P., Finite Element Methods for Maxwell’s Equations, Clarendon Press, Oxford, 2003.
[34]Morin, P., Nochetto, R.H., and Siebert, K., Convergence of adaptive finite element methods, SIAM Review, 44 (2002), 631658.
[35] MUMPS: a MUltifrontal Massively Parallel sparse direct Solver. http://mumps.enseeiht.fr.
[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), 20852106.
[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), 3740.
[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.
[39]Schöberl, J., A posteriori error estimates for Maxwell equations, Math. Comp., 77 (2008), 633649.
[40]Verfürth, R., A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Rechniques, Wiley-Teubner, Chichester, Stuttgart, 1996.
[41]Zhang, L., A Parallel Algorithm for Adaptive Local Refinement of Tetrahedral Meshes Using Bisection, Numer. Math.: Theor. Method Appl., 2 (2009), 6589.
[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), 667689.

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

  • Xue Jiang (a1), Linbo Zhang (a1) and Weiying Zheng (a1)

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