Superconvergence analysis for Maxwell’s equations in dispersive media

Lin, Qun; Li, Jichun

doi:10.1090/S0025-5718-07-02039-X

Superconvergence analysis for Maxwell’s equations in dispersive media
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by Qun Lin and Jichun Li PDF

Math. Comp. 77 (2008), 757-771 Request permission

Abstract:

In this paper, we consider the time dependent Maxwell’s equations in dispersive media on a bounded three-dimensional domain. Global superconvergence is obtained for semi-discrete mixed finite element methods for three most popular dispersive media models: the isotropic cold plasma, the one-pole Debye medium, and the two-pole Lorentz medium. Global superconvergence for a standard finite element method is also presented. To our best knowledge, this is the first superconvergence analysis obtained for Maxwell’s equations when dispersive media are involved.

References

Mark Ainsworth, Penny Davies, Dugald Duncan, Paul Martin, and Bryan Rynne (eds.), Topics in computational wave propagation, Lecture Notes in Computational Science and Engineering, vol. 31, Springer-Verlag, Berlin, 2003. Direct and inverse problems. MR 2032865, DOI 10.1007/978-3-642-55483-4
Alain Bossavit, Computational electromagnetism, Electromagnetism, Academic Press, Inc., San Diego, CA, 1998. Variational formulations, complementarity, edge elements. MR 1488417
Carsten Carstensen, Stefan Funken, Wolfgang Hackbusch, Ronald H. W. Hoppe, and Peter Monk (eds.), Computational electromagnetics, Lecture Notes in Computational Science and Engineering, vol. 28, Springer-Verlag, Berlin, 2003. MR 1995092, DOI 10.1007/978-3-642-55745-3
C.M. Chen and Y.Q. Huang, High Accuracy Theory of Finite Element Methods (in Chinese), Hunan Science Press, China, 1995.
Min-Hung Chen, Bernardo Cockburn, and Fernando Reitich, High-order RKDG methods for computational electromagnetics, J. Sci. Comput. 22/23 (2005), 205–226. MR 2142195, DOI 10.1007/s10915-004-4152-6
Q. Chen, M. Katsurai and P.H. Aoyagi, An FDTD formulation for dispersive media using a current density, IEEE Trans. Antennas Propagat. 46 (1998) 1739-1746.
Zhiming Chen, Qiang Du, and Jun Zou, Finite element methods with matching and nonmatching meshes for Maxwell equations with discontinuous coefficients, SIAM J. Numer. Anal. 37 (2000), no. 5, 1542–1570. MR 1759906, DOI 10.1137/S0036142998349977
P. Ciarlet Jr. and Jun Zou, Fully discrete finite element approaches for time-dependent Maxwell’s equations, Numer. Math. 82 (1999), no. 2, 193–219 (English, with English and French summaries). MR 1685459, DOI 10.1007/s002110050417
Bernardo Cockburn, Fengyan Li, and Chi-Wang Shu, Locally divergence-free discontinuous Galerkin methods for the Maxwell equations, J. Comput. Phys. 194 (2004), no. 2, 588–610. MR 2034859, DOI 10.1016/j.jcp.2003.09.007
Gary C. Cohen, Erkki Heikkola, Patrick Joly, and Pekka Neittaanmäki (eds.), Mathematical and numerical aspects of wave propagation—WAVES 2003, Springer-Verlag, Berlin, 2003. MR 2077970, DOI 10.1007/978-3-642-55856-6
S.A. Cummer, An analysis of new and existing FDTD methods for isotropic cold plasma and a method for improving their accuracy, IEEE Trans. Antennas Propagat. 45 (1997) 392-400.
L. Demkowicz, Fully automatic $hp$-adaptivity for Maxwell’s equations, Comput. Methods Appl. Mech. Engrg. 194 (2005), no. 2-5, 605–624. MR 2105184, DOI 10.1016/j.cma.2004.05.023
L. Demkowicz, P. Monk, Ch. Schwab, and L. Vardapetyan, Maxwell eigenvalues and discrete compactness in two dimensions, Comput. Math. Appl. 40 (2000), no. 4-5, 589–605. MR 1772657, DOI 10.1016/S0898-1221(00)00182-6
Richard E. Ewing, Yanping Lin, Tong Sun, Junping Wang, and Shuhua Zhang, Sharp $L^2$-error estimates and superconvergence of mixed finite element methods for non-Fickian flows in porous media, SIAM J. Numer. Anal. 40 (2002), no. 4, 1538–1560. MR 1951906, DOI 10.1137/S0036142900378406
R. Hiptmair, Finite elements in computational electromagnetism, Acta Numer. 11 (2002), 237–339. MR 2009375, DOI 10.1017/S0962492902000041
Paul Houston, Ilaria Perugia, and Dominik Schötzau, Energy norm a posteriori error estimation for mixed discontinuous Galerkin approximations of the Maxwell operator, Comput. Methods Appl. Mech. Engrg. 194 (2005), no. 2-5, 499–510. MR 2105178, DOI 10.1016/j.cma.2004.02.025
D. Jiao and J.-M. Jin, Time-domain finite-element modeling of dispersive media, IEEE Microwave and Wireless Components Letters 11 (2001) 220-222.
Jianming Jin, The finite element method in electromagnetics, 2nd ed., Wiley-Interscience [John Wiley & Sons], New York, 2002. MR 1903357
Michal Křížek and Pekka Neittaanmäki, Bibliography on superconvergence, Finite element methods (Jyväskylä, 1997) Lecture Notes in Pure and Appl. Math., vol. 196, Dekker, New York, 1998, pp. 315–348. MR 1602730
Jichun Li, Error analysis of finite element methods for 3-D Maxwell’s equations in dispersive media, J. Comput. Appl. Math. 188 (2006), no. 1, 107–120. MR 2192630, DOI 10.1016/j.cam.2005.03.060
Jichun Li and Yitung Chen, Analysis of a time-domain finite element method for 3-D Maxwell’s equations in dispersive media, Comput. Methods Appl. Mech. Engrg. 195 (2006), no. 33-36, 4220–4229. MR 2229840, DOI 10.1016/j.cma.2005.08.002
J. Li, Error analysis of fully discrete mixed finite element schemes for 3-D Maxwell’s equations in dispersive media, preprint, Sept. 2005.
Jichun Li and Mary F. Wheeler, Uniform convergence and superconvergence of mixed finite element methods on anisotropically refined grids, SIAM J. Numer. Anal. 38 (2000), no. 3, 770–798. MR 1781203, DOI 10.1137/S0036142999351212
Q. Lin, J. Li and A. Zhou, rectangle test for Ciarlet-Raviart scheme, Prof. of Sys. Sci. & Sys. Engrg., Great Wall (H.K.) Culture Publish Co., 1991, pp. 230-231.
Q. Lin, J. Li and A. Zhou, A rectangle test for the Stokes equations, Prof. of Sys. Sci. & Sys. Engrg., Great Wall (H.K.) Culture Publish Co., 1991, pp. 240-241.
Qun Lin and Jia Fu Lin, High accuracy approximation of mixed finite element for 2-D Maxwell equations, Acta Math. Sci. Ser. A (Chinese Ed.) 23 (2003), no. 4, 499–503 (Chinese, with English and Chinese summaries). MR 2090553
Qun Lin and Jinchao Xu, Linear finite elements with high accuracy, J. Comput. Math. 3 (1985), no. 2, 115–133. MR 854355
Qun Lin and Ning Ning Yan, Superconvergence of mixed element methods for Maxwell’s equations, Gongcheng Shuxue Xuebao 13 (1996), no. 5, suppl., 1–10, 66 (Chinese, with English and Chinese summaries). MR 1437477
Qun Lin and Ningning Yan, Global superconvergence for Maxwell’s equations, Math. Comp. 69 (2000), no. 229, 159–176. MR 1654029, DOI 10.1090/S0025-5718-99-01131-X
Q. Lin and N. Yan, The Construction and Analysis of High Accurate Finite Element Methods (in Chinese), Hebei University Press, Hebei, China, 1996.
Q. Lin, N. Yan and A. Zhou, A rectangle test for interpolated finite elements, Prof. of Sys. Sci. & Sys. Engrg., Great Wall (H.K.) Culture Publish Co., 1991, pp. 217-229.
Tiao Lu, Pingwen Zhang, and Wei Cai, Discontinuous Galerkin methods for dispersive and lossy Maxwell’s equations and PML boundary conditions, J. Comput. Phys. 200 (2004), no. 2, 549–580. MR 2095277, DOI 10.1016/j.jcp.2004.02.022
Peter Monk, An analysis of Nédélec’s method for the spatial discretization of Maxwell’s equations, J. Comput. Appl. Math. 47 (1993), no. 1, 101–121. MR 1226366, DOI 10.1016/0377-0427(93)90093-Q
Peter Monk, Superconvergence of finite element approximations to Maxwell’s equations, Numer. Methods Partial Differential Equations 10 (1994), no. 6, 793–812. MR 1298123, DOI 10.1002/num.1690100611
Peter Monk, Finite element methods for Maxwell’s equations, Numerical Mathematics and Scientific Computation, Oxford University Press, New York, 2003. MR 2059447, DOI 10.1093/acprof:oso/9780198508885.001.0001
J.-C. Nédélec, Mixed finite elements in $\textbf {R}^{3}$, Numer. Math. 35 (1980), no. 3, 315–341. MR 592160, DOI 10.1007/BF01396415
Alfio Quarteroni and Alberto Valli, Numerical approximation of partial differential equations, Springer Series in Computational Mathematics, vol. 23, Springer-Verlag, Berlin, 1994. MR 1299729
Allen Taflove, Computational electrodynamics, Artech House, Inc., Boston, MA, 1995. The finite-difference time-domain method; With contributions by Stephen D. Gedney, Faiza S. Lansing, Thomas G. Jurgens, Gregory W. Saewert, Melinda J. Piket-May, Eric T. Thiele and Stephen T. Barnard. MR 1338377
Lars B. Wahlbin, Superconvergence in Galerkin finite element methods, Lecture Notes in Mathematics, vol. 1605, Springer-Verlag, Berlin, 1995. MR 1439050, DOI 10.1007/BFb0096835
Zhimin Zhang, Ningning Yan, and Tong Sun, Superconvergent derivative recovery for the intermediate finite element family of the second type, IMA J. Numer. Anal. 21 (2001), no. 3, 643–665. MR 1844362, DOI 10.1093/imanum/21.3.643