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Nonconforming Mixed Finite Element Method for Time-dependent Maxwell's Equations with ABC

Published online by Cambridge University Press:  24 May 2016

Changhui Yao  and
Dongyang Shi
Changhui Yao*
Affiliation:
School of Mathematics and Statistics, Zhengzhou University, Zhengzhou, 450001, China
Dongyang Shi*
Affiliation:
School of Mathematics and Statistics, Zhengzhou University, Zhengzhou, 450001, China
*
*Corresponding author. Email addresses: [email protected] (C. -H. Yao), shi [email protected] (D. -Y. Shi)
*Corresponding author. Email addresses: [email protected] (C. -H. Yao), shi [email protected] (D. -Y. Shi)
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Abstract

In this paper, a nonconforming mixed finite element method (FEM) is presented to approximate time-dependent Maxwell's equations in a three-dimensional bounded domain with absorbing boundary conditions (ABC). By employing traditional variational formula, instead of adding penalty terms, we show that the discrete scheme is robust. Meanwhile, with the help of the element's typical properties and derivative transfer skills, the convergence analysis and error estimates for semidiscrete and backward Euler fully-discrete schemes are given, respectively. Numerical tests show the validity of the proposed method.

Keywords

65N3065N1535J25Maxwell's equationsAbsorbing boundary conditionNonconforming mixed FEMSemi-discrete and fully discrete schemes
Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 9 , Issue 2 , May 2016 , pp. 193 - 214
Copyright
Copyright © Global-Science Press 2016 

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References

[1]Huang, Y. Q. and Li, J. and Yang, W., Interior penalty DG methods for Maxwell's equations in dispersive media, J. Comput. Phys., 230:12 (2011), pp. 4559–4570.CrossRefGoogle Scholar
[2]Monk, P., A mixed method for approximating Maxwell's equations, SIAM J. Numer. Anal., 28:6 (1999), pp. 1610–1634.CrossRefGoogle Scholar
[3]Monk, P., A comparison of three mixed methods for the time-dependent Maxwell's equations, SIAM J. Sci. Statist. Comput., 13:5 (1992), pp. 1097–1122.CrossRefGoogle Scholar
[4]Monk, P., Analysis of a finite element method for Maxwell's equations, SIAM J. Numer. Anal., 29:3 (1992), pp. 714–729.CrossRefGoogle Scholar
[5]Monk, P., A finite element method for approximating the time-harmonic Maxwell equations, Numer. Math., 63:2 (1992), pp. 243–261.CrossRefGoogle Scholar
[6]Monk, P., A comparison of three mixed methods for the time-dependent Maxwell's equations, SIAM J. Sci. Statist. Comput., 13:5(1992), pp. 1097–1122.CrossRefGoogle Scholar
[7]Monk, P., Finite Element Methods for Maxwell's Equations, Oxford University Press, New York, 2003.Google Scholar
[8]Monk, P. and Suli, Endre, A convergence analysis of Yee's scheme on nonuniform grids, SIAM J. Numer. Anal., 31:2(1994), pp. 393–412.CrossRefGoogle Scholar
[9]Nédélec, J.-C., Mixed finite element in 3D in H(div) and H(curl), Lecture Notes in Math., 1192(1986), pp. 321–325.Google Scholar
[10]Nédélec, J.-C., A new family of mixed finite elements in R3, Numer. Math., 50:1(1986), pp. 57–81.CrossRefGoogle Scholar
[11]Bonnet, A.-S., Dhia, B., Hazard, C. and Lohrengel, S., A sigular field method for the solution of Maxwell's equations in polyhedral domains, SIAM J. Appl. Math., 59(1999), pp. 2028–2044.Google Scholar
[12]Hazard, C. and Lenoir, M., On the solution of time-harmonic scattering problems for Maxwell's equations, SIAM J. Math. Anal., 27(1996), pp. 1559–1630.CrossRefGoogle Scholar
[13]Duan, H. Y., Jia, F., Lin, P. and Roger Tan, C. E., The Local L2 Projected C0 Finite Element Method for Maxwell Problem, SIAM J. Numer. Anal., 47:2(2009), pp,1274–1303.Google Scholar
[14]Brenner, S. C., Li, F. and Sung, L.-Y., A local divergence-free interior penalty method for two-dimensional curl-curl problem, SIAM J. Numer. Anal., 46:3 (2008), pp, 1190–1211.CrossRefGoogle Scholar
[15]Brenner, S. C., Cui, J., Li, F. and Sung, L.-Y., A nonconforming finite element method for a two-dimensional curlCcurl and grad-div problem, Numer. Math., 109 (2008), pp,509–533.Google Scholar
[16]Brenner, S. C., Li, F. and Sung, L.-Y., A locally divergence-free nonconforming finite element method for the time-harmonic maxwell equations, Math. Comp., 76:258 (2007), pp, 573–595.CrossRefGoogle Scholar
[17]Li, J., Error analysis of finite element methods for 3-D Maxwell's equations in dispersive media, J. Comput. Appl. Math., 188:1 (2006), pp, 107–120.CrossRefGoogle Scholar
[18]Li, J. and Chen, Y., Analysis of a time-domain finite element method for 3-D Maxwell's equations in dispersive media, Comput. Methods Appl. Mech. Engrg., 195:33-36, (2006), pp, 4220–4229.Google Scholar
[19]Li, J. and Wood, A. H., Finite element analysis for wave propagation in double negative metamaterials, J. Sci. Comput., 32:2 (2007), pp, 263–286.Google Scholar
[20]Ciarlet, P. Jr., Garcia, E. and Zou, J., Solving Maxwell equations in 3D prismatic domains, C. R. Math. Acad. Sci., 339:10(2004), pp, 721–726.Google Scholar
[21]Chen, J., Xu, Y. and Zou, J., Convergence analysis of an adaptive edge element method for Maxwell's equations, Appl. Numer. Math., 59:12(2009), pp, 2950–2969.Google Scholar
[22]Chen, J., Xu, Y. F. and Zou, J., An adaptive inverse iteration for Maxwell eigenvalue problem based on edge elements, J. Comput. Phys., 229:7(2010), pp, 2649–2658.Google Scholar
[23]Shi, D. Y. and Pei, L. F., Low order Crouzeix-Raviart type nonconforming finite element methods for approximating Maxwell's equations, Int. J. Numer. Anal. Model., 5:3(2008), pp, 373–385.Google Scholar
[24]Shi, D. Y. and Pei, L. F., Low order Crouzeix-Raviart type nonconforming finite element methods for the 3D time-dependent Maxwell's equations, Appl. Math. Comput., 211:1(2009), pp, 1–9.Google Scholar
[25]Shi, D. Y., Pei, L. F. and Chen, S.C., A nonconforming arbitrary quadrilateral finite element method for approximating Maxwell's equations, Numer. Math. J. Chin. Univ. (Engl. Ser.), 16:4(2007), pp, 289–299.Google Scholar
[26]Qiao, Z., Yao, C.H. and Jia, S.J., Superconvergence and extrapolation analysis of a nonconforming mixed finite element approximation for time-harmonic Maxwell's equations, J. Sci. Comput., 46:1 (2011), pp 1–19.Google Scholar
[27]Qiao, Z., Numerical Investigations of the dynamical behaviors and instabilities for the Gierer-Meinhardt system, Commun. Comput. Phys., 3 (2008), pp, 406–426.Google Scholar
[28]Qiao, Z., Zhang, Z. and Tang, T., An adaptive time-stepping strategy for the molecular beam epitaxy models, SIAM J. Sci. Comput., 33 (2011), pp, 1395–1414.Google Scholar
[29]Santos, J. E. and Sheen, D., On the existence and uniqueness of solutions to Maxwell’s equations in bounded domains with application to magnetotellurics, Math. Models Methods Appl. Sci., 10:4 (2000), pp, 593–613.CrossRefGoogle Scholar
[30]Feng, X., Absorbing boundary conditions for electromagnetic wave propagation, Math. Comp., 68:225 (1999), pp, 145–168.Google Scholar
[31]Namiki, T., A new FDTD algorithm based on alternating direction implicit method, Microwave Theory and Techniques IEEE Transactions on, 47:10 (1999), pp, 2003–2007.Google Scholar
[32]Mackie, R. L., Madden, T. R. and Wannamaker, P. E., A 3-Dimensinal magnetotelluric modeling using difference-equations -theorem and comparisions to integral-equation solutions, Geophysics, 58:2 (1993), pp, 215–226.CrossRefGoogle Scholar
[33]Yee, K. S., Numerical solution of inital boundary value problem involving Maxwell's equations in isotropic media, IEEE Trans. Antennas Propagat., 14 (1996), pp, 302–307.Google Scholar