Abstract
We propose a Finite Element Heterogeneous Multiscale Method (FE-HMM) for time dependent Maxwell’s equations in second-order formulation in locally periodic materials. This method can approximate the effective behavior of an electromagnetic wave traveling through a highly oscillatory material without the need to resolve the microscopic details of the material. To prove an a-priori error bound for the semi-discrete FE-HMM scheme, we need a new generalization of a Strang-type lemma for second-order hyperbolic equations. Finally, we present a numerical example that is in accordance with the theoretical results.
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A. Abdulle, The finite element heterogeneous multiscale method: a computational strategy for multiscale PDEs. GAKUTO Int. Ser. Math. Sci. Appl. 31, 133–181 (2009)
A. Abdulle, M.J. Grote, Finite element heterogeneous multiscale method for the wave equation. Multiscale Model. Simul. 9(2), 766–792 (2011)
A. Abdulle, W. E, B. Engquist, E. Vanden-Eijnden, The heterogeneous multiscale method. Acta Numer. 21, 1–87 (2012)
G. Allaire, Homogenization and two-scale convergence. SIAM J. Math. Anal. 23(6), 1482–1518 (1992)
A. Bensoussan, J.L. Lions, G. Papanicolaou, Asymptotic analysis for periodic structures, in Studies in Mathematics and its Applications, vol. 5 (North-Holland Publishing Co., Amsterdam/New York, 1978)
V.T. Chu, V.H. Hoang, High-dimensional finite elements for multiscale Maxwell-type equations. IMA J. Numer. Anal. drx001 (2017, Online). doi:https://doi.org/10.1093/imanum/drx001
P.G. Ciarlet, The finite element method for elliptic problems, in Classics in Applied Mathematics, vol. 40 (SIAM, Philadelphia, 2002)
P. Ciarlet Jr., J. Zou, Fully discrete finite element approaches for time-dependent Maxwell’s equations. Numer. Math. 82(2), 193–219 (1999)
P. Ciarlet Jr., S. Fliss, C. Stohrer, On the approximation of electromagnetic fields by edge finite elements. Part 2: a heterogeneous multiscale method for Maxwell’s equations. Comput. Math. Appl. 73(9), 1900–1919 (2017)
L.C. Evans, Partial differential equations, Graduate Studies in Mathematics, vol. 19 (American Mathematical Society, Providence, 1998)
F. Hecht, New development in FreeFem++. J. Numer. Math. 20(3–4), 25–265 (2012)
P. Henning, M. Ohlberger, B. Verfürth, A new heterogeneous multiscale method for time-harmonic Maxwell’s equations. SIAM J. Numer. Anal. 54(6), 3493–3522 (2016)
V.V. Jikov, S.M. Kozlov, O.A. Oleinik, Homogenization of Differential Operators and Integral Functionals (Springer, Berlin, 1994)
J.L. Lions, E. Magenes, Non-homogeneous boundary value problems and applications. Vol. I, Die Grundlehren der mathematischen Wissenschaften, vol. 181 (Springer, New York/Heidelberg, 1972)
P.A. Markowich, F. Poupaud, The Maxwell equation in a periodic medium: Homogenization of the energy density. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 23(2), 301–324 (1996)
P. Monk, Analysis of a finite element method for Maxwell’s equations. SIAM J. Numer. Anal. 29(3), 714–729 (1992)
P. Monk, Finite Element Methods for Maxwell’s Equations (Oxford University Press, Oxford, 2003)
N. Wellander, Homogenization of the Maxwell equations: Case I. Linear theory. Appl. Math. 46(1), 29–51 (2001)
Acknowledgements
We gratefully acknowledge financial support by the Deutsche Forschungsgemeinschaft (DFG) through CRC 1173 and the Klaus Tschira Stiftung. In addition we thank the anonymous referee for helpful suggestions.
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Hochbruck, M., Stohrer, C. (2017). Finite Element Heterogeneous Multiscale Method for Time-Dependent Maxwell’s Equations. In: Bittencourt, M., Dumont, N., Hesthaven, J. (eds) Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2016. Lecture Notes in Computational Science and Engineering, vol 119. Springer, Cham. https://doi.org/10.1007/978-3-319-65870-4_18
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DOI: https://doi.org/10.1007/978-3-319-65870-4_18
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