| Title:
|
Numerical homogenization for indefinite H(curl)-problems (English) |
| Author:
|
Verfürth, Barbara |
| Language:
|
English |
| Journal:
|
Proceedings of Equadiff 14 |
| Volume:
|
Conference on Differential Equations and Their Applications, Bratislava, July 24-28, 2017 |
| Issue:
|
2017 |
| Year:
|
|
| Pages:
|
137-146 |
| . |
| Category:
|
math |
| . |
| Summary:
|
In this paper, we present a numerical homogenization scheme for indefinite, timeharmonic Maxwell’s equations involving potentially rough (rapidly oscillating) coefficients. The method involves an H(curl)-stable, quasi-local operator, which allows for a correction of coarse finite element functions such that order optimal (w.r.t. the mesh size) error estimates are obtained. To that end, we extend the procedure of [D. Gallistl, P. Henning, B. Verfürth, Numerical homogenization for H(curl)-problems, arXiv:1706.02966, 2017] to the case of indefinite problems. In particular, this requires a careful analysis of the well-posedness of the corrector problems as well as the numerical homogenization scheme. (English) |
| Keyword:
|
Multiscale method, wave propagation, Maxwell’s equations, finite element method |
| MSC:
|
35Q61 |
| MSC:
|
65N12 |
| MSC:
|
65N15 |
| MSC:
|
65N30 |
| MSC:
|
78M10 |
| . |
| Date available:
|
2019-09-27T07:48:44Z |
| Last updated:
|
2019-09-27 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/703038 |
| . |
| Reference:
|
[1] Abdulle, A., Henning, P.: Localized orthogonal decomposition method for the wave equation with a continuum of scales., Math. Comp., 86 (2017), pp. 549–587. MR 3584540, 10.1090/mcom/3114 |
| Reference:
|
[2] Babuška, I. M., Sauter, S. A.: Is the pollution effect avoidable for the Helmholtz equation considering high wave numbers?., SIAM Rev., 42 (2000), pp. 451–484. MR 1786934 |
| Reference:
|
[3] Jr., P. Ciarlet, Fliss, S., HASH(0x2ad2750), Stohrer, C.: On the approximation of electromagnetic fields by edge finite elements.. Part 2: A heterogeneous multiscale method for Maxwell’s equations, Comput. Math. Appl., 73 (2017), pp. 1900–1919. MR 3634959, 10.1016/j.camwa.2017.02.043 |
| Reference:
|
[4] Falk, R. S., Winther, R.: Local bounded cochain projections., Math. Comp., 83 (2014), pp. 2631–2656. MR 3246803, 10.1090/S0025-5718-2014-02827-5 |
| Reference:
|
[5] Gallistl, D., Henning, P., HASH(0x2ad4f60), Verf\"urth, B.: Numerical homogenization of H(curl)-problems., arXiv:1706.02966 (2017), preprint. MR 3810505 |
| Reference:
|
[6] Gallistl, D., Peterseim, D.: Stable multiscale Petrov-Galerkin finite element method for high frequency acoustic scattering., Comp. Appl. Mech. Eng., 295 (2015), pp. 1–17. MR 3388822, 10.1016/j.cma.2015.06.017 |
| Reference:
|
[7] Hellman, F., P.Henning, HASH(0x2ad7fe0), M{\aa}lqvist, A.: Multiscale mixed finite elements., Discr. Contin. Dyn. Syst. Ser. S, 9 (2016), pp. 1269–1298. MR 3591945, 10.3934/dcdss.2016051 |
| Reference:
|
[8] Henning, P., M{\aa}lqvist, A.: Localized orthogonal decomposition techniques for boundary value problems., SIAM J. Sci. Comput., 36 (2014), pp. A1609–A1634. MR 3240855, 10.1137/130933198 |
| Reference:
|
[9] Henning, P., Ohlberger, M., HASH(0x2adb738), Verf\"urth, B.: A new Heterogeneous Multiscale Method for time-harmonic Maxwell’s equations., SIAM J. Numer. Anal., 54 (2016), pp. 3493–3522. MR 3578028, 10.1137/15M1039225 |
| Reference:
|
[10] Henning, P., Ohlberger, M., HASH(0x2adc158), Verf\"urth, B.: Analysis of multiscale methods for time harmonic Maxwell’s equations., Pro. Appl. Math. Mech., 16 (2016), pp. 559–560. MR 3578028, 10.1002/pamm.201610268 |
| Reference:
|
[11] Hiptmair, R.: Maxwell equations: continuous and discrete., in Computational Electromagnetism, A. Bermúdez de Castro and A. Valli, eds., Lecture Notes in Mathematics, Springer, Cham, 2015, pp. 1–58. MR 3382059 |
| Reference:
|
[12] M{\aa}lqvist, A., Peterseim, D.: Localization of elliptic multiscale problems., Math. Comp., 83 (2014), pp. 2583–2603. MR 3246801, 10.1090/S0025-5718-2014-02868-8 |
| Reference:
|
[13] Moiola, A.: Trefftz-Discontinuous Galerkin methods for time-harmonic wave problems., PhD thesis, ETH Z\"urich, 2011. |
| Reference:
|
[14] Monk, P.: Finite element methods for Maxwell’s equation., Numerical Mathematics and Scientific Computation, Oxford University Press, New York, 2003. MR 2059447 |
| Reference:
|
[15] Ohlberger, M., Verf\"urth, B.: Localized Orthogonal Decomposition for two-scale Helmholtz-type problems., AIMS Mathematics, 2 (2017), pp. 458–478. 10.3934/Math.2017.2.458 |
| Reference:
|
[16] Peterseim, D.: Eliminating the pollution effect by local subscale correction., Math. Comp., 86(2017), pp. 1005–1036. MR 3614010, 10.1090/mcom/3156 |
| Reference:
|
[17] Wellander, N., Kristensson, G.: Homogenization of the Maxwell equations at fixed frequency., AIAM J. Appl. Math., 64 (2003), pp. 170–195. MR 2029130, 10.1137/S0036139902403366 |
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