Implicit a posteriori error estimates for the Maxwell equations

Izsák, Ferenc; Harutyunyan, Davit; van der Vegt, Jaap

doi:10.1090/S0025-5718-08-02046-2

Implicit a posteriori error estimates for the Maxwell equations
HTML articles powered by AMS MathViewer

by Ferenc Izsák, Davit Harutyunyan and Jaap J.W. van der Vegt PDF

Math. Comp. 77 (2008), 1355-1386 Request permission

Abstract:

An implicit a posteriori error estimation technique is presented and analyzed for the numerical solution of the time-harmonic Maxwell equations using Nédélec edge elements. For this purpose we define a weak formulation for the error on each element and provide an efficient and accurate numerical solution technique to solve the error equations locally. We investigate the well-posedness of the error equations and also consider the related eigenvalue problem for cubic elements. Numerical results for both smooth and non-smooth problems, including a problem with reentrant corners, show that an accurate prediction is obtained for the local error, and in particular the error distribution, which provides essential information to control an adaptation process. The error estimation technique is also compared with existing methods and provides significantly sharper estimates for a number of reported test cases.

References

Mark Ainsworth and J. Tinsley Oden, A posteriori error estimation in finite element analysis, Pure and Applied Mathematics (New York), Wiley-Interscience [John Wiley & Sons], New York, 2000. MR 1885308, DOI 10.1002/9781118032824
C. Amrouche, C. Bernardi, M. Dauge, and V. Girault, Vector potentials in three-dimensional non-smooth domains, Math. Methods Appl. Sci. 21 (1998), no. 9, 823–864 (English, with English and French summaries). MR 1626990, DOI 10.1002/(SICI)1099-1476(199806)21:9<823::AID-MMA976>3.0.CO;2-B
Ivo Babuška and Theofanis Strouboulis, The finite element method and its reliability, Numerical Mathematics and Scientific Computation, The Clarendon Press, Oxford University Press, New York, 2001. MR 1857191
Wolfgang Bangerth and Rolf Rannacher, Adaptive finite element methods for differential equations, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2003. MR 1960405, DOI 10.1007/978-3-0348-7605-6
Randolph E. Bank and R. Kent Smith, A posteriori error estimates based on hierarchical bases, SIAM J. Numer. Anal. 30 (1993), no. 4, 921–935. MR 1231320, DOI 10.1137/0730048
Rudi Beck, Ralf Hiptmair, Ronald H. W. Hoppe, and Barbara Wohlmuth, Residual based a posteriori error estimators for eddy current computation, M2AN Math. Model. Numer. Anal. 34 (2000), no. 1, 159–182 (English, with English and French summaries). MR 1735971, DOI 10.1051/m2an:2000136
R. Beck, R. Hiptmair, and B. Wohlmuth, Hierarchical error estimator for eddy current computation, Numerical mathematics and advanced applications (Jyväskylä, 1999) World Sci. Publ., River Edge, NJ, 2000, pp. 110–120. MR 1936173
Susanne C. Brenner and L. Ridgway Scott, The mathematical theory of finite element methods, 2nd ed., Texts in Applied Mathematics, vol. 15, Springer-Verlag, New York, 2002. MR 1894376, DOI 10.1007/978-1-4757-3658-8
F. Brezzi, L. P. Franca, T. J. R. Hughes, and A. Russo, $b=\int g$, Comput. Methods Appl. Mech. Engrg. 145 (1997), no. 3-4, 329–339. MR 1456019, DOI 10.1016/S0045-7825(96)01221-2
A. Buffa and P. Ciarlet Jr., On traces for functional spaces related to Maxwell’s equations. II. Hodge decompositions on the boundary of Lipschitz polyhedra and applications, Math. Methods Appl. Sci. 24 (2001), no. 1, 31–48. MR 1809492, DOI 10.1002/1099-1476(20010110)24:1<9::AID-MMA191>3.0.CO;2-2
A. Buffa, M. Costabel, and D. Sheen, On traces for $\textbf {H}(\textbf {curl},\Omega )$ in Lipschitz domains, J. Math. Anal. Appl. 276 (2002), no. 2, 845–867. MR 1944792, DOI 10.1016/S0022-247X(02)00455-9
W. Cecot, W. Rachowicz, and L. Demkowicz, An $hp$-adaptive finite element method for electromagnetics. III. A three-dimensional infinite element for Maxwell’s equations, Internat. J. Numer. Methods Engrg. 57 (2003), no. 7, 899–921. MR 1984585, DOI 10.1002/nme.713
Robert Dautray and Jacques-Louis Lions, Mathematical analysis and numerical methods for science and technology. Vol. 3, Springer-Verlag, Berlin, 1990. Spectral theory and applications; With the collaboration of Michel Artola and Michel Cessenat; Translated from the French by John C. Amson. MR 1064315
L. Demkowicz, A posteriori error analysis for steady-state Maxwell’s equations, Advances in adaptive computational methods in mechanics (Cachan, 1997) Stud. Appl. Mech., vol. 47, Elsevier Sci. B. V., Amsterdam, 1998, pp. 513–526. MR 1644507, DOI 10.1016/S0922-5382(98)80029-9
L. Demkowicz. Finite element methods for Maxwell equations. In Encyclopedia of Computational Mechanics, pages 723–737. Jonh Wiley & Sons, Chicester, 2004.
L. Demkowicz and L. Vardapetyan, Modeling of electromagnetic absorption/scattering problems using $hp$-adaptive finite elements, Comput. Methods Appl. Mech. Engrg. 152 (1998), no. 1-2, 103–124. Symposium on Advances in Computational Mechanics, Vol. 5 (Austin, TX, 1997). MR 1602771, DOI 10.1016/S0045-7825(97)00184-9
Alexandre Ern and Jean-Luc Guermond, Theory and practice of finite elements, Applied Mathematical Sciences, vol. 159, Springer-Verlag, New York, 2004. MR 2050138, DOI 10.1007/978-1-4757-4355-5
Michael B. Giles and Endre Süli, Adjoint methods for PDEs: a posteriori error analysis and postprocessing by duality, Acta Numer. 11 (2002), 145–236. MR 2009374, DOI 10.1017/S096249290200003X
F. Izsák, D. Harutyunyan, and J. J. W. van der Vegt. Implicit a posteriori error estimates for the Maxwell equations. Internal report, Dept. of Appl. Math., University of Twente, Netherlands, 2007. http://eprints.eemcs.utwente.nl/8448/.
Jerrold E. Marsden and Thomas J. R. Hughes, Mathematical foundations of elasticity, Dover Publications, Inc., New York, 1994. Corrected reprint of the 1983 original. MR 1262126
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
S. Nicaise and E. Creusé, A posteriori error estimation for the heterogeneous Maxwell equations on isotropic and anisotropic meshes, Calcolo 40 (2003), no. 4, 249–271. MR 2025712, DOI 10.1007/s10092-003-0077-y
W. Rachowicz and L. Demkowicz. An $hp$-adaptive finite element method for electromagnetics. part I: data structure and constrained approximation. Comput. Methods Appl. Mech. Engrg., 187(1-2):307–337, 2000.
W. Rachowicz and L. Demkowicz, An $hp$-adaptive finite element method for electromagnetics. II. A 3D implementation, Internat. J. Numer. Methods Engrg. 53 (2002), no. 1, 147–180. $p$ and $hp$ finite element methods: mathematics and engineering practice (St. Louis, MO, 2000). MR 1870737, DOI 10.1002/nme.396
W. Rachowicz and A. Zdunek, An $hp$-adaptive finite element method for scattering problems in computational electromagnetics, Internat. J. Numer. Methods Engrg. 62 (2005), no. 9, 1226–1249. MR 2120293, DOI 10.1002/nme.1227
Garry Rodrigue and Daniel White, A vector finite element time-domain method for solving Maxwell’s equations on unstructured hexahedral grids, SIAM J. Sci. Comput. 23 (2001), no. 3, 683–706. MR 1860960, DOI 10.1137/S1064827598343826
R. Verfürth. A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques. Advances in Numerical Mathematics. Wiley - Teubner, Chichester - Stuttgart, 1996.
A. Zdunek and W. Rachowicz. Application of the $hp$ adaptive finite element method to the three-dimensional scattering of time-harmonic electromagnetic waves. In Proceedings of the European Conference on Computational Mechanics (Cracow, Poland, 2001), 2001.