|
Implicit a posteriori error estimates for the Maxwell equations
Authors:
Ferenc Izsák, Davit Harutyunyan and Jaap J.W. van der Vegt
Journal:
Math. Comp. 77 (2008), 1355-1386
MSC (2000):
Primary 65N15, 65N30, 65R20
Posted:
February 20, 2008
MathSciNet review:
2398772
Full-text PDF
Abstract |
References |
Similar Articles |
Additional Information
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
- 1.
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
(2003b:65001)
- 2.
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
(99e:35037), http://dx.doi.org/10.1002/(SICI)1099-1476(199806)21:9<823::AID-MMA976>3.0.CO;2-B
- 3.
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
(2002k:65001)
- 4.
Wolfgang
Bangerth and Rolf
Rannacher, Adaptive finite element methods for differential
equations, Lectures in Mathematics ETH Zürich, Birkhäuser
Verlag, Basel, 2003. MR 1960405
(2004b:65002)
- 5.
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
(95f:65212), http://dx.doi.org/10.1137/0730048
- 6.
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
(2000k:65203), http://dx.doi.org/10.1051/m2an:2000136
- 7.
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
- 8.
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
(2003a:65103)
- 9.
F.
Brezzi, L.
P. Franca, T.
J. R. Hughes, and A.
Russo, 𝑏=∫𝑔, Comput. Methods Appl. Mech.
Engrg. 145 (1997), no. 3-4, 329–339. MR 1456019
(98g:65086), http://dx.doi.org/10.1016/S0045-7825(96)01221-2
- 10.
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
(2002b:78025), http://dx.doi.org/10.1002/1099-1476(20010110)24:1<9::AID-MMA191>3.0.CO;2-2
- 11.
A.
Buffa, M.
Costabel, and D.
Sheen, On traces for
𝐻(𝑐𝑢𝑟𝑙,Ω) in Lipschitz
domains, J. Math. Anal. Appl. 276 (2002), no. 2,
845–867. MR 1944792
(2004i:35045), http://dx.doi.org/10.1016/S0022-247X(02)00455-9
- 12.
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
(2004f:78039), http://dx.doi.org/10.1002/nme.713
- 13.
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
(91h:00004a)
- 14.
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, http://dx.doi.org/10.1016/S0922-5382(98)80029-9
- 15.
L. Demkowicz.
Finite element methods for Maxwell equations.
In Encyclopedia of Computational Mechanics, pages 723-737. Jonh Wiley & Sons, Chicester, 2004.
- 16.
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
(99b:78003), http://dx.doi.org/10.1016/S0045-7825(97)00184-9
- 17.
Alexandre
Ern and Jean-Luc
Guermond, Theory and practice of finite elements, Applied
Mathematical Sciences, vol. 159, Springer-Verlag, New York, 2004. MR 2050138
(2005d:65002)
- 18.
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
(2005d:65190), http://dx.doi.org/10.1017/S096249290200003X
- 19.
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/.
- 20.
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 (95h:73022)
- 21.
Peter
Monk, Finite element methods for Maxwell’s equations,
Numerical Mathematics and Scientific Computation, Oxford University Press,
New York, 2003. MR 2059447
(2005d:65003)
- 22.
J.-C.
Nédélec, Mixed finite elements in
𝑅³, Numer. Math. 35 (1980), no. 3,
315–341. MR
592160 (81k:65125), http://dx.doi.org/10.1007/BF01396415
- 23.
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
(2004j:65133), http://dx.doi.org/10.1007/s10092-003-0077-y
- 24.
W. Rachowicz and L. Demkowicz.
An  -adaptive finite element method for electromagnetics. part I: data structure and constrained approximation.
Comput. Methods Appl. Mech. Engrg., 187(1-2):307-337, 2000.
- 25.
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
(2002j:78030), http://dx.doi.org/10.1002/nme.396
- 26.
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
(2005k:78010), http://dx.doi.org/10.1002/nme.1227
- 27.
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
(electronic). MR
1860960 (2002h:78036)
- 28.
R. Verfürth.
A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques.
Advances in Numerical Mathematics. Wiley - Teubner, Chichester - Stuttgart, 1996.
- 29.
A. Zdunek and W. Rachowicz.
Application of the  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.
Similar Articles
Retrieve articles in Mathematics of Computation
with MSC (2000):
65N15,
65N30,
65R20
Retrieve articles in all journals
with MSC (2000):
65N15,
65N30,
65R20
Additional Information
Ferenc Izsák
Affiliation:
Department of Applied Mathematics, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands
Address at time of publication:
ELTE TTK, Department of Applied Analysis and Computational Mathematics, P.O. Box 120, 1518 Budapest, Hungary
Email:
izsakf@cs.elte.hu
Davit Harutyunyan
Affiliation:
Department of Applied Mathematics, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands
Address at time of publication:
Department of Mathematics and Computer Science, Technische Universiteit Eindhoven, P.O. Box 513, 5600 MB Eindhoven, The Netherlands
Email:
d.harutyunyan@tue.nl
Jaap J.W. van der Vegt
Affiliation:
Department of Applied Mathematics, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands
Email:
j.j.w.vandervegt@math.utwente.nl
DOI:
http://dx.doi.org/10.1090/S0025-5718-08-02046-2
PII:
S 0025-5718(08)02046-2
Keywords:
Maxwell equations,
a posteriori error estimates,
implicit estimates,
N\'ed\'elec edge finite elements.
Received by editor(s):
February 1, 2006
Received by editor(s) in revised form:
February 24, 2007
Posted:
February 20, 2008
Additional Notes:
This research was supported by the Dutch government through the national program BSIK: knowledge and research capacity, in the ICT project BRICKS (http://www.bsik-bricks.nl), theme MSV1.
Article copyright:
© Copyright 2008 American Mathematical Society
|
