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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
Published electronically: February 20, 2008
MathSciNet review: 2398772
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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.

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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

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

Jaap J.W. van der Vegt
Affiliation: Department of Applied Mathematics, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands

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
Published electronically: 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 (, theme MSV1.
Article copyright: © Copyright 2008 American Mathematical Society

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