A P-HIERARCHICAL ERROR ESTIMATOR FOR A FEM-BEM COUPLING OF AN EDDY CURRENT PROBLEM IN ℝ3 -DEDICATED TO PROFESSOR WOLFGANG L. WENDLAND ON THE OCCASION OF HIS 75TH BIRTHDAY

• Leydecker, Florian (INSTITUTE FOR APPLIED MATHEMATICS, LEIBNIZ UNIVERSITAT HANNOVER) ;
• Maischak, Matthias (BRUNEL UNIVERSITY) ;
• Stephan, Ernst P. (INSTITUTE FOR APPLIED MATHEMATICS, LEIBNIZ UNIVERSITAT HANNOVER) ;
• Teltscher, Matthias (INSTITUTE FOR APPLIED MATHEMATICS, LEIBNIZ UNIVERSITAT HANNOVER)
• Received : 2011.08.24
• Accepted : 2013.04.30
• Published : 2013.09.25

Abstract

We extend a p-hierarchical decomposition of the second degree finite element space of N$\acute{e}$d$\acute{e}$lec for tetrahedral meshes in three dimensions given in [1] to meshes with hexahedral elements, and derive p-hierarchical decompositions of the second degree finite element space of Raviart-Thomas in two dimensions for triangular and quadrilateral meshes. After having proved stability of these subspace decompositions and requiring certain saturation assumptions to hold, we construct a local a posteriori error estimator for fem and bem coupling of a time-harmonic electromagnetic eddy current problem in $\mathbb{R}^3$. We perform some numerical tests to underline reliability and efficiency of the estimator and test its usefulness in an adaptive refinement scheme.

References

1. B. Beck, R. Hiptmair and B.Wohlmuth, Hierarchical error estimator for eddy current computation, Numerical mathematics and advanced applications (Jyvaskyla, 1999), 110-120, World Sci. Publishing, River Edge, NJ, 2000.
2. R. C. MacCamy and E. P. Stephan, A boundary element method for an exterior problem for three-dimensional Maxwell's equations, Applicable Anal. 16 (1983), no. 2, 141-163. https://doi.org/10.1080/00036818308839466
3. R. C. MacCamy and E. P. Stephan, A skin effect approximation for eddy current problems, Arch. Rational Mech. Anal. 90 (1985), no. 1, 87-98. https://doi.org/10.1007/BF00281588
4. R. C. MacCamy and E. P. Stephan, A simple layer potential method for three-dimensional eddy current problems, Ordinary and partial differential equations (Dundee, 1982), 477-484, Lecture Notes in Math., 964, Springer-Verlag, 1982.
5. R. C. MacCamy and E. P. Stephan, Solution procedures for three-dimensional eddy current problems, J. Math. Anal. Appl. 101 (1984), no. 2, 348-379. https://doi.org/10.1016/0022-247X(84)90108-2
6. J.-C. Nedelec, Computation of eddy currents on a surface in $\mathbb{R}^{3}$ by finite element methods, SIAM J. Numer. Anal. 15 (1978), no. 3, 580-594. https://doi.org/10.1137/0715038
7. J.-C. Nedelec, Integral equations with non-integrable kernels, Integral Equations Operator Theory 5 (1982), 562 - 572. https://doi.org/10.1007/BF01694054
8. A. Bossavit, The computation of eddy-currents in dimension 3 by using mixed finite elements and boundary elements in association, Math. Comput. Modelling 15 (1991), 33-42.
9. M. Costabel and E. P. Stephan, Strongly elliptic boundary integral equations for electromagnetic transmission problems, Proc. Royal Soc. Edinburgh 109 A (1988), 271-296.
10. H. Ammari and J.-C. Nedelec, Couplage elements finis/equations integrales pour la resolution des equations de Maxwell en milieu heterogene, Equations aux derivees partielles et applications, 19-33, Gauthier-Villars, Ed. Sci. Med. Elsevier, Paris, 1998.
11. H. Ammari and J.-C. Nedelec, Coupling integral equations method and finite volume elements for the resolution of the Leontovich boundary value problem for the time-harmonic Maxwell equations in three-dimensional heterogeneous media, Mathematical aspects of boundary element methods (Palaiseau, 1998), 11-22, Chapman & Hall/CRC Res. Notes Math., 414, Chapman & Hall/CRC, Boca Raton, FL, 2000.
12. H. Ammari and J.-C. Nedelec, Coupling of finite and boundary element methods for the time-harmonic Maxwell equations. Part II: a symmetric formulation, Oper. Theory Adv. Appl. 110 (1999), Birkhauser Verlag, 23 -32.
13. V. Levillain, Couplage elements finis-equations integrales pour la resolution des equations de Maxwell en milieu heterogene, PhD-Thesis, Ecole Polytechnique, 1991.
14. R. Hiptmair, Symmetric Coupling for Eddy Current Problems, SIAM J. Numer. Anal., 40 (2002), no. 1, 41-65. https://doi.org/10.1137/S0036142900380467
15. R. Hiptmair, Coupling of finite elements and boundary elements in electromagnetic scattering, SIAM J. Numer. Anal. 41 (2003), no. 3, 919-944. https://doi.org/10.1137/S0036142901397757
16. H. Ammari, A. Buffa and J.-C. Nedelec, A justification of eddy currents model for the Maxwell equations, SIAM J. Appl. Math. 60 (2000), no. 5, 1805-1823. https://doi.org/10.1137/S0036139998348979
17. M. COSTABEL, Symmetric methods for the coupling of finite elements and boundary elements (invited contribution), in Boundary elements IX, Vol. 1 (Stuttgart, 1987), Comput. Mech., Southampton, 1987, pp. 411-420.
18. E. P. Stephan and M. Maischak, A posteriori error estimates for fem-bem couplings of three-dimensional electromagnetic problems, Comput. Methods Appl. Mech. Engrg., 194 (2005), pp. 441-452. https://doi.org/10.1016/j.cma.2004.03.017
19. J.-C. Nedelec, Mixed finite elements in $\mathbb{R}^{3}$, Numer. Math. 35 (1980), 315-341. https://doi.org/10.1007/BF01396415
20. F. Leydecker and M. Maischak and E.P. Stephan and M. Teltscher, Adaptive FE-BE coupling for an electromagnetic problem in $\mathbb{R}^{3}$ - a residual error estimator, Math. Methods appl. Sci. 33 (2010), no. 18, 2162-2186. https://doi.org/10.1002/mma.1389
21. R. Bank, Hierarchical Bases and the Finite Element Method, Acta Numer. 5 (1996), 1 - 43. https://doi.org/10.1017/S0962492900002610
22. R. Verfurth, A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques, Wiley-Teubner, 1996.
23. P. Mund and E. P. Stephan, An adaptive two-level method for the coupling of nonlinear FEM-BEM equations, SIAM J. Numer. Anal. 36 (1999), no. 4, 1001-1021. https://doi.org/10.1137/S0036142997316499
24. F. Leydecker and M. Maischak and E.P. Stephan and M. Teltscher, A p-hierarchical error estimator for a fe-be coupling formulation applied to electromagnetic scattering problems in $\mathbb{R}^{3}$, Appl. Anal. 91 (2012), no. 2, 277-293. https://doi.org/10.1080/00036811.2011.614605
25. U. Brink and E. P. Stephan, Implicit residual error estimators for the coupling of finite elements and boundary elements, Math. Meth. Appl. Sci. 22 (1999), 923-936. https://doi.org/10.1002/(SICI)1099-1476(19990725)22:11<923::AID-MMA27>3.0.CO;2-Y
26. C. Carstensen, A posteriori error estimate for the symmetric coupling of finite elements and boundary elements, Computing 57 (1996), no. 4, 301-322. https://doi.org/10.1007/BF02252251
27. C. Carstensen, S. A. Funken and E. P. Stephan, On the adaptive coupling of FEM and BEM in 2-d-elasticity, Numer. Math. 77 (1997), 187-221. https://doi.org/10.1007/s002110050283
28. C. Carstensen and E. P. Stephan, Adaptive coupling of boundary elements and finite elements, RAIRO Modelisation Math. Anal. Numer. 29 (1995), 779-817. https://doi.org/10.1051/m2an/1995290707791
29. B. Beck, P. Deuflhard, R. Hiptmair, R. H. W. Hoppe and B. Wohlmuth, Adaptive multilevel methods for edge element discretizations of Maxwell's equations, Surveys Math. Indust. 8 (1999), no. 3-4, 271-312.
30. B. Beck, R. Hiptmair, R. H.W. Hoppe and B.Wohlmuth, Residual based a posteriori error estimator for eddy current computation, M2AN Math. Model. Numer. Anal. 34 (2000), no. 1, 159-182. https://doi.org/10.1051/m2an:2000136
31. R. H. W. Hoppe and B. I. Wohlmuth, Hierarchical basis error estimators for Raviart-Thomas discretizations of arbitrary order, Finite Element Methods: Superconvergence, Post-processing and A Posteriori Estimates, P. et al., ed., Marcel Dekker, New York (1997), 155-167.
32. R. H. W. Hoppe and B. I. Wohlmuth, A comparison of a posteriori error estimators of mixed finite element discretizations by Raviart-Thomas elements, Math. Comp. 68 (1999), no. 228, 1347-1378. https://doi.org/10.1090/S0025-5718-99-01125-4
33. P. Grisvard, Elliptic Problems in Nonsmooth Domains, Pitman Publishing, Boston-London, 1985.
34. A. Buffa and P. Ciarlet, On traces for functional spaces related to Maxwell's equations. Part I: An integration by parts formula in Lipschitz polyhedra, Math. Methods Appl. Sci. 24 (2001), no. 1, 9-30. https://doi.org/10.1002/1099-1476(20010110)24:1<9::AID-MMA191>3.0.CO;2-2
35. A. Buffa and P. Ciarlet, On traces for functional spaces related to Maxwell's equations. Part II: Hodge decompositions on the boundary of Lipschitz polyhedra an applications, Math. Methods Appl. Sci. 24 (2001), no. 1, 31-48. https://doi.org/10.1002/1099-1476(20010110)24:1<31::AID-MMA193>3.0.CO;2-X
36. R. Hiptmair, Canonical Construction of finite elements, Math. Comp. 68 (1999), no. 228, 1325-1346. https://doi.org/10.1090/S0025-5718-99-01166-7
37. R. Hiptmair, Multigrid method for Maxwell's equation, SIAM J. Numer. Anal. 36 (1998), no. 1, 204-225. https://doi.org/10.1137/S0036142997326203
38. F. Brezzi and M. Fortin, Mixed And Hybrid Finite Element Methods, Springer Series in Computational Mathematics, Band 15, Springer-Verlag, 1991.
39. L. R. Scott and S. Zhang, Finite element interpolation of nonsmooth functions satisfying boundary conditions, Math. Comp. 54 (1990), no. 190, 483-493. https://doi.org/10.1090/S0025-5718-1990-1011446-7
40. A. Alonso and A. Valli, An optimal domain decomposition preconditioner for low-frequency time-harmonic Maxwell equations, Math. Comp., 68 (1999), pp. 607-631. https://doi.org/10.1090/S0025-5718-99-01013-3
41. M. Maischak, Webpage of the software package maiprogs, www.ifam.uni-hannover.de/-maiprogs.