The ℎ𝑝-local discontinuous Galerkin method for low-frequency time-harmonic Maxwell equations

Perugia, Ilaria; Schötzau, Dominik

doi:10.1090/S0025-5718-02-01471-0

			Journals Home Search Author Info Subscribe Tech Support My Subscriptions Browser Compatibility Info Help
ISSN 1088-6842(online) ISSN 0025-5718(print)

The -local discontinuous Galerkin method for low-frequency time-harmonic Maxwell equations

Authors: Ilaria Perugia and Dominik Schötzau
Journal: Math. Comp. 72 (2003), 1179-1214
MSC (2000): Primary 65N30
Posted: October 18, 2002
MathSciNet review: 1972732
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The local discontinuous Galerkin method for the numerical approximation of the time-harmonic Maxwell equations in a low-frequency regime is introduced and analyzed. Topologically nontrivial domains and heterogeneous media are considered, containing both conducting and insulating materials. The presented method involves discontinuous Galerkin discretizations of the curl-curl and grad-div operators, derived by introducing suitable auxiliary variables and so-called numerical fluxes. An -analysis is carried out and error estimates that are optimal in the meshsize and slightly suboptimal in the approximation degree are obtained.

1. Ana Alonso, A mathematical justification of the low-frequency heterogeneous time-harmonic Maxwell equations, Math. Models Methods Appl. Sci. 9 (1999), no. 3, 475–489. MR 1686609 (2000c:78002), http://dx.doi.org/10.1142/S0218202599000245
2. Ana Alonso and Alberto Valli, A domain decomposition approach for heterogeneous time-harmonic Maxwell equations, Comput. Methods Appl. Mech. Engrg. 143 (1997), no. 1-2, 97–112. MR 1442391 (98b:78020), http://dx.doi.org/10.1016/S0045-7825(96)01144-9
3. Ana Alonso and Alberto Valli, An optimal domain decomposition preconditioner for low-frequency time-harmonic Maxwell equations, Math. Comp. 68 (1999), no. 226, 607–631. MR 1609607 (99i:78002), http://dx.doi.org/10.1090/S0025-5718-99-01013-3
4. 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
5. Douglas N. Arnold, An interior penalty finite element method with discontinuous elements, SIAM J. Numer. Anal. 19 (1982), no. 4, 742–760. MR 664882 (83f:65173), http://dx.doi.org/10.1137/0719052
6. D.N. Arnold, F. Brezzi, B. Cockburn, and L.D. Marini, Unified analysis of discontinuous Galerkin methods for elliptic problems, SIAM J. Numer. Anal. 39 (2001/02), 1749-1779. CMP 2002:09
7. I. Babuška and Manil Suri, The ℎ-𝑝 version of the finite element method with quasi-uniform meshes, RAIRO Modél. Math. Anal. Numér. 21 (1987), no. 2, 199–238 (English, with French summary). MR 896241 (88d:65154)
8. Carlos Erik Baumann and J. Tinsley Oden, A discontinuous ℎ𝑝 finite element method for convection-diffusion problems, Comput. Methods Appl. Mech. Engrg. 175 (1999), no. 3-4, 311–341. MR 1702201 (2000d:65171), http://dx.doi.org/10.1016/S0045-7825(98)00359-4
9. Anne-Sophie Bonnet-Ben Dhia, Christophe Hazard, and Stephanie Lohrengel, A singular field method for the solution of Maxwell’s equations in polyhedral domains, SIAM J. Appl. Math. 59 (1999), no. 6, 2028–2044 (electronic). MR 1709795 (2000g:78032), http://dx.doi.org/10.1137/S0036139997323383
10. A. Bossavit, A rationale for edge-elements in 3D fields computation, IEEE Trans. on Magnetics 24 (1988), 74-79.
11. Alain Bossavit, Computational electromagnetism, Electromagnetism, Academic Press Inc., San Diego, CA, 1998. Variational formulations, complementarity, edge elements. MR 1488417 (99m:78001)
12. A. Bossavit, “Hybrid” electric-magnetic methods in eddy-current problems, Comput. Methods Appl. Mech. Engrg. 178 (1999), no. 3-4, 383–391. MR 1711045 (2000h:78006), http://dx.doi.org/10.1016/S0045-7825(99)00027-4
13. C.F. Bryant, C.R.I. Emson, P. Fernandes, and C.W. Trowbridge, Lorentz gauge eddy current formulations for multiply connected piece-wise homogeneous conductors, COMPEL 17 (1998), 732-740.
14. A. Buffa, Hodge decomposition on the boundary of a polyhedron: the nonsimply connected case, Math. Models Methods Appl. Sci. 11 (2001), pp. 1491-1504.
15. A. Buffa and P. Ciarlet Jr., On traces for functional spaces related to Maxwell’s equations. I. An integration by parts formula in Lipschitz polyhedra, Math. Methods Appl. Sci. 24 (2001), no. 1, 9–30. MR 1809491 (2002b:78024), http://dx.doi.org/10.1002/1099-1476(20010110)24:1<9::AID-MMA191>3.0.CO;2-2
16. 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
17. P. Castillo, Local discontinuous Galerkin methods for convection-diffusion and elliptic problems, Ph.D. thesis, University of Minnesota, Minneapolis, MN, 2001.
18. P. Castillo, Performance of discontinuous Galerkin methods for elliptic partial differential equations, Tech. Report 1764, IMA, University of Minnesota, 2001.
19. P. Castillo, B. Cockburn, I. Perugia, and D. Schötzau, An a priori error analysis of the local discontinuous Galerkin method for elliptic problems, SIAM J. Numer. Anal. 38 (2000), 1676-1706. CMP 2001:08
20. P. Castillo, B. Cockburn, D. Schötzau, and C. Schwab, Optimal a priori error estimates for the -version of the local discontinuous Galerkin method for convection-diffusion problems, Math. Comp. 71 (2002), 455-478. CMP 2002:09
21. Zhiming Chen, Qiang Du, and Jun Zou, Finite element methods with matching and nonmatching meshes for Maxwell equations with discontinuous coefficients, SIAM J. Numer. Anal. 37 (2000), no. 5, 1542–1570. MR 1759906 (2001h:78044), http://dx.doi.org/10.1137/S0036142998349977
22. Philippe G. Ciarlet, The finite element method for elliptic problems, North-Holland Publishing Co., Amsterdam, 1978. Studies in Mathematics and its Applications, Vol. 4. MR 0520174 (58 #25001)
23. Bernardo Cockburn and Clint Dawson, Some extensions of the local discontinuous Galerkin method for convection-diffusion equations in multidimensions, The mathematics of finite elements and applications, X, MAFELAP 1999 (Uxbridge), Elsevier, Oxford, 2000, pp. 225–238. MR 1801979 (2001j:65142), http://dx.doi.org/10.1016/B978-008043568-8/50014-6
24. B. Cockburn, G. Kanschat, I. Perugia, and D. Schötzau, Superconvergence of the local discontinuous Galerkin method for elliptic problems on Cartesian grids, SIAM J. Numer. Anal. 39 (2001), 264-285.
25. Bernardo Cockburn and Chi-Wang Shu, The local discontinuous Galerkin method for time-dependent convection-diffusion systems, SIAM J. Numer. Anal. 35 (1998), no. 6, 2440–2463 (electronic). MR 1655854 (99j:65163), http://dx.doi.org/10.1137/S0036142997316712
26. -, Runge-Kutta discontinuous Galerkin methods for convection-dominated problems, J. Sci. Comput. 16 (2001), 173-261.
27. Martin Costabel, Monique Dauge, and Serge Nicaise, Singularities of Maxwell interface problems, M2AN Math. Model. Numer. Anal. 33 (1999), no. 3, 627–649. MR 1713241 (2001g:78005), http://dx.doi.org/10.1051/m2an:1999155
28. M. Dauge, M. Costabel, and D. Martin, Weighted regularization of Maxwell equations in polyhedral domains, Preprint, University of Rennes, 2001.
29. 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
30. Paolo Fernandes and Gianni Gilardi, Magnetostatic and electrostatic problems in inhomogeneous anisotropic media with irregular boundary and mixed boundary conditions, Math. Models Methods Appl. Sci. 7 (1997), no. 7, 957–991. MR 1479578 (98h:78006), http://dx.doi.org/10.1142/S0218202597000487
31. P. Fernandes and I. Perugia, Vector potential formulation for magnetostatics and modelling of permanent magnets, IMA J. Appl. Math. 66 (2001), no. 3, 293–318. MR 1852930 (2002e:78003), http://dx.doi.org/10.1093/imamat/66.3.293
32. E.H. Georgoulis and E. Süli, -DGFEM on shape-irregular meshes: reaction-diffusion, Tech. Report NA 01-09, Oxford University Computing Laboratory, 2001.
33. R. Hiptmair, Symmetric coupling for eddy-current problems, Tech. Report 148, SFB382, Universität Tübingen, 2000.
34. P. Houston, C. Schwab, and E. Süli, Discontinuous finite element methods for advection-diffusion problems, SIAM J. Numer. Anal., to appear.
35. J.-L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications. Vol. 1, Travaux et Recherches Mathématiques, No. 17, Dunod, Paris, 1968 (French). MR 0247243 (40 #512)
J.-L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications. Vol. 2, Travaux et Recherches Mathématiques, No. 18, Dunod, Paris, 1968 (French). MR 0247244 (40 #513)
36. Peter Monk, A finite element method for approximating the time-harmonic Maxwell equations, Numer. Math. 63 (1992), no. 2, 243–261. MR 1182977 (94b:65134), http://dx.doi.org/10.1007/BF01385860
37. 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
38. J.-C. Nédélec, Éléments finis mixtes incompressibles pour l’équation de Stokes dans 𝑅³, Numer. Math. 39 (1982), no. 1, 97–112 (French, with English summary). MR 664539 (83g:65111), http://dx.doi.org/10.1007/BF01399314
39. J. Tinsley Oden, Ivo Babuška, and Carlos Erik Baumann, A discontinuous ℎ𝑝 finite element method for diffusion problems, J. Comput. Phys. 146 (1998), no. 2, 491–519. MR 1654911 (99m:65173), http://dx.doi.org/10.1006/jcph.1998.6032
40. I. Perugia and D. Schötzau, On the coupling of local discontinuous Galerkin and conforming finite element methods, J. Sci. Comp. (2001), no. 16, 411-433.
41. S. Prudhomme, F. Pascal, J.T. Oden, and A. Romkes, Review of a priori error estimation for discontinuous Galerkin methods, Tech. Report 2000-27, TICAM, University of Texas at Austin, 2000.
42. Béatrice Rivière and Mary F. Wheeler, A discontinuous Galerkin method applied to nonlinear parabolic equations, Discontinuous Galerkin methods (Newport, RI, 1999) Lect. Notes Comput. Sci. Eng., vol. 11, Springer, Berlin, 2000, pp. 231–244. MR 1842177 (2002d:65105), http://dx.doi.org/10.1007/978-3-642-59721-3_17
43. Béatrice Rivière, Mary F. Wheeler, and Vivette Girault, Improved energy estimates for interior penalty, constrained and discontinuous Galerkin methods for elliptic problems. I, Comput. Geosci. 3 (1999), no. 3-4, 337–360 (2000). MR 1750076 (2001d:65145), http://dx.doi.org/10.1023/A:1011591328604
44. Ch. Schwab, 𝑝- and ℎ𝑝-finite element methods, Numerical Mathematics and Scientific Computation, The Clarendon Press Oxford University Press, New York, 1998. Theory and applications in solid and fluid mechanics. MR 1695813 (2000d:65003)
45. 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

	Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
\|