Superconvergence analysis for Maxwell's equations in dispersive media

Mathematics of Computation

Volumn 77, Issue 262, 2008, Pages 757-771

  Lin, Qun ^a   Li, Jichun ^b  

^a CHINESE ACADEMY OF SCIENCES   (China)

^b UNIVERSITY OF NEVADA   (United States)

Author keywords

Dispersive media; Maxwell's equations; Superconvergence analysis

Indexed keywords

EID: 42349109473     PISSN: 00255718     EISSN: None     Source Type: Journal    
DOI: 10.1090/S0025-5718-07-02039-X     Document Type: Article

References (40)

1. M. Ainsworth, P. Davies, D. Duncan, P. Martin and B. Rynne, Topics in computational wave propagation, Lecture Notes in Computational Science and Engineering 31, Springer-Verlag, Berlin, 2003. MR2032865 (2004h:65008)
2. A. Bossavit, Computational Electromagnetism, Academic Press, San Diego, 1998. MR1488417 (99m:78001)
3. C. Carstensen, S. Funken, W. Hackbusch, Ronald H.W. Hoppe and P. Monk (eds.), Computational Electromagnetics, Lecture Notes in Computational Science and Engineering 28, Springer-Verlag, Berlin, 2003. MR1995092 (2004b:78001)
4. C.M. Chen and Y.Q. Huang, High Accuracy Theory of Finite Element Methods (in Chinese), Hunan Science Press, China, 1995.
5. M.-H. Chen, B. Cockburn and F. Reitich, High-order RKDG methods for computational electromagnetics, J. Sci. Comp. 22/23 (2005) 205-226. MR2142195 (2005m:65208)
6. Q. Chen, M. Katsurai and P.H. Aoyagi, An FDTD formulation for dispersive media using a current density, IEEE Trans. Antennas Propagat. 46 (1998) 1739-1746.
7. Z. Chen, Q. Du and J. Zou, Finite element methods with matching and nonmatching meshes for Maxwell equations with discontinuous coefficients, SIAM J. Numer. Anal. 37 (2000) 1542-1570. MR1759906 (2001h:78044)
8. P. Ciarlet, Jr. and J. Zou, Fully discrete finite element approaches for time-dependent Maxwell's equations, Numer. Math. 82 (1999) 193-219. MR1685459 (2000c:65083)
9. B. Cockburn, F. Li and C.-W. Shu, Locally divergence-free discontinuous Galerkin methods for the Maxwell equations, J. Comp. Phys. 194 (2004) 588-610. MR2034859 (2004j:78024)
10. G.C. Cohen, E. Heikkola, P. Joly and P. Neittaanmäki (eds.), Mathematical and numerical aspects of wave propagation - WAVES 2003, Springer-Verlag, Berlin, 2003. MR2077970 (2005b:00020)
11. S.A. Cummer, An analysis of new and existing FDTD methods for isotropic cold plasma and a method for improving their accuracy, IEEE Trans. Antennas Propagat. 45 (1997) 392-400.
12. L. Demkowicz, Fully automatic hp-adaptivity for Maxwell's equations, Comput. Methods Appl. Mech. Engrg. 194 (2005) 605-624. MR2105184 (2005m:65296)
13. L. Demkowicz, P. Monk, Ch. Schwab and L. Vardapetyan, Maxwell eigenvalues and discrete compactness in two dimensions, Comput. Math. Appl. 40 (2000) 589-605. MR1772657 (2002e:65166)
14. 2-error estimates and super-convergence of mixed finite element methods for non-Fickian flows in porous media, SIAM J. Numer. Anal. 40 (2002) 1538-1560. MR1951906 (2003m:65177)
15. R. Hiptmair, Finite elements in computational electromagnetism, Acta Numerica 11 (2002) 237-339. MR2009375 (2004k:78028)
16. P. Houston, I. Perugia and D. Schötzau, Energy norm a posteriori error estimation for mixed discontinuous Galerkin approximations of the Maxwell operator, Comput. Methods Appl. Mech. Engrg. 194 (2005) 499-510. MR2105178 (2005h:78027)
17. D. Jiao and J.-M. Jin, Time-domain finite-element modeling of dispersive media, IEEE Microwave and Wireless Components Letters 11 (2001) 220-222.
18. J. Jin, The Finite Element Method in Electromagnetics, 2nd ed., John Wiley & Sons, Inc., New York, 2002. MR1903357 (2004b:78019)
19. M. Krizek and P. Neittaanmaki, Bibliography on Superconvergence, in "Finite Element Methods", Lecture Notes in Pure and Applied Math., Vol. 196, 1998, pp. 315-348. MR1602730
20. J. Li, Error analysis of finite element methods for 3-D Maxwell's equations in dispersive media, Journal of Computational and Applied Mathematics, 188 (2006) 107-120. MR2192630 (2006m:65279)
21. J. Li and Y. Chen, Analysis of a time-domain finite element method for 3-D Maxwell's equations in dispersive media, Comput. Methods Appl. Mech. Engrg. 195 (2006) 4220-4229. MR2229840
22. J. Li, Error analysis of fully discrete mixed finite element schemes for 3-D Maxwell's equations in dispersive media, preprint, Sept. 2005.
23. J. Li and M.F. Wheeler, Uniform convergence and superconvergence of mixed finite element methods on anisotropically refined grids, SIAM J. Numer. Anal. 38 (2000) 770-798. MR1781203 (2001f:65137)
24. Q. Lin, J. Li and A. Zhou, rectangle test for Ciarlet-Raviart scheme, Prof. of Sys. Sci. & Sys. Engrg., Great Wall (H.K.) Culture Publish Co., 1991, pp. 230-231.
25. Q. Lin, J. Li and A. Zhou, A rectangle test for the Stokes equations, Prof. of Sys. Sci. & Sys. Engrg., Great Wall (H.K.) Culture Publish Co., 1991, pp. 240-241.
26. Q. Lin and J.F. Lin, High accuracy approximation of mixed finite element for 2-D Maxwell equations. (Chinese) Acta Math. Sci. Ser. A Chin. Ed. 23 (2003), no. 4, 499-503. MR2090553 (2005i:65145)
27. Q. Lin and J. Xu, Linear finite elements with high accuracy, J. Comp. Math. 3 (1985) 115-133. MR854355 (87k:65141)
28. Q. Lin and N. Yan, Superconvergence of mixed element methods for Maxwell's equations (in Chinese), Gongcheng Shuxue Xuebao 13 (1996), suppl., 1-10. MR1437477 (98c:65186)
29. Q. Lin and N. Yan, Global superconvergence for Maxwell's equations, Math. Comp. 69 (1999) 159-176. MR1654029 (2000i:65131)
30. Q. Lin and N. Yan, The Construction and Analysis of High Accurate Finite Element Methods (in Chinese), Hebei University Press, Hebei, China, 1996.
31. Q. Lin, N. Yan and A. Zhou, A rectangle test for interpolated finite elements, Prof, of Sys. Sci. & Sys. Engrg., Great Wall (H.K.) Culture Publish Co., 1991, pp. 217-229.
32. T. Lu, P. Zhang and W. Cai, Discontinuous Galerkin methods for dispersive and lossy Maxwell's equations and PML boundary conditions, J. Comp. Phys. 200 (2004) 549-580. MR2095277 (2005e:78025)
33. P. Monk, An analysis of Nédélec's method for the special discretization of Maxwell's equations, J. Comp. Appl. Math. 47 (1993) 101-121. MR1226366 (94g:65105)
34. P. Monk, Superconvergence of finite element approximations to Maxwell's equations, Numer. Methods Partial Differential Equations 10 (1994) 793-812. MR1298123 (95h:65090)
35. P. Monk, Finite element methods for Maxwell's equations, Oxford University Press, 2003. MR2059447 (2005d:65003)
36. 3, Numer. Math. 35 (1980) 315-341. MR592160 (81k:65125)
37. A. Quarteroni and A. Valli, Numerical Approximation of Partial Differential Equations, Springer-Verlag, Berlin, 1994. MR1299729 (95i:65005)
38. A. Taflove and C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 2nd ed., Artech House, Norwood, MA, 2000. MR1338377 (96f:78001)
39. L.B. Wahlbin, Superconvergence in Galerkin Finite Element Methods, Springer, Berlin, 1995. MR1439050 (98j:65083)
40. Z. Zhang, N. Yan and T. Sun, Superconvergent derivative recovery for the intermediate finite element family of the second order, IMA Journal of Numerical Analysis 21 (2001) 643-665. MR1844362 (2002f:65171)