Multigrid computation of axisymmetric electromagnetic fields

Advances in Computational Mathematics

Volumn 16, Issue 4, 2002, Pages 331-356

  BoOrm S ^a   Hiptmair, R ^b  

^a UNIVERSITY OF KIEL   (Germany)

^b UNIVERSITY OF TÜBINGEN   (Germany)

Author keywords

Computational electromagnetism; Cylindrical symmetry; Degenerate problems; Edge elements; Multigrid; Semi coarsening; Vector valued problems

Indexed keywords

EID: 0036110351     PISSN: 10197168     EISSN: None     Source Type: Journal    
DOI: 10.1023/A:1014533409747     Document Type: Article

References (49)

1. D. Arnold, R. Falk and R. Winther, Preconditioning in H(div) and applications, Math. Comp. 66 (1997) 957-984.
2. D. Arnold, R. Falk and R. Winther, Multigrid in H(div) and H(curl), Numer. Math. 85 (2000) 175-195.
3. R. Beck, P. Deuflhard, R. Hiptmair, R. Hoppe and B. Wohlmuth, Adaptive multilevel methods for edge element discretizations of Maxwell's equations, Surveys Math. Indust. 8 (1998) 271-312.
4. S. Börm and R. Hiptmair, Analysis of tensor product multigrid, Numer. Algorithms 26 (2001) 219-234.
5. A. Bossavit, A new viewpoint on mixed elements, Meccanica 27 (1992) 3-11.
6. D. Braess and W. Hackbusch, A new convergence proof for the multigrid method including the V-cycle, SIAM J. Numer. Anal. 20 (1983) 967-975.
7. J. Bramble, Multigrid Methods, Pitman Research Notes in Mathematics Series, Vol. 294 (Longman, London, 1993).
8. J. Bramble and J. Pasciak, The analysis of smoothers for multigrid algorithms, Math. Comp. 58 (1992) 467-488.
9. J. Bramble, J. Pasciak and J. Xu, The analysis of multigrid algorithms with non-nested spaces or non-inherited quadratic forms, Math. Comp. 56 (1991) 1-34.
10. P. Ciarlet, The Finite Element Method for Elliptic Problems, Studies in Mathematics and its Applications, Vol. 4 (North-Holland, Amsterdam, 1978).
11. F. Cooray and G. Costache, An overview of the absorbing boundary conditions, J. Electromagn. Waves Appl. 5 (1991) 1041-1054.
12. J. Dendy and C. Tazartes, Grandchild of the frequency decomposition multigrid method, SIAM J. Sci. Comput. 2 (1995) 307-319.
13. R. Ewing and J. Wang, Analysis of the Schwarz algorithm for mixed finite element methods, Math. Modelling Numer. Anal. 26 (1992) 739-756.
14. P. Grisvard, Singularities in Boundary Value Problems, Research Notes in Applied Mathematics, Vol. 22 (Springer, New York, 1992).
15. W. Hackbusch, Multi-grid Methods and Applications (Springer, Berlin, 1985).
16. W. Hackbusch, Theorie und Numerik elliptischer Differentialgleichungen (Teubner, Stuttgart, 1986).
17. W. Hackbusch, Iterative Solution of Large Sparse Systems of Equations, Applied Mathematical Sciences, Vol. 95 (Springer, New York, 1993).
18. W. Hackbusch, The frequency decomposition multi-grid method, in: Multigrid Methods IV. Proc. of the 4th European Multigrid Conf., Amsterdam, Netherlands, 6-9 July 1993, ed. P. Hemker (Birkhäuser, Basel, 1994) pp. 43-56.
19. R. Hiptmair, Multigrid method for H(div) in three dimensions, ETNA 6 (1997) 133-152.
20. R. Hiptmair, Multigrid method for Maxwell's equations, SIAM J. Numer. Anal. 36 (1999) 204-225.
21. R. Hiptmair and A. Toselli, Overlapping and multilevel Schwarz methods for vector valued elliptic problems in three dimensions, in: Parallel Solution of Partial Differential Equations, eds. P. Bjorstad and M. Luskin, IMA Volumes in Mathematics and its Applications, Vol. 120 (Springer, Berlin, 1999) pp. 181-202.
22. J. Hyman and M. Shashkov, Natural discretizations for the divergence, gradient, and curl on logically rectangular grids, Internal. J. Comput. Math. Appl. 33 (1997) 81-104.
23. K. Mahadevan and R. Mitta, Use of Whitney's edge and face elements for efficient finite element time domain solution of Maxwell's equations, J. Electromagn. Waves Appl. 8 (1994) 1173-1191.
24. G. Makridakis and P. Monk, Time-discrete finite element schemes for Maxwell's equations, RAIRO Math. Modelling Numer. Anal. 29 (1995) 171-197.
25. D. Mavriplis, Directional coarsening and smoothing for anisotropic Navier-Stokes problems, Electron. Trans. Numer. Anal. 6 (1997) 182-197.
26. P. Monk, A mixed method for approximating Maxwell's equations, SIAM J. Numer. Anal. 28 (1991) 1610-1634.
27. P. Monk, Analysis of a finite element method for Maxwell's equations, SIAM J. Numer. Anal. 29 (1992) 714-729.
28. P. Monk, A comparison of three mixed methods for the time dependent Maxwell equations, SIAM J. Sci. Statist. Comput. 13 (1992) 1097-1122.
29. P. Monk and E. Süli, A convergence analysis of Yee's scheme on nonuniform grids, SIAM J. Numer. Anal. 31 (1994) 393-412.
30. P. Monk and S. Zhang, Multigrid computation of vector potentials, J. Comput. Appl. Math. 62 (1995) 301-320.
31. W. Mulder, A new multigrid approach to convection problems, J. Comput. Phys. 83 (1989) 303-323.
32. 3, Numer. Math. 35 (1980) 315-341.
33. 3, Numer. Math. 50 (1986) 57-81.
34. C. Oosterlee, F. Gaspar, T Washio and R. Wienands, Multigrid line smoothers for higher order upwind discretizations of convection dominated problems, J. Comput. Phys. 139 (1998) 274-307.
35. C. Pflaum, A robust multilevel algorithm for anisotropic elliptic equations, Technical Report 224, Mathematisches Institut, Universität Würzburg, Würzbug, Germany (1998).
36. C. Pflaum, Fast and robust multilevel algorithms, Habilitationsschrift, Mathematische Institute, Universität Würzburg, Würzburg, Germany (1999).
37. P. Raviart and W. Sonnendrücker, A hierarchy of approximate models for the Maxwell equations, Numer. Math. 73 (1996) 329-372.
38. J. Schöberl, Multigrid methods for parameter dependent problems 1: The Stokes-type case, in: Proc. of the 5th European Multigrid Conf., Vol. 199, eds. W. Hackbusch and G. Wittum (Springer, 1998) to appear.
39. J. Schöberl, Multigrid methods for a parameter dependent problem in primal variables, Numer. Math. 84 (1999) 97-119.
40. B. Smith, P. Bjoørstad and W. Gropp, Domain Decomposition (Cambridge Univ. Press, Cambridge, 1996).
41. A. Taflove, The Finite Difference in Time Domain Method (Artech House, Boston/London, 1995).
42. P. Vassilevski and J. Wang, Multilevel iterative methods for mixed finite element discretizations of elliptic problems, Numer. Math. 63 (1992) 503-520.
43. T. Washio and C. Oosterlee, Flexible multiple semicoarsening for 3-dimensional singularly perturbed problems, SIAM J. Sci. Comput. 19 (1998) 1646-1666.
44. T. Weiland, Time domain electromagnetic field computation with finite difference methods, Internat. J. Numer. Modelling 9 (1996) 295-319.
45. P. Wesseling, An Introduction to Multigrid Methods (Wiley, Chichester, 1992).
46. J. Xu, Iteritive methods by space decomposition and subspace correction, SIAM Review 34 (1992) 581-613.
47. K. Yee, Numerical solution of initial boundary value problems involving Maxwell's equations in isotropic media, IEEE Trans. Antennas Propagation 16 (1966) 302-307.
48. H. Yserentant, Old and new convergence proofs for multigrid methods, Acta Numerica (1993) 285-326.
49. X. Zhang, Multilevel Schwarz methods, Numer. Math. 63 (1992) 521-539.