Multilevel Gauging for Edge Elements

Computing (Vienna/New York)

Volumn 64, Issue 2, 2000, Pages 97-122

  Hiptmair, Ralf ^a  

^a UNIVERSITY OF TÜBINGEN   (Germany)

Author keywords

Computational electromagnetism; Edge elements; Gauging; Hierarchical bases; Multilevel decomposition; Vector potentials

Indexed keywords

EID: 0345813580     PISSN: 0010485X     EISSN: None     Source Type: Journal    
DOI: 10.1007/s006070050005     Document Type: Article

References (51)

1. Albanese, R., Rubinacci, G.: Integral formulation for 3D eddy-current computation using edge elements. IEE Proc. A 135, 457-462 (1988).
2. Albanese, R., Rubinacci, G.: Fomulation of the eddy-current problem. IEE Proc. A 137, 16-22 (1990).
3. Amrouche, C., Bernardi, C., Dauge, M., Girault, V.: Vector potentials in three-dimensional nonsmooth domains. Math. Methods Appl. Sci. 21, 823-864 (1998).
4. Arnold, D., Falk, R., Winther, R.: Multigrid in H(div) and H(curl). Numer. Math. (2000) (to appear).
5. Arnold, D., Mukherjee, A., Pouly, L.: Locally adapted tetrahedral meshes using bisection. SIAM J. Sci. Comput. (2000) (to appear).
6. Bank, R.: Hierarchical bases and the finite element method. Acta Numer. 5, 1-43 (1996).
7. Bank, R., Dupont, T., Yserentant, H.: The hierarchical basis multigrid method. Numer. Math. 52, 427-458 (1988).
8. Bänsch, E.: Local mesh refinement in 2 and 3 dimensions. IMPACT Comput. Sci. Eng. 3, 181-191 (1991).
9. Bastian, P., Birken, K., Johannsen, K., Lang, S., Neuss, N., Rentz-Reichert, H., Wieners, C.: UG-A flexible software toolbox for solving partial differential equations. Comput. Visual. Sci. I, 27-40 (1997).
10. Beck, R., Deuflhard, P., Hiptmair, R., Hoppe, R., Wohlmuth, B.: Adaptive multilevel methods for edge element discretizations of Maxwell's equations. Surv. Math. Ind. 8, 271-312 (1999).
11. Beck, R., Erdmann, B., Roitzsch, R.: KASKADE 3.0. An object-oriented adaptive finite element code. Tech. Rep. TR 95-4, ZIB, Berlin, Germany, June 1995.
12. 2AN.
13. Bey, J.: Tetrahedral grid refinement. Computing 55, 355-378 (1995).
14. Biro, O., Richter, K.: CAD in electromagnetism. In: Advances in Electronics and Electron Physics, vol. 82 (Hawkes, P., ed.) pp. 1-96. New York: Academic Press, 1991.
15. Bossavit, A.: A rationale for edge elements in 3D field computations. IEEE Trans. Mag. 24, 74-79 (1988).
16. Bossavit, A.: Whitney forms: A class of finite elements for three-dimensional computations in electromagnetism. IEE Proc. A 135, 493-500 (1988).
17. Bossavit, A.: Computational electromagnetism. Variational formulation, complementarity, edge elements. Academic Press Electromagnetism Series, vol. 2. Academic Press, San Diego, 1998.
18. Brandt, A.: Multigrid techniques: 1984 guide with applications, of GMD-Studien, vol. 85. Bonn: GMD, 1984.
19. Brezzi, F., Fortin, M.: Mixed and hybrid finite element methods. Berlin Heidelberg New York Tokyo: Springer, 1991.
20. Cendes, C., Shenton, D.: Adaptive mesh refinement in the finite element computation of magnetic fields. IEEE Trans. Mag. 21, 1811-1816 (1985).
21. Ciarlet, P.: The finite element method for elliptic problems. Studies in Mathematics and its Applications, vol. 4 North-Holland, Amsterdam, 1978.
22. Ciarlet, P., Jr., Zou, J.: Fully discrete finite element approaches for time-dependent Maxwell equations. Numer. Math. 82, 193-219 (1999).
23. Dubois, F.: Discrete vector potential representation of a divergence free vector field in three dimensional domains: Numerical analysis of a model problem. SIAM J. Numer. Anal. 27, 1103-1141 (1990).
24. Dular, P., Hody, J.-Y., Nicolet, A., Genon, A., Legros, W.: Mixed finite elements associated with a collection of tetrahedra, hexahedra and prisms. IEEE Trans. Magnetics, MAG-30, 2980-2983 (1994).
25. Erdmann, B., Lang, J., Roitzsch, R.: Kaskade manual, version 2.0 - fem for 2 and 3 space dimensions. Tech. Rep. TR 93-5, ZIB-Berlin, 1993.
26. 2AN Math. Modell. Numer. Anal. 26, 739-756 (1992).
27. Girault, V., Raviart, P.: Finite element methods for Navier-Stokes equations. Berlin Heidelberg New York Tokyo: Springer, 1986.
28. Gradinaru, V., Hiptmair, R.: Discrete differential forms for pyramidal elements. Tech. Rep. 113, SFB 382, Universität Tübingen, Tübingen, Germany, 1999.
29. Graglia, R., Wilton, D., Peterson, A.: Higher order interpolatory vector bases for computational electromagnetics. IEEE Trans. Antennas and Propagation 45, 329-342 (1997).
30. Hackbusch, W.: Multi-grid methods and applications. Berlin Heidelberg New York Tokyo: Springer, 1985.
31. Hiptmair, R.: Multigrid method for H(div) in three dimensions. ETNA 6, 7-77 (1997).
32. Hiptmair, R.: Canonical construction of finite elements. Math. Comp. 68, 1325-1346 (1999).
33. Hiptmair, R.: Multigrid method for Maxwell's equations. SIAM J. Numer. Anal. 36, 204-225 (1999).
34. Hiptmair, R., Hoppe, R.: Multilevel preconditioning for mixed problems in three dimension. Numer. Math. 82, 253-279 (1999).
35. Hiptmair, R., Toselli, A.: Overlapping and multilevel Schwarz methods for vector valued elliptic problems in three dimensions. In: Parallel Solution of PDEs, IMA Volumes in Mathematics and its Applications, Vol. 120 (Bjorstad, P., Luskin, M., eds.), pp. 181-202. Berlin Heidelberg New York Tokyo: Springer, 1999.
36. Jackson, J.: Classical Electrodynamics. Wiley, New York, 2nd ed., 1975.
37. Kettunen, L., Forsman, K., Bossavit, A.: Formulation of the eddy current problems in multiply connected regions in terms of h. Int. J. Numer. Meth. Eng. 41, 935-954 (1998).
38. Kettunen, L., Forsman, K., Bossavit, A.: Gauging in Whitney spaces. IEEE Trans. Magnetics 35, 1466-1469 (1999).
39. Leinen, P.: Data structures and concepts for adaptive finite element methods. Computing 55, 325-354 (1995).
40. Mitchell, W.: Optimal multilevel iterative methods for adaptive grids. SIAM J. Sci. Stat. Comput. 13, 146-167 (1992).
41. 3. Numer. Math. 35, 315-341 (1980).
42. Peng, G., Dyczij-Edlinger, R., Lee, J.-F.: Hierarchical methods for solving matrix equations from TVFEMs for microwave components. IEEE Trans. Mag. 35, 1474-1477 (1998).
43. Perugia, I.: A mixed formulation for 3D magnetostatic problems: Theoretical analysis and face-edge finite element approximation. Numer. Math. (1999) (online).
44. Raviart, P. A., Thomas, J. M.: A mixed finite element method for second order elliptic problems. Springer Lecture Notes in Mathematics, vol. 606, pp. 292-315. New York: Springer, 1977.
45. Savage, J., Peterson, A.: Higher order vector unite elements for tetrahedral cells. IEEE Trans. Microwave Theory Technol. 44, 874-879 (1996).
46. Schwarz, H.: Methode der finiten Elemente. 3. Aufl. Leitfäden der angewandten Mathematik und Mechanik, vol. 47, Stuttgart: Teubner B. G., 1991.
47. Subramaniam, S., Ratnajeevan, S., Hoole, S.: Edge elements. In: Finite Elements, Electro-magnetics and Design. (Hoole, S., Ratnajeevan, S., eds.). pp. 342-393. Amsterdam: Elsevier, 1995.
48. Verfürth, R.: A review of a posteriori error estimation and adaptive mesh-refinement techniques. Teubner-Verlag, Stuttgart, 1996.
49. Wohlmuth, B.: Adaptive Multilevel-Finite-Elemente Methoden zur Lösung elliptischer Randwertprobleme. PhD thesis, TU München, 1995.
50. Wohlmuth, B., Toselli, A., Widlund, O.: An iterative substructuring method for Raviart-Thomas vector fields in three dimensions. Tech. Rep. 773, Courant Institute, New York, 1998.
51. Yserentant, H.: On the multi-level splitting of unite element spaces. Numer. Math. 58, 379-412 (1986).