volume | PIER Journals

Abstract

The calculus of differential forms can be used to devise a unified description of discrete differential forms of any order and polynomial degree on simplicial meshes in any spatial dimension. A general formula for suitable degrees of freedom is also available. Fundamental properties of nodal interpolation can be established easily. It turns out that higher order spaces, including variants with locally varying polynomial order, emerge from the usual Whitneyforms by local augmentation. This paves the way for an adaptive pversion approach to discrete differential forms.

1. Andersen, L. and J. Volakis, "Development and application of a novel class of hierarchical tangential vector finite elements for electromagnetics," IEEE Trans. Antennas and Propagation, Vol. 47, 112-120, 1999.
doi:10.1109/8.753001 Google Scholar

2. Babuska, I., M. Griebel, and J. Pitkaranta, "The problem of selecting the shape functions for a p-type finite element," Int. J. Num. Meth. Engin., Vol. 28, 1891-1908, 1988.
doi:10.1002/nme.1620280813 Google Scholar

3. Baldomir, D. and P. Hammond, Geometry of Electromagnetic Systems, Clarendon Press, Oxford, 1996.

4. Beck, R., R. Hiptmair, and B. Wohlmuth, "A hierarchical error estimator for eddy current computation," ENUMATH 99 — Proceedings of the 3rd European Conference on Numerical Mathematics and Advanced Applications, P. Neittaanmaki and T. Tiihonen (eds.), July 26–30, Jyvskyl, Finland, 110–120, World Scientific, Singapore, 2000. Google Scholar

5. Boffi, D., "Discrete compactness and Fortin operator for edge elements," Tech. Rep. AM187, Department of Mathematics, Pennsylvania State University, State College, USA, April 1999. To appear in Numerische Mathematik. Google Scholar

6. Bossavit, A., "Mixed finite elements and the complex of Whitney forms," The Mathematics of Finite Elements and Applications VI, J. Whiteman (ed.), 137–144, Academic Press, London, 1988. Google Scholar

7. Bossavit, A., "Whitney forms: A class of finite elements for threedimensional computations in electromagnetism," IEE Proc. A, Vol. 135, 493-500, 1988.
doi:10.1049/ip-d.1988.0075 Google Scholar

8. Bossavit, A., "On the geometry of electromagnetism IV: “Maxwell’s house," J. Japan Soc. Appl. Electromagnetics & Mech., Vol. 6, 318-326, 1998. Google Scholar

9. Brenner, S. and R. Scott, Mathematical Theory of Finite Element Methods, Texts in Applied Mathematics, Springer-Verlag, New York, 1994.
doi:10.1007/978-1-4757-4338-8

10. Caorsi, S., P. Fernandes, and M. Raffetto, "On the convergence of Galerkin finite element approximations of electromagnetic eigenproblems," SIAM J. Numer. Anal., To appear. Google Scholar

11. Cartan, H., Formes Differentielles, Hermann, Paris, 1967.

12. Ciarlet, P., "The finite element method for elliptic problems," Studies in Mathematics and its Applications, Vol. 4, North-Holland, Amsterdam, 1978. Google Scholar

13. Demkowicz, L., P. Monk, L. Vardapetyan, and W. Rachowicz, "De Rham diagram for hp finite element spaces," Tech. Rep., 99-06, TICAM, University of Texas, Austin, TX, 1999. Google Scholar

14. Deschamps, G., "Electromagnetics and differential forms," Proc. IEEE, Vol. 69, 676-695, 1981.
doi:10.1109/PROC.1981.12048 Google Scholar

15. Elmkies, A. and P. Joly, "Elements finis et condensation de masse pour les equations des Maxwell: le cas 3D," Tech. Rep., 3381, INRIA, Roucquencourt, Domaine de Voluveau, France, May 1998. Google Scholar

16. Graglia, R., D. Wilton, and A. Peterson, "Higher order interpolatory vector bases for computational electromagnetics," IEEE Trans. Antennas and Propagation, Vol. 45, 329-342, 1997.
doi:10.1109/8.558649 Google Scholar

17. Graglia, R., D. Wilton, A. Peterson, and I.-L. Gheorma, "Higher order interpolatory vector bases on prism elements," IEEE Trans. Antennas and Propagation, Vol. 46, 442-450, 1998.
doi:10.1109/8.662664 Google Scholar

18. Hiptmair, R., "Canonical construction of finite elements," Math. Comp., Vol. 68, 1325-1346, 1999.
doi:10.1090/S0025-5718-99-01166-7 Google Scholar

19. Iwaniec, T., "Nonlinear differential forms,” “Lectures notes of the International Summer School in Jyvaskyla,", 1998 80, University of Jyvaskyla, Department of Mathematics, Jyvaskyla, Finland, 1999. Google Scholar

20. Monk, P., "On the p and hp-extension of Nedelec’s conforming elements," J. Comp. Appl. Math., Vol. 53, 117-137, 1994.
doi:10.1016/0377-0427(92)00127-U Google Scholar

21. Monk, P. and L. Demkowicz, "Discrete compactness and the approximation of Maxwell’s equations in R3," Math. Comp., 1999, to appear.
doi:10.1016/0377-0427(92)00127-U Google Scholar

22. Nedelec, J., "Mixed finite elements in R3," Numer. Math., Vol. 35, 315-341, 1980.
doi:10.1007/BF01396415 Google Scholar

23. Nedelec, J., "A new family of mixed finite elements in R3," Numer. Math., Vol. 50, 57-81, 1986.
doi:10.1007/BF01389668 Google Scholar

24. Peng, G., R. Dyczij-Edlinger, and J.-F. Lee, "Hierarchical methods for solving matrix equations from TVFEMs for microwave components," IEEE Trans. Mag., Vol. 35, 1474-1477, 1998.
doi:10.1109/20.767245 Google Scholar

25. Raviart, P. A. and J. M. Thomas, "A mixed finite element method for second order elliptic problems," Springer Lecture Notes in Mathematics, Vol. 606, 292-315, Springer-Verlag, New York, 1977.
doi:10.1007/BFb0064470 Google Scholar

26. Savage, J. and A. Peterson, "Higher order vector finite elements for tetrahedral cells," IEEE Trans. Microwave Theory and Techniques, Vol. 44, 874-879, 1996.
doi:10.1109/22.506446 Google Scholar

27. Whitney, H., Geometric Integration Theory, Princeton Univ. Press, Princeton, 1957.
doi:10.1515/9781400877577

28. Yiailtsis, T. and T. Tsiboukis, "A systematic approach to the construction of higher order vector finite elements for three-dimensional electromagnetic field computation," COMPEL, Vol. 14, 49-53, 1995.
doi:10.1108/eb051912 Google Scholar

Dr. R. Hiptmair