A characterization of second-order differential operators on finite element spaces. (English) Zbl 1076.83008
The paper is concerned with the development of the Regge calculus [Nuovo Cimento 10, 558–571 (1961)]. A very pedagogical review of the Regge calculus in General Relativity is given by M. Piso in [“Rotationally invariant discrete space-time structure”, in: Studies in Gravitation theory, I. Gottlieb and N. I. Ionescu-Pallas (eds.), CIP Press, Bucharest 34–42 (1988)], which, due to its obscure origin, is difficult to find in western libraries and databases. The authors refer to well-known papers by R. Sorkin and D. Weingarten, which develop the Regge calculus in Electrodynamics mainly, or combine the Electrodynamics and General Relativity.
In this article the author provides a new link between Regge calculus and finite element theory obtained via the tensor product interpretation of bilinear forms arising from differential operators. The author intends to demonstrate that this link provides guidelines for various constructions of finite element spaces. He tries to prove that linear forms on Regge elements and Nédélec elements J.-C. Nédélec [Numer. Math. 35, 315–341 (1980; Zbl 0419.65069)] are in one-to one correspondence respectively.
In this article the author provides a new link between Regge calculus and finite element theory obtained via the tensor product interpretation of bilinear forms arising from differential operators. The author intends to demonstrate that this link provides guidelines for various constructions of finite element spaces. He tries to prove that linear forms on Regge elements and Nédélec elements J.-C. Nédélec [Numer. Math. 35, 315–341 (1980; Zbl 0419.65069)] are in one-to one correspondence respectively.
Reviewer: Alex Gaina (Chisinau)
MSC:
| 83C27 | Lattice gravity, Regge calculus and other discrete methods in general relativity and gravitational theory |
| 58J99 | Partial differential equations on manifolds; differential operators |
| 58J60 | Relations of PDEs with special manifold structures (Riemannian, Finsler, etc.) |
| 65Z05 | Applications to the sciences |
| 65L60 | Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations |
Citations:
Zbl 0419.65069References:
| [1] | D. N. Arnold, Differential complexes and numerical stability, Plenary address delivered at ICM 2002; International Congress of Mathematicians, Beijing 2002. |
| [2] | Arnold, D. N.Winther, R., Numer. Math.92, 401 (2002). · Zbl 1090.74051 |
| [3] | Babuska, I., Numer. Math.16, 322 (1971). · Zbl 0214.42001 |
| [4] | Barrett, J. W.et al., Int. J. Theor. Phys.36, 815 (1997). |
| [5] | Boffi, D., Appl. Math. Lett.14, 33 (2001). · Zbl 0983.65125 |
| [6] | Bossavit, A., The Mathematics of Finite Elements and Applications, VI (Uxbridge, 1987) (Academic Press, 1988) pp. 137-144. · Zbl 0692.65053 |
| [7] | Brezzi, F., RAIRO Anal. Numér.8, 129 (1974). · Zbl 0338.90047 |
| [8] | ; Brezzi, F.; Fortin, M., Mixed and Hybrid Finite Element Methods, 1991, Springer-Verlag · Zbl 0788.73002 |
| [9] | Buffa, A.Christiansen, S. H., Numer. Math.94, 229 (2003). · Zbl 1027.65188 |
| [10] | Cheeger, J.Müller, W.Schrader, R., Commun. Math. Phys.92, 405 (1984). · Zbl 0559.53028 |
| [11] | Coifman, R.et al., J. Math. Pures Appl.72, 247 (1993). · Zbl 0864.42009 |
| [12] | Gentle, A. P., Gen. Rel. Grav.34, 1701 (2002). · Zbl 1019.83006 |
| [13] | ; Hatcher, A., Algebraic Topology, 2002, Cambridge Univ. Press · Zbl 1044.55001 |
| [14] | Hiptmair, R., Math. Comp.68, 1325 (1999). · Zbl 0938.65132 |
| [15] | ; Lang, S., Differential and Riemannian Manifolds, 1995, Springer-Verlag · Zbl 0824.58003 |
| [16] | Lehner, L., Class. Quantum Grav.18, 25 (2001). · Zbl 0987.83001 |
| [17] | ; Monk, P., Finite Element Methods for Maxwell’s Equations, 2003, Oxford Univ. Press · Zbl 1024.78009 |
| [18] | Nédélec, J.-C., Numer. Math.35, 315 (1980). · Zbl 0419.65069 |
| [19] | Raviart, P. A.Thomas, J.-M., Mathematical Aspects of the Finite Element Method, 606, eds. Galligani, I.Magenes, E. (Springer-Verlag, 1977) pp. 292-315. · Zbl 0362.65089 |
| [20] | Regge, T., Nuovo Cimento10, 558 (1961). |
| [21] | Regge, T.Williams, R. M., J. Math. Phys.41, 3964 (2000). · Zbl 0984.83016 |
| [22] | ; DeRham, G., Variétés Différentiables. Formes, Courants, Formes Narmoniques, 1973, Hermann · Zbl 0284.58001 |
| [23] | ; Ryan, R. A., Introduction to Tensor Products of Banach Spaces, 2002, Springer · Zbl 1090.46001 |
| [24] | Sorkin, R., J. Math. Phys.16, 2432 (1975). |
| [25] | ; Taylor, M., Partial Differential Equations, Vol. III Nonlinear Equations, 1996, Springer-Verlag · Zbl 0869.35004 |
| [26] | Weingarten, D., J. Math. Phys.18, 165 (1977). |
| [27] | ; Whitney, H., Geometric Integration Theory, 1957, Princeton Univ. Press · Zbl 0083.28204 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
