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High order edge elements on simplicial meshes

Published online by Cambridge University Press:  15 December 2007

Francesca Rapetti
Francesca Rapetti*
Affiliation:
Laboratoire J.-A. Dieudonné, C.N.R.S. & Université de Nice Sophia-Antipolis, Parc Valrose, 06108 Nice Cedex 02, France. [email protected]
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Abstract

Low order edge elements are widely used for electromagnetic field problems. Higher order edge approximations are receiving increasing interest but their definition become rather complex.In this paper we propose a simple definition for Whitney edge elements of polynomial degree higher than one. We give a geometrical localization of all degrees of freedom over particular edges and provide a basis for these elements on simplicial meshes.As for Whitney edge elements of degree one, the basis is expressedonly in terms of the barycentric coordinates of the simplex.

Keywords

Maxwell equationshigher order edge elementssimplicial meshes.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 41 , Issue 6 , November 2007 , pp. 1001 - 1020
Copyright
© EDP Sciences, SMAI, 2007

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References

Ainsworth, M., Dispersive properties of high order Nédélec/edge element approximation of the time-harmonic Maxwell equations. Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 362 (2004) 471–491. CrossRef
Ainsworth, M. and Coyle, J., Hierarchic finite element bases on unstructured tetrahedral meshes. Int. J. Numer. Meth. Engng. 58 (2003) 2103–2130. CrossRef
Ainsworth, M., Coyle, J., Ledger, P.D. and Morgan, K., Computation of Maxwell eigenvalues using higher order edge elements in three-dimensions. IEEE Trans. Magn. 39 (2003) 2149–2153. CrossRef
M.A. Armstrong, Basic Topology. Springer-Verlag, New York (1983).
Arnold, D., Falk, R. and Winther, R., Finite element exterior calculus, homological techniques, and applications. Acta Numer. 15 (2006) 1–155. CrossRef
D. Boffi, M. Costabel, M. Dauge and L.F. Demkowicz, Discrete compactness for the hp version of rectangular edge finite elements. ICES Report 04–29 (2004).
A. Bossavit, Computational Electromagnetism. Academic Press, New York (1998).
Bossavit, A., Generating Whitney forms of polynomial degree one and higher. IEEE Trans. Magn. 38 (2002) 341–344. CrossRef
A. Bossavit and F. Rapetti, Whitney forms of higher degree. Preprint.
V. Girault and P.A. Raviart, Finite element methods for Navier-Stokes equations. Springer-Verlag, Berlin (1986).
J. Gopalakrishnan, L.E. Garcia-Castillo and L.F. Demkowicz, Nédélec spaces in affine coordinates. ICES Report 03–48 (2003).
Graglia, R.D., Wilton, D.R. and Peterson, A.F., Higher order interpolatory vector bases for computational electromagnetics. IEEE Trans. on Ant. and Propag. 45 (1997) 329–342. CrossRef
Hiptmair, R., Canonical construction of finite elements. Math. Comp. 68 (1999) 1325–1346. CrossRef
Hiptmair, R., High order Whitney forms. Prog. Electr. Res. (PIER) 32 (2001) 271–299. CrossRef
G.E. Karniadakis and S.J. Sherwin, Spectral hp element methods for CFD. Oxford Univ. Press, London (1999).
Melenk, J.M., On condition numbers in hp-FEM with Gauss-Lobatto-based shape functions. J. Comput. Appl. Math. 139 (2002) 21–48. CrossRef
P. Monk, Finite Element Methods for Maxwell's Equations. Oxford University Press (2003).
Nédélec, J.C., Mixed finite elements in $\mathbb R^3$ . Numer. Math. 35 (1980) 315–341. CrossRef
Rapetti, F. and Bossavit, A., Geometrical localization of the degrees of freedom for Whitney elements of higher order. IEE Sci. Meas. Technol. 1 (2007) 63–66. CrossRef
Schöberl, J. and Zaglmayr, S., High order Nédélec elements with local complete sequence properties. COMPEL 24 (2005) 374–384. CrossRef
J. Stillwell, Classical topology and combinatorial group theory, Graduate Text in Mathematics 72. Springer-Verlag (1993).
Webb, J.P. and Forghani, B., Hierarchal scalar and vector tetrahedra. IEEE Trans. on Magn. 29 (1993) 1495–1498. CrossRef
H. Whitney, Geometric integration theory. Princeton Univ. Press (1957).