no. 6
High order edge elements on simplicial meshes
ESAIM: Modélisation mathématique et analyse numérique, Tome 41 (2007) no. 6, pp. 1001-1020.

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 expressed only in terms of the barycentric coordinates of the simplex.

DOI : 10.1051/m2an:2007049
Classification : 78M10, 65N30, 68U20
Mots clés : Maxwell equations, higher order edge elements, simplicial meshes 
@article{M2AN_2007__41_6_1001_0,
     author = {Rapetti, Francesca},
     title = {High order edge elements on simplicial meshes},
     journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique},
     pages = {1001--1020},
     publisher = {EDP-Sciences},
     volume = {41},
     number = {6},
     year = {2007},
     doi = {10.1051/m2an:2007049},
     mrnumber = {2377104},
     zbl = {1141.78014},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/m2an:2007049/}
}
TY  - JOUR
AU  - Rapetti, Francesca
TI  - High order edge elements on simplicial meshes
JO  - ESAIM: Modélisation mathématique et analyse numérique
PY  - 2007
SP  - 1001
EP  - 1020
VL  - 41
IS  - 6
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/m2an:2007049/
DO  - 10.1051/m2an:2007049
LA  - en
ID  - M2AN_2007__41_6_1001_0
ER  - 
%0 Journal Article
%A Rapetti, Francesca
%T High order edge elements on simplicial meshes
%J ESAIM: Modélisation mathématique et analyse numérique
%D 2007
%P 1001-1020
%V 41
%N 6
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/m2an:2007049/
%R 10.1051/m2an:2007049
%G en
%F M2AN_2007__41_6_1001_0
Rapetti, Francesca. High order edge elements on simplicial meshes. ESAIM: Modélisation mathématique et analyse numérique, Tome 41 (2007) no. 6, pp. 1001-1020. doi : 10.1051/m2an:2007049. http://www.numdam.org/articles/10.1051/m2an:2007049/

[1] M. Ainsworth, 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. | Zbl

[2] M. Ainsworth and J. Coyle, Hierarchic finite element bases on unstructured tetrahedral meshes. Int. J. Numer. Meth. Engng. 58 (2003) 2103-2130. | Zbl

[3] M. Ainsworth, J. Coyle, P.D. Ledger and K. Morgan, Computation of Maxwell eigenvalues using higher order edge elements in three-dimensions. IEEE Trans. Magn. 39 (2003) 2149-2153.

[4] M.A. Armstrong, Basic Topology. Springer-Verlag, New York (1983). | MR | Zbl

[5] D. Arnold, R. Falk and R. Winther, Finite element exterior calculus, homological techniques, and applications. Acta Numer. 15 (2006) 1-155.

[6] 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). | Zbl

[7] A. Bossavit, Computational Electromagnetism. Academic Press, New York (1998). | MR | Zbl

[8] A. Bossavit, Generating Whitney forms of polynomial degree one and higher. IEEE Trans. Magn. 38 (2002) 341-344.

[9] A. Bossavit and F. Rapetti, Whitney forms of higher degree. Preprint.

[10] V. Girault and P.A. Raviart, Finite element methods for Navier-Stokes equations. Springer-Verlag, Berlin (1986). | MR | Zbl

[11] J. Gopalakrishnan, L.E. Garcia-Castillo and L.F. Demkowicz, Nédélec spaces in affine coordinates. ICES Report 03-48 (2003). | Zbl

[12] R.D. Graglia, D.R. Wilton and A.F. Peterson, Higher order interpolatory vector bases for computational electromagnetics. IEEE Trans. on Ant. and Propag. 45 (1997) 329-342.

[13] R. Hiptmair, Canonical construction of finite elements. Math. Comp. 68 (1999) 1325-1346. | Zbl

[14] R. Hiptmair, High order Whitney forms. Prog. Electr. Res. (PIER) 32 (2001) 271-299.

[15] G.E. Karniadakis and S.J. Sherwin, Spectral hp element methods for CFD. Oxford Univ. Press, London (1999). | MR | Zbl

[16] J.M. Melenk, On condition numbers in hp-FEM with Gauss-Lobatto-based shape functions. J. Comput. Appl. Math. 139 (2002) 21-48. | Zbl

[17] P. Monk, Finite Element Methods for Maxwell's Equations. Oxford University Press (2003). | Zbl

[18] J.C. Nédélec, Mixed finite elements in ℝ 3 . Numer. Math. 35 (1980) 315-341. | Zbl

[19] F. Rapetti and A. Bossavit, Geometrical localization of the degrees of freedom for Whitney elements of higher order. IEE Sci. Meas. Technol. 1 (2007) 63-66.

[20] J. Schöberl and S. Zaglmayr, High order Nédélec elements with local complete sequence properties. COMPEL 24 (2005) 374-384. | Zbl

[21] J. Stillwell, Classical topology and combinatorial group theory, Graduate Text in Mathematics 72. Springer-Verlag (1993). | MR | Zbl

[22] J.P. Webb and B. Forghani, Hierarchal scalar and vector tetrahedra. IEEE Trans. on Magn. 29 (1993) 1495-1498.

[23] H. Whitney, Geometric integration theory. Princeton Univ. Press (1957). | MR | Zbl

Cité par Sources :