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
Keywords: 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},
     year = {2007},
     publisher = {EDP Sciences},
     volume = {41},
     number = {6},
     doi = {10.1051/m2an:2007049},
     mrnumber = {2377104},
     zbl = {1141.78014},
     language = {en},
     url = {https://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  - https://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 https://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

[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). | Zbl | MR

[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). | Zbl | MR

[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). | Zbl | MR

[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). | Zbl | MR

[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). | Zbl | MR

[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). | Zbl | MR

Cité par Sources :