High order edge elements on simplicial meshes
ESAIM: Mathematical Modelling and Numerical Analysis (2007)
- Volume: 41, Issue: 6, page 1001-1020
- ISSN: 0764-583X
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topRapetti, Francesca. "High order edge elements on simplicial meshes." ESAIM: Mathematical Modelling and Numerical Analysis 41.6 (2007): 1001-1020. <http://eudml.org/doc/250061>.
@article{Rapetti2007,
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 expressed
only in terms of the barycentric coordinates of the simplex.
},
author = {Rapetti, Francesca},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Maxwell equations; higher order edge elements; simplicial meshes.; simplicial meshes},
language = {eng},
month = {12},
number = {6},
pages = {1001-1020},
publisher = {EDP Sciences},
title = {High order edge elements on simplicial meshes},
url = {http://eudml.org/doc/250061},
volume = {41},
year = {2007},
}
TY - JOUR
AU - Rapetti, Francesca
TI - High order edge elements on simplicial meshes
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2007/12//
PB - EDP Sciences
VL - 41
IS - 6
SP - 1001
EP - 1020
AB -
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.
LA - eng
KW - Maxwell equations; higher order edge elements; simplicial meshes.; simplicial meshes
UR - http://eudml.org/doc/250061
ER -
References
top- 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.
- M. Ainsworth and J. Coyle, Hierarchic finite element bases on unstructured tetrahedral meshes. Int. J. Numer. Meth. Engng.58 (2003) 2103–2130.
- 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.
- M.A. Armstrong, Basic Topology. Springer-Verlag, New York (1983).
- D. Arnold, R. Falk and R. Winther, Finite element exterior calculus, homological techniques, and applications. Acta Numer.15 (2006) 1–155.
- D. Boffi, M. Costabel, M. Dauge and L.F. Demkowicz, Discrete compactness for the hp version of rectangular edge finite elements. ICES Report04–29 (2004).
- A. Bossavit, Computational Electromagnetism. Academic Press, New York (1998).
- A. Bossavit, Generating Whitney forms of polynomial degree one and higher. IEEE Trans. Magn.38 (2002) 341–344.
- 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 Report03–48 (2003).
- 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.
- R. Hiptmair, Canonical construction of finite elements. Math. Comp.68 (1999) 1325–1346.
- R. Hiptmair, High order Whitney forms. Prog. Electr. Res. (PIER)32 (2001) 271–299.
- G.E. Karniadakis and S.J. Sherwin, Spectral hp element methods for CFD. Oxford Univ. Press, London (1999).
- J.M. Melenk, On condition numbers in hp-FEM with Gauss-Lobatto-based shape functions. J. Comput. Appl. Math.139 (2002) 21–48.
- P. Monk, Finite Element Methods for Maxwell's Equations. Oxford University Press (2003).
- J.C. Nédélec, Mixed finite elements in . Numer. Math.35 (1980) 315–341.
- 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.
- J. Schöberl and S. Zaglmayr, High order Nédélec elements with local complete sequence properties. COMPEL24 (2005) 374–384.
- J. Stillwell, Classical topology and combinatorial group theory, Graduate Text in Mathematics72. Springer-Verlag (1993).
- J.P. Webb and B. Forghani, Hierarchal scalar and vector tetrahedra. IEEE Trans. on Magn.29 (1993) 1495–1498.
- H. Whitney, Geometric integration theory. Princeton Univ. Press (1957).
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