2370IEEE TRANSACTIONS ON MAGNETICS, VOL. 38, NO. 5, SEPTEMBER 2002
Third-Order Nédélec Curl-Conforming
L. E. García-Castillo, Member, IEEE, A. J. Ruiz-Genovés, I. Gómez-Revuelto,
M. Salazar-Palma, Senior Member, IEEE, and T. K. Sarkar, Fellow, IEEE
Abstract—The third-order version of Nédélec’s first family of
curl-conforming elements over simplices is presented. Following
the definition of the element given by Nédélec, the third-order
vector basis functions are deduced. The elements thus ob-
tained exhibit some important differences with respect to other
higher-order curl-conforming elements appeared in the literature.
Among other features, the proposed third-order curl-conforming
finite element leads to better conditioned finite-element method
Index Terms—Curl-conforming element, finite element calcula-
tions, Nédélec elements, third-order element.
a wide variety of physical problems including electrostatics,
magnetostatics, eddy current problems and general electro-
magnetics. Specifically, vector finite elements have shown
to be the appropriate choice for the approximation of elec-
tromagnetic field quantities. Two families of vector elements
may be distinguished: div-confoming and curl-conforming
elements, providing continuity across element interfaces in the
sense of the divergence operator (i.e., normal continuity) and
curl operator (i.e., tangential continuity), respectively. Thus,
div-conforming elements are suitable for the discretization
of the electric and magnetic inductions,
curl-conforming elements are appropriate to approximate the
electric and magnetic fields,
Among the curl-conforming elements appeared in the litera-
ture, it is worth mentioning the so called mixed-order elements
proposed by Nédélec . Implementations of first and second-
order Nédélec elements have been proposed by the authors pre-
viously (e.g., see ). In this paper, the FEM implementation
of the third-order Nédélec element over two-dimensional (2-D)
HE finite-element method (FEM) has proven to be a
powerful and flexible tool for the numerical solution of
and , whereas
Manuscript received February 15, 2002; revised May 28, 2002. This work
was supported in part by the Ministerio de Ciencia y Tecnología of Spain under
Project TIC2001-1019 and Project TIC1999-1172-C02-01/02.
L. E. García-Castillo is with the Departamento de Teoría de la Señal
y Comunicaciones, Universidad de Alcalá, 28871 Madrid, Spain (e-mail:
A. J. Ruiz-Genovés and M. Salazar-Palma are with the Departamento de
Señales, Sistemas y Radiocomunicaciones, Universidad Politécnica de Madrid,
28040 Madrid, Spain (e-mail: firstname.lastname@example.org).
I. Gómez-Revuelto is with the Departamento de Ingeniería Audiovisual y
Comunicaciones, Universidad Politécnica de Madrid, 28037 Madrid, Spain
T. K. Sarkar is with the Department of Electrical Engineering and Com-
puter Science, Syracuse University, Syracuse, NY 13210 USA (e-mail:
Digital Object Identifier 10.1109/TMAG.2002.803577.
emphasized that the third-order element presented here, in con-
rigorously the element definition given in , leading, among
other features, to better conditioned FEM matrices.
II. THIRD-ORDER NéDÉLEC ELEMENT
The finite-element definition given in  is made in terms of
(dof) acting as linear functionals on that space of functions. In
in the 3-D context, i.e., the domain
dron. The third-order triangular element is easily obtained spe-
cializing the 3-D definitions to a face of the tetrahedron.
The space of functions for the third-order element is denoted
, which is the space of vector polynomials of order 3
that satisfy certain constraints (i.e., the so called Nédélec con-
straints). Thus, the polynomials of
be seen that the vector polynomials of
shown at the bottom of the next page.
Note that for a given polynomial of
efficients have to be determined, i.e., the dimension of
45. Thus, the number of degrees of freedom of the tetrahedral
element is 45, and 45 is also the number of vector basis func-
For the 2-D case, vector polynomials of
(1) but considering only the first two vector components and
discarding any term with the
coordinate, leading to 15 coeffi-
cients. Thus, the number of degrees of freedom of the triangular
element is 15.
It should be emphasized that in this paper the vector basis
are obtained from the element dof definitions. The
definition of the degrees of freedom of a finite element (that has
been neglected by other authors) is a fundamental issue in order
to understand how boundary conditions should be imposed. It
also allows to simplify given post-processes. Specifically, the
vector basis functions are obtained by imposing the interpola-
tory character of the basis functions with respect to the defini-
tion of the degrees of freedom [given in (3)–(5)], i.e.,
considered is a tetrahe-
are not complete. It may
are given by (1), as
, 45 independent co-
would be given by
for 3-D (or 15 for 2-D) (2)
the element. Note that (2) represents a system of equations for
basis function where the unknowns are the coefficients
Once the coefficients are obtained for
stands for the th functional defining the th dof of
0018-9464/02$17.00 © 2002 IEEE
GARCA-CASTILLO et al.: THIRD-ORDER NéDéLEC CURL-CONFORMING FINITE ELEMENT2371
functions are completely determined. It should be stressed that
the unisolvency of the finite element  implies that the basis
functions obtained following the above procedure are linearly
The definition of the
dof functionals of the third-order
Nédélec tetrahedral curl-conforming element is as follows: 18
dof associated to the six edges of the tetrahedron (i.e., 3 dof per
edge) given by
24 dof associated to the four faces of the tetrahedron (i.e., 6 dof
per face) given by
and 3 dof associated to the volume of the tetrahedron given by
for the space of polynomials of order
for the unit vector along the direction of the edge and
the unit vector normal to the face. It is worth noting that the dof
associated to the volume are internal to the element and, thus,
they will not be involved in the FEM assembly procedure. For
the triangular case and specializing the above dof definitions to
a face of the tetrahedron, 9 dof associated to the edges (3 dof per
edge) and 6 dof associated to the triangle area itself (internal to
the element) are obtained. Obviously, there are no degrees of
freedom of the third type (5).
It is worth noting that the previous definitions of degrees of
freedom are not ready to be used in a finite element code. They
the choice of a basis for each one of the polynomial spaces,
, appearing in (3)–(5). Thus, the practical FEM dof
definition is made in terms of momentums (of different order)
over the adequate components of the unknown . Specifically,
the components involved in the definitions of the degrees of
freedom associated to the boundary of the element [i.e., (3) and
(4)] are the components tangential to that boundary. Thus, the
tangential continuity between elements may be easily imposed
in the FEM assembly procedure. However, care must be taken
with the local definitions of the vector quantities involved in the
dof definition, specially when the basis functions are obtained
in the parent element. Namely, unit vectors along the directions
of the edges, , normal to the faces,
and the directions of the
be adequately chosen.
It is worth noting that some freedom exists for the choice of
the basis for
,, . An example of a basis for
over the faceis the choice of the monomial basis (1, ,
) for each one of the two components of the polynomial vec-
built with the Lagrange polynomials (
erarchical face vector basis functions may be obtained by the
choice of a suitable hierarchical basis of polynomials in
(e.g., Legendre polynomials). In addition, some directions must
be chosen for the vectors
in its volume.
An alternate definition for the dof associated to the edges (3)
made in terms of the value of the component of the unknown
tangential to the edges ingiven points (nodes) along the edges
may be used
of dof associated to faces, should
, , ). Also, hi-
over each face of the tetra-
This alternate definition for the edge dof is a consequence of
a property satisfied by the polynomial vectors
edges a second order polynomial, this polynomial is completely
determined by 3 momentums of up to order 2, but also by its
value in 3 given points. This alternate definition corresponds
to (6) where a weight
and the length of the edge
been added for normalization purposes. Freedom exists in the
location of the nodes.
The authors have focused on Lagrange bases for
and the edge dof definition of (6) with nodes located at the three
Gauss–Legendre integration points, which provide maximum
cationis alsoadvantageousfor givenpost-processes.Theparent
element of the third-order tetrahedral curl-conforming element
presented here is shown in Fig. 1. It corresponds to the choice
of orthogonal directions for vectors
and Gauss–Legendre points for the nodes on edges. The parent
element of the third-order triangular version is obtained by con-
sidering any of the four faces of the tetrahedron.
Plots of one of the edge basis function and one of the face
plots (and also the results shown in Table I) correspond to basis
functions obtained considering a Lagrange basis for
Two dimensional plots are shown for clarity purposes, specifi-
cally the plots show the component tangential to the face
Thus, theyalso correspond toexamples ofplots ofthe two types
of triangular basis functions (edge and area—internal—basis
2372 IEEE TRANSACTIONS ON MAGNETICS, VOL. 38, NO. 5, SEPTEMBER 2002 Download full-text
Fig. 1.Third-order tetrahedron (parent element).
component to the face shown—. (a) ? . (b) ?
Examples of edge and face third-order basis functions—Tangential
Fig. 3.Example of volume third-order basis functions (? ?
functions). A plot of one of the volume basis functions of the
tetrahedral element is shown in Fig. 3.
CONDITION NUMBER COMPARISON
It is worth noting that the basis functions of the third-order
element presented here are different from those proposed in the
literature, which in most cases have been derived by inspection
of the properties of polynomials in simplex coordinates, rather
than from Nédélec’s principles.
III. RESULTS AND CONCLUSION
A comparison of the condition number provided by the basis
functions of the tetrahedral element presented in this paper
and some other implementations appeared in the literature
for tetrahedra is shown in Table I. It refers to the condition
number (defined as the ratio of the maximum and minimum
eigenvalue) of the matrices corresponding to the inner products
of the vector basis functions in a finite element and of their
curls, i.e., element mass matrix
and stiffness matrix
in FEM terminology, respectively. Specifically, Table I shows
the condition number of the preconditioned matrices
and obtained by normalizing the basis functions, i.e.,
is a diagonal matrix with
obtained discarding the zero eigenvalues of the stiffness matrix.
It is worth noting that analogous results are obtained with FEM
assembled matrices provided that a diagonal preconditioning is
performed (a common procedure for any FEM practitioner). In
order to show how the condition number varies with the aspect
ratio of the elements, Table I compares results corresponding
to the parent element of Fig. 1 and two elongated elements
(obtained by stretching the parent element in the
a factor of 4 and 8).
As it may be seen in Table I, the condition number corre-
sponding to the third-order element proposed in this paper is
significantly lower than for the other third-order elements, thus
providing a beneficial result in the solution stage of the large
system of equations arising from FEM discretization.
). The results have been
 J. C. Nédélec, “Mixed finite elements in ? ,” Numerische Mathematik,
vol. 35, pp. 315–341, 1980.
 M. Salazar-Palma, T. K. Sarkar, L. E. García-Castillo, T. Roy, and A. R.
Djordjevic, Iterative and Self-Adaptive Finite-Elements in Electromag-
netic Modeling. Norwood, MA: Artech House, 1998, p. 808.
 A. F. Peterson, S. L. Ray, and R. Mittra, Computational Methods for
Electromagnetics.New York: IEEE Press, 1998.
 L. S. Andersen and J. L. Volakis, “Hierarchical tangential vector finite
elements for tetrahedra,” IEEE Microwave Guided Wave Lett., vol. 8,
pp. 127–129, Mar. 1998.
 R. D. Graglia, D. R. Wilton, and A. F. Peterson, “Higher order inter-
polatory vector bases for computational electromagnetics,” IEEE Trans.
Antennas Propagat., vol. 45, pp. 329–342, Mar. 1997.