Acta Numerica (2002), pp. 237–339
c
Cambridge University Press, 2002
DOI: 10.1017/S0962492902000041 Printed in the United Kingdom
Finite elements in computational
electromagnetism
R. Hiptmair
Sonderforschungsbereich 382,
Universit¨at T¨ubingen,
D-72076 T¨ubingen, Germany
E-mail: ralf@hiptmair.de
This article discusses finite element Galerkin schemes for a number of lin-
ear model problems in electromagnetism. The finite element schemes are in-
troduced as discrete di↵erential forms, matching the coordinate-independent
statement of Maxwell’s equations in the calculus of di↵erential forms. The
asymptotic convergence of discrete solutions is investigated theoretically. As
discrete di↵erential forms represent a genuine generalization of c onventional
Lagrangian finite elements, the analysis is based upon a judicious adaptation
of established techniques in the theory of finite elements. Risks and difficulties
haunting finite element schemes that do not fit the framework of discrete dif-
ferential forms are highlighted.
CONTENTS
1Introduction 238
2Maxwell’sequations 240
3 Discrete di↵erential forms 255
4Maxwelleigenvalueproblem 291
5Maxwellsourceproblem 306
6Regularizedformulations 318
7Conclusionandfurtherissues 327
References 328
Appendix: Symbols and notation 337
See also
HIP02_notes.xoj
238 R. Hiptmair
1. Introduction
Most modern technology is inconceivable without harnessing electromag-
netic phenomena. Hence the design and analysis of schemes for the approx-
imate solution of electromagnetic field problems can claim a rightful place as
a core discipline of numerical mathemati c s and scientific c o m p u ti n g . How-
ever, for a long time it received far less attention a m o n g numerical analysts
than, for instance, computational fluid dynamics and solid mechanics.
One reason might be that electromagnetism is described by a generically
linear theory, in the sense that linear equa t i o n s arise from basic physical
principles. This i s in stark contrast to continuum mechanics, where linear
models only emerge through linearization of inherently nonlinear governing
principles. Being linear, the fundamental laws of electromagnetism might
have struck many mathematicians as ‘dull’. This view might also have been
fostered by the misconc e p t ion that electro m a g n etism basically boils down
to plain second-order elliptic equations, which have been amply studied and
are well understood.
It is one objective of this survey article to refute the idea that one can
cope with el e c t r o m a g n e t i c s once one knows how to solve Laplace equati o n s
numerically. I aim to convey the richness in subtle mathematical features
displayed by apparently ‘simple’ pr o b lems in computational electro m a gnet-
ism. The problems I have in mind arise from the spatial discretization of
electromagnetic fields by means of finite elements. Yet I will not settle for
merely specifying and describing the finite element spaces. To gain insight,
acomprehensiveviewismandatory,encompassingthestructuralaspects
of the physical model, a thorough knowledge of function spa c e s as well as
familiarity with classical finite element techniques. All these issues will be
addressed in the paper, and an attempt is made to convince the reader that
understanding all of them is necessary for succes s f u l l y tackling electromag-
netic field problems.
Many readers might object t o my regular delving into technical details. In
my opinion, major breakt h r o u g h s in computa t i o n a l electromagnetism have
often been brought about by successfully addressing technical issues. This
neatly fits my desire to embrac e a formal ‘rigorous’ treatment. Therefore,
space permitting, and in order to make the article self-contained, I will
not skip proofs. Yet, someti m e s I will put forth ‘views’ – even at the risk
of sounding fuzzy and arcane – in a possibly doomed attempt to inspire
‘intuitive understanding’.
Plenty of references to original papers and related work will be given. Of
course, they can never be exhaustive and will reflect my personal biases and
history. In particul ar , s c or e s of engineering public a ti o ns that a d dr e s s issues
also covered in t his article could be cited, but will not be mentioned. My
Finite elements in computational electromagnetism 239
emphasis on theory i s reflected by the almost complete absence of numerical
results. They can be found in abundance in research papers.
Iwaspleasedtowitnessasurgeinresearchactivitiesintomathematical
aspects of computational electromagnetism in recent years. Now the field is
rapidly evolving, which means that this article can hardly be more than a
snapshot of the knowledge as of 2001. Many of the results covered are likely
to experience significant improvement and extension in years to come. It
also means that there is much left to be done. In a sense, I will not balk
at stating incomplete results and even conjectures. M aybe this will trigger
some fresh research.
Even with a focus on finite eleme nt schemes, all that can be covered in a
survey article are model problems. Admittedly, the y fall way short of match-
ing the complexity of typical engineering applications. For instance, in light
of the linear nature of electromagnetism, I will complet ely restrict my at -
tention to linear problems, that is, only simpl e ‘linear’ materials will be con-
sidered. In this setting it is possible to skirt any issues of temporal discret-
ization by switching to the frequency domain:allquantitiesaresupposedto
show a sinusoidal dependence on time with a fixed an g u l a r frequency !>0.
Thus, t h a n ks to linearity, temporal deri vation @
t
can be replaced by the
multiplication operator i!·.Thisconvertsallequationsintorelationships
between complex amplitudes depending on space only. If u = u(x), x 2 R
3
standing for the independent space variable, is such a complex amplitude,
the related physical quantity u can be recovered through
u(x,t)=Re(u(x) · exp(i!t)).
The classical notion of finit e elements is tied to bounded computat i o nal do-
mains. Yet many central problems in computational electromagnetism are
posed on unbounded domai n s . The most promi n e nt example is the scatter-
ing of electromagnetic waves. Not all of th ese problems will be full y treated
in this articl e . Still, when combined w i t h other techniques, for instance
boundary element methods, finite elements can play an important role even
in these cases. Thus the results reported in th i s article remain of interest.
Instead of an outline, I am only listing a few points of view that I embrace.
They can o↵er guidance when negotiating through this articl e .
• In order t o discretize the fundamental laws of electromagneti s m prop-
erly, i t is important to appreciate their link with di↵erential geometry
and algebraic topology (cohomology theory).
• There is a close relationship between second-order elliptic equations
and the governing equations of electromagnetism, but the lack of strong
ellipticity introduces subtle new challenges.
• Suitable finite elements for electromagnetic fields should be introduced
and understood as discrete di↵erential forms.
240 R. Hiptmair
• Discrete di↵erential forms are a generalization of H
1
(⌦)-conforming
Lagrangian finite elements. Th e i r analysis can often use and adapt the
tools devel o ped for the latter.
• Finite elements that lack an interpretation as discrete di↵erential forms
have to be used with great care.
2. Maxwell’s equations
The fundamental governing equations of electromagnetism are Maxwell’s
equations. Mathematicians usually encounter them i n the form of the two
first-order partial di↵erential equations
Faraday’s law: curl e = i! b,
Amp`ere’s law: curl h = i! d + j,
(2.1)
posed over all of affine space A(R
3
). The equations link (the complex amp-
litudes of) the electric field e,themagneticinductionb,themagneticfield
h, and the displacement current d.Here,j denotes a (formal) excitation
supplied by an imposed current. The equations have to be supplemented by
the material laws (also called constitutive laws)
d = ✏e, b = µh, (2.2)
where the dielectric tensor ✏ and the magnetic permeability tensor µ are
usually introduced as L
1
-functions m apping into the real symmetric, posit-
ive definite 3 ⇥ 3matricessuchthat
min
(✏(x)) >✏
0
> 0 and
min
(µ(x)) >
µ
0
> 0almosteverywhere.Suchmatrix-valuedfunctionswillbereferredto
as metric tensors in the following. If good conductors are involved, a part
of the source current may be given through Ohm’s law
j = e + j
0
, (2.3)
where stands for the symmetric, positive semi-definite conductivity tensor,
yet another metric tensor.
2.1. Fields and forms
Is there more to the unknowns of (2.1) than being plain vector-fields w it h
three components? To a n s wer this quest i o n , it is useful to remember the
physicists’ favourite way of writing Maxwell’s equations, namely the integ-
ral form :
Faraday’s law:
R
@⌃
e · ds = i!
R
⌃
b · n dS,
Amp`ere’s law:
R
@⌃
h · ds = i!
R
⌃
d · n dS +
R
⌃
j · n dS.
(2.4)
Finite elements in computational electromagnetism 241
This is to hold for any bounded, two-dimensional, piecewise smooth sub-
manifold ⌃ of A(R
3
), equipped with oriented
1
unit normal vector-field n.
First, the integral form (2.4) reveals that the fields e, h and b, d have an
entirely di↵erent nature, as Maxwell remarked in his ‘Treatise on Electricity
and Magnetism’ (Maxwell 1891, Chapter 1):
Physical vector quantities may be divided into two classes, in one of which the
quantity is defined with reference to a line, while in the other the quantity is
defined with reference to an area.
Laconically speaking, electromagnetic fields are an abstraction for associat-
ing ‘voltages’ a nd ‘fluxes’ to directed paths and oriented surfac e s; they are
integral forms in the sense of the following definition, which is deliberately
kept fuzzy because it targets some ‘intuitive concepts’.
2
Definition 1. An integral form of degree l 2 N
0
,0 l n, n 2 N,
on a piecewise sm ooth n-dimensional manifold M is a continuous
3
additive
mapping from the set S
l
(M)ofcompact,oriented,piecewisesmooth,l-
dimensional sub-manifolds of M into the com p l e x numbers. These so-calle d
integral l-forms on M form the vector space F
l
(M) (which is t o be trivial
for l<0orl>n).
Here, by ‘additive’, we mean that the i ntegral form assigns the sum of
the respective numbers to the union of dis j o i nt sub-m a n i f o l d s . Further,
flipping the orientation of a sub-manifold should change the sign of the
assigned value. This is what we should expect from the integrals occurring
in (2.4). Therefore the evaluation !(⌃), ! 2F
l
(M), ⌃ 2S
l
(M)isdubbed
‘integrating ! over ⌃’, in symbols
R
⌃
!.Now,bymerelylookingat(2.4),we
identify e and h as integral 1-forms, whereas b, d, and j should be regarded
as integral 2-forms.
We are accustomed to referring to the ‘field at a point in space’, that is, a
local perspective. Measurement procedures adopt it : measuring an e l e c t r i c
field amounts to det ermining the virtual work
w = q e(x) · x (2.5)
needed for the tiny displacement x of a test charge q at x,with· desig-
nating the inner product in Euclidean space R
3
.Themagneticinductionis
1
Taking for granted an orientation of the ambient space A(
3
), we need not distinguish
between interior and exterior orientation of manifolds.
2
It is the subject of geometric measure theory to come up with a more rigorous ap-
proach. See Morgan (1995) for an introduction and Federer (1969) for a comprehensive
exposition.
3
Continuity refers to a sort of ‘deformation t opology’ on sets of piecewise smooth man-
ifolds.
