A posteriori error indicators for Maxwell’s Equations
Peter Monk∗
Department of Mathematical Sciences, University of Delaware, 501 Ewing Hall, Newark, DE 18716-2553, USA
Received 29 May 1998
Abstract
In this paper we shall analyze a class of a posteriori error indicators for an electromagnetic scattering problem for Maxwell’s equations in the presence of a bounded, inhomogeneous and anisotropic scatterer. Problems of this type arise when computing the interaction of electromagnetic radiation with biological tissue. We briey recall existence and unique-ness theory associated with this problem. Then we show how a posteriori error indicators can be derived using an adjoint equation approach. The error indicators use both the jump in normal and tangential components of the eld across faces in the mesh. c1998 Elsevier Science B.V. All rights reserved.
Keywords:Maxwell equations; Residual error indicators; Finite elements
1. Introduction
Suppose a bounded, anisotropic and inhomogeneous body is illuminated by a time-harmonic in-cident electromagnetic wave. The inin-cident eld will interact with the inhomogeneity to produce a scattered eld. The total eld, which is the sum of the incident and scattered elds, satises Maxwell’s equations and the approximation of this eld (or equivalently the scattered eld) will be the subject of this paper. Such problems arise, for example, in computing the interaction of electromagnetic radiation with biological tissue. A schematic of the scattering problem is shown in Fig. 1.
To write down the equations satised by the elds, we need some denitions. We suppose that the medium is characterized by a permittivity ”=”(x), a permeability =(x) and a conductivity =(x). All these are assumed to be uniformly bounded, symmetric matrix functions of position, and in addition ” and are supposed to be uniformly positive denite while is positive semi-denite. For simplicity, in this paper, we shall assume that all the coecients are smooth functions
∗E-mail: [email protected]
Fig. 1. A schematic of the scattering problem showing the bounded scatterer. A known incident eld interacts with the scatterer and produces a scattered eld.
of position (this can be relaxed considerably see [19, 17, 26]). Of course, in practice, discontinuous coecients are of considerable interest.
Since the inhomogeneity is bounded, we assume that there are positive real constants b, ”0 and
0 such that
”(x) =”0I; (x) =0I and(x) = 0
if |x|¿b where I is the identity matrix.
We wish to compute time harmonic solutions with frequency !. We dene the wave-number k and functions (x) and n(x) by
k=√”00!; (x) =0
−1(x); n(x) =”−1 0
”(x) + i(x) !
:
The known incident electric eld is denoted by Ei and is assumed to satisfy the homogeneous
Maxwell system
∇ × ∇ ×Ei−k2Ei= 0 inR3: A typical choice for Ei is the plane wave
Ei= (d
×p)×d exp (ikx· d)
where d is a unit vector describing the direction of propagation and p is the constant polarization vector (see [8]). More generally we might be interested in incident elds from point sources. These incident elds can also be accommodated by the theory provided the source is outside a sphere containing the scatterer.
If we denote by E the total electric eld and Es the scattered electric eld, then E satises the
Maxwell system:
∇ ×∇ ×E − k2nE= 0 in R3: (1.1a)
The incident, scattered and total elds are related by
and the following condition at innity is needed to complete the problem
lim
r→∞((∇ ×E
s)
×x−ikrEs) = 0 (1.1c)
where r=|x|. Eq. (1.1c) is the Silver–Muller radiation condition which guarantees that the scattered eld is outgoing.
To approximate (1.1) by nite elements we need to reduce the problem to a problem on a bounded domain. We adopt the Dirichlet to Neumann map approach (see [17] for details). Let BR be the ball of radius R centered at the origin with R ¿ b and let R=@BR. For a given tangential eld on
R). For the simple case of a spherical articial boundary considered here a series expansion for u is well known (see for example [8]). Using this expansion we can see that
ˆ
Since we shall need the expansion later, we summarize it now (see [17] for details). Let {Ym n( ˆx)}, m=−n;· · ·; n be an orthonormal sequence of spherical harmonics of order n on the unit sphere, normalized such that hYm
n ; Ym
Here ∇ is the gradient on the unit sphere. On the sphere of radius R these functions are not orthonormal, but only orthogonal (with a factor of R2 for the nonzero inner products).
Given ∈H−1=2(Div;
R) we can show that
Ge: = ˆx×Hs= ∞ X
n=1
n X
m=−n
−ikRbn; m n
Unm+an; mn ikR V
m n
: (1.5)
where
n=kR (h(1)
n ) ′
(kR) h(1)n (kR)
+ 1: (1.6)
Proceeding formally, we can use Ge to reduce (1.1) to a problem on a bounded domain. For two vector elds u and C we dene
(u;C) =
Z
BR
u·C dV and hu;Ci=
Z
R
u·C dA
(we will also use h·;·i to denote suitable duality pairings on R). Let ˆx=x=r (this is the unit outer normal to BR). Then using (1.1b) in (1.1a) we have
∇ ×∇ ×Es
−k2nEs=f
where f =−(∇ ×∇ ×Ei−k2nEi) (note that f = 0 if |x|¿ b). Now multiplying this equation by
a smooth test function, integrating over BR and using integration by parts we obtain
(f;)
=(∇ ×∇ ×Es;)−k2(nEs;
)
=(∇ ×Es;∇ ×)−k2(nEs;) +hxˆ×∇ ×Es;( ˆx×)×xˆi: (1.7)
Using Ge and the fact that Es satises the homogeneous Maxwell equations (1.2a) outside BR and the radiation condition (1.2c) we know that
ˆ
x×∇ ×Es=ikGe( ˆx×Es):
We are now ready to state a precise variational formulation for the Maxwell system (1.1) that we shall study in the remainder of the paper. Let
H(curl;BR) = n
u∈ L2(BR) 3
| ∇ ×u∈ L2(BR) 3o
;
then we seek Es∈H(curl;B
R) such that
(∇ ×Es;∇ ×)−k2(nEs;)
+ikhGe( ˆx×Es);( ˆx×)×xˆi= (f;) (1.8)
for all ∈H(curl;BR). Note that if a function u∈H(curl; BR), then ( ˆx ×u)×xˆ is in the dual space of H−1=2(Div;
R) so (1.8) is well dened (see [17] for details).
Keller–Givolli scheme approximatesGe using a series expansion [14]. The approach of [11] is some-what dierent and replaces the term Ge( ˆx×Es) by an integral representation of the eld computed from data within BR. All these techniques involve a nonlocal boundary operator which complicates implementation [16].
2. Continuous problem
The analysis of (1.8) is complicated by three factors. First, in common with many scattering problems, the bilinear form is not coercive due to the term −k2(nEs;). Second the bilinear form (∇ ×Es;∇ ×) is not coercive on H(curl;B
R) and nally n is a complex symmetric matrix func-tion. These problems can be handled by a combination of standard tricks which we shall outline in this section.
The rst step in the analysis of (1.8) is to prove uniqueness of solutions of (1.1). Most uniqueness results to date are for scalar coecients. For real matrix valued coecients [19] proves uniqueness if the coecients are twice continuously dierentiable. More recently Hazard [11] has proved unique-ness if,and are piecewise Lipschitz scalar functions of position. Potthast [26] proves uniqueness for complex, matrix values n(x) and it is his theorem we quote:
Theorem 2.1. (Potthast [26]). Suppose =0 and assume n(x) is three times continuously dier-entiable (and satises the other conditions outlined in the introduction) then there is at most one solution to the Maxwell problem (1:1).
Obviously there is still some room for improving the uniqueness result for matrix valued coe-cients to include less smooth coecoe-cients (piecewise analytic for example) and variable .
Once uniqueness is veried, the analysis of Kirsch and Monk [17] shows that (1.8) has a unique solution. Unfortunately [17] has an error which was corrected in a later notice [15]. The basis of the corrected argument is to write the problem as an operator equation to which the Fredholm alternative theorem applies. In doing this it is useful to factor out the null-space of the curl operator. This can be done from the point of view of mixed methods [1] but we prefer to use the classical Helmholtz decomposition, which must be extended to allow for a complex valued weight matrix n(x) and boundary operator. We dene the spaces
Y : =
∇p:p∈H1(B
R) ;
Y+: =
u∈H(curl;BR) :−k2(nu;∇q) +ikhGe( ˆx×u);∇ qi= 0
∀q∈H1(BR) (2.1)
= {u∈H(curl;BR): ∇ ·(nu) = 0 in BR and
k2xˆ·u=ik∇ ·Ge( ˆx×u) on R : (2.2)
and show that
First, we determine p∈H1(B
R)=R with
a(∇p;∇q) =b(∇q) for all q∈H˜1(BR); (2.3)
which is exactly the problem of determining p∈H1(B
R)=R such that
−k2(n
∇p;∇q) + ik
R hGe( ˆx× ∇p);∇ qi= (f;∇q) (2.4)
for allq∈H1(B
R)=R. This problem can be uniquely solved as was shown in [17]. Having determined p complete the determination of E by solving w∈Y+ from the equation
a(w; ) =b( )−a(∇p; ) for all ∈Y+: (2.5)
The factor 1=R arises because
∇q= 1
R∇ q+ @q @xˆxˆ:
Assuming that this problem has a unique solution, the desired eld is E=w+∇p.
To determine w we use the Fredholm theory. We split the bilinear form into a= ˆa1+ ˆ1 where
ˆ
a1(u;C) = (∇ ×u;∇ ×C) + (nu;C) +ikhGe2( ˆx×u);C⊤i; ˆ
1(u;C) =−(k2+ 1) (nu;C) +ikhG1e( ˆx×u);C⊤i:
where
Ge = ∞ X
n=0
n X
m=−n "
−ikRbnm n
umn + anm(n− ˜ n)
ikR V
m n
#
+ 1 ikR
∞ X
n=0
n X
m=−n
anm˜nVnm
= :G1
e+Ge2: (2.6)
and
˜ n=kR
(h(1)
n ) ′(iR)
h(1)n (iR)
+ 1: (2.7)
Now, ˆa1 is coercive over Y+ and ˆ1 denes a compact operator on the same space. Fredholm
theory can then be applied [17] (using the uniqueness result) to prove the following theorem:
Theorem 2.2. (Kirsch and Monk [17]). Under the hypotheses of Theorem (2:1) and assuming
f ∈(L2(B
R))3; there exists a unique solution to (1:8).
3. Finite element approximation
A nite element discretization of (1.8) now proceeds by constructing a nite element subspace of H(curl;BR). There are many possibilities. For example it is possible to use standard continuous elements (see for example [18, 6, 3, 9]) although explicit “ellipticization” of the bilinear form is often felt necessary (this consists of adding divergence terms to the bilinear form). This approach works well if n is continuous (a similar approach computing with the magnetic eld can be applied if is continuous). We will instead describe the use of edge elements to discretize (1.8) which have the advantage of handling, in principle, discontinuous , and without modication of the bilinear form (although the theoretical analysis of problems with discontinuous coecients is not covered by this paper). We shall use the edge elements of [24]. A second family of edge elements due to [23, 25, 22] could also be used.
Suppose that BR is meshed using a regular and quasi-uniform tetrahedral mesh denoted h (with curvilinear tetrahedra near the outer boundary R [10]). Suppose the maximum diameter of the elements in the mesh in h. We shall describe only the lowest order edge element space (due to [24]). These elements are also referred to as Whitney elements after on earlier use in geometry. In the brief description of these elements given next we shall ignore the curvilinear nature of the mesh near R. There are a number of ways of handling the curved domain. Since the boundary condition is natural there, we can simply allow curvilinear elements (as in [21]) or map the elements to t the boundary [10]. Both these methods of handling the boundary yield spaces that have the properties and estimates used in this paper.
For an element K∈h let
RK=
|(x) =aK+bK×x; aK;bK∈C3
then the nite element subspace Vh⊂H(curl;BR is given by
Vh={uh∈H(curl; BR)|uh|K∈RK ∀K∈h}:
Note that Vh does not contain all linear polynomials. The degrees of freedom for Vh are dened elementwise [24] by
K= Z
e
u·ds|e an edge of K with unit tangent vector
:
Since the degrees of freedom are associated with edges, the elements are called edge elements (Fig. 2).
Using these degrees of freedom it is possible to construct an interpolation operator on functions in (H2(B
R))3 where H2(BR) is the Sobolev space of twice dierentiable functions in the L2 sense (a
weaker space, but not (H1(B
R))3, is possible). The interpolation operator rh has the following error estimate:
ku−rhukH(curl;BR): =
q
ku−rhuk2L2(BR)+k∇ ×(−rhu)k2L2(BR)
Fig. 2. Here we depict the degrees of freedom of the linear edge element. The degrees of freedom are the average values of the tangential component of the eld along each edge.
The discrete problem is to nd Es
h∈Vh such that
(∇ ×Ehs;∇ ×h)−k2(nEhs;h)
+ikhGe( ˆx×Ehs);( ˆx×h)×xˆi= (f;h) (3.2)
for all h∈Vh and we can prove the following optimal estimate [17]:
Theorem 3.1. The discrete solution Es
h∈Vh is well dened for all h suciently small and
kEs −Es
hkH(curl;BR)6ChkE
s kH2(BR):
4. A Posteriori error estimates
In practice we would like to have a posteriori error estimates to help with selective mesh rene-ment. Note that there is a limit on the maximum diameter of the elements in a grid imposed by the non-coercivity of the bilinear form (or the oscillatory nature of the solution). This requires thathk be suciently small for the interpolant to have reasonable accuracy (although even this is not sucient for accuracy [4, 12]). However selective mesh renement to deal with changes in parameters or geometric features may still be needed to guarantee accuracy.
Theorem 4.1. Under the conditions of this paper there is a constant C independent of Es, Es
1. For higher-order edge schemes the weighting factors for the rst three terms on the right-hand side would change due to the superior approximation properties of the discrete spaces (higher powers of h could be used). The weighting of the remaining three terms would not change because of the regularity property used in the proof.
2. The theorem shows that it is necessary to use both jumps in the tangential and normal components (suitably weighted by n) across element faces. This agrees with the analysis of Beck et al. [5] where they analyze error estimators for the coercive constant coecient problem on a bounded domain that arises in the numerical solution of the time dependent Maxwell equations in a cavity. In this case they are able to provide asymptotically exact estimators.
3. The constants in this error estimate are dicult to estimate and depend on k. It is likely that as k increases the constants increase.
The adjoint approach to a posteriori error estimation rst denes a suitable adjoint solution driven by the actual error in the solution. Let z∈H(curl;BR) satisfy
a(;z) = (n;Es−Ehs); ∀∈H(curl;BR): (4.1)
Lemma 4.2. There exists a unique solution z to(4.1). Furthermorez=u+∇swhereu∈H(curl;BR)
Furthermore s satises the following a priori estimate:
k∇skL2(BR)6CkEs−Es
hkL2(BR): (4.2)
In addition, if the coecients in and are smooth functions of position
kukH2(BR)6CkEs−EhskL2(BR): (4.3)
Remark. The reason for splitting z into two parts is to obtain a smooth part u and a correction ∇s. This allows the proof of the a priori estimate (4.3). A glance at the proof shows that all that is really needed is that and are smooth enough for the a priori estimate
X
K∈h
kuk2H2(K)6CkEs−Ehsk2L2(BR)
to hold.
To prove this lemma we rst need to study the adjoint of the operator Ge denoted G∗e. To do this we need to recall that the space
H−1=2(Curl;
Lemma 4.3. The operator G∗
e is a bounded operator from H
and h(2)
n is the spherical Hankel function of second type. Furthermore G ∗
e may be characterized
as follows. Let z satisfy
∇ × ∇ ×z−k2z= 0 in R3\R; (4.4)
zT= on R; (4.5)
(∇ ×z)×x+ikrzT →0 as r→ ∞ (4.6)
where zT = ( ˆx×z)×xˆ then Ge∗= (1=ik)(∇ ×u)T on R.
Proof of Lemma 4.3. G∗
e is dened by duality as follows (using the orthogonality of the surface basis functions (the factor of R2 comes from the ratio of the area of
R to the area of the unit
This proves the series denition of G∗
e since (2)n =n. But if ∈H−1=2(Curl; R),
This proves the claimed boundedness of the operator. The proof of the representation of G∗
e in terms of the solution of a scattering problem can also be done directly by a series argument. Let the volume basis functions Mm
n and Nnm be dened by
Mnm(x) =∇ ×x(2)n (k|x|)Ynm( ˆx); and Nnm(x) = 1
then following [8] the solution z of (4.4) and (4.6) can be written as
we deduce from (4.5) that
n; m=−
restricting to R and using (4.7) and (4.8) again proves the representation of G∗e.
Proof of Lemma 4.2. Using the denition of z (4.1) z∈H(curl;BR) satises
(∇ ×;∇ ×z)−k2(n;z) +ik ¡ Ge( ˆx×;( ˆx×)×xˆ¿=(n;Es−Ehs)∈H(curl;BR):
Hence via the denition of adjoint operator G∗
e, and the fact that is real symmetric, we can see that z is the weak solution of the Maxwell system:
∇ ×∇ ×z−k2nz=n(Es−Ehs) in BR; (∇ ×z)T =ikG
∗
e(zT) on R: Hence, using the characterization of G∗
e in Lemma 4.3, we see that z=w|BR where w is the weak
solution of the scattering problem:
Taking the complex conjugate of this equation reduces us to a standard scattering problem as studied in Section 2. Hence the existence of a unique z is veried by the theory outlined in the previous section, and the estimate (4.2) follows from the same analysis.
The eld u (extended to all of R3) satises
∇ ×∇ ×u−k2nu=F inR3; lim
r→∞((∇ ×u)×x+ikru) = 0; where F∈L2(B
R) is dened by
(n ;F) = (n ;(Es−Ehs))−a( ;∇s); ∀ ∈H(curl;BR)
and is extended to R3 by zero. Note that if we choose =∇ for any ∈H1(B
R)=R we have (n∇;F) = 0 for any ∈H1(B
R)=R so that ˆx·F= 0 on R and so ∇ ·(nF) = 0 in all of R3. Thus
u satises a standard scattering problem and because nF is divergence free (see [19, p. 168]):
kukH2(BR)6CkFkL2(BR)6CkEs−EhskL2(BR): (4.9)
Proof of Theorem 4.1. Using Lemma (4.2), we know that there is a unique solution z∈H(curl;BR) to the adjoint problem
a(;z) = (n;(Es −Es
h)); ∀∈H(curl;BR): (4.10)
Furthermore
z=u+∇s
where s∈H1(B
R)=R’s and z∈H(curl;BR), Thus, choosing =Es−Es
h in (4.10) and using the fact that Es−Ehs satises the nite element “orthogonality” relation
a(Es−Ehs;h) = 0; ∀h∈Sh;
we have
(n(Es−Ehs);(Es−Ehs)) =a(Es−Ehs;u) +a(Es−Ehs;∇s)
=a(Es−Ehs;u−uh) +a(Es−Ehs;∇(s−sh)); (4.11)
for any uh∈Vh and any sh∈Sh.
We shall estimate the error in two parts. We start with the rst term on the right-hand side of (4.11). Writing the inner product as a sum of element-wise inner products, then using the fact that Es satises (1.8), and nally integrating by parts on each element we have
a(Es−Ehs;u−uh) = (∇ ×(Es−Ehs);∇ ×(u−uh))−k2(n(Es−Ehs);(u−uh)) +ikhGe( ˆx×(Es−Ehs));( ˆx×(u−uh))×xˆi
= (f;u−uh)−(∇ ×Ehs;∇ ×(u−uh)) +k2(nEhs;(u−uh))
= quantity across a face f, we may rewrite (4.12) as
a(Es
Next we chooseuh=rhu (this is possible sinceu∈H2()). Then using the Cauchy–Schwarz inequal-ity and the error estimate for u−rhu from (3.1) together with a similar estimate for the interpolant on the faces of the mesh, we get
Next we use the following estimate which is valid for any q∈H1(K
f) with constantC independent of the mesh parameter hK (for a regular mesh)
kqk2L2(f)6C kqkL2(Kf)k∇qkL2(Kf)+
where Kf is a tetrahedron with f as one face. With this estimate we have
kukH1(f)6Ch
−1=2
Kf kukH2(Kf):
Using this estimate in (4.13) followed by the a priori estimate (4.9) provides the following estimate for |a(Es−Es
The next step is to estimate the error ina(Es −Es
h;∇(s−sh)). This is the error due to approximating the divergence of the electric eld. Again using (1.8), rewriting the integrals element-wise and integrating by parts we obtain
Next we integrate by parts on the boundary R and rewrite the face integrals by grouping the integrals over the face of each element:
a(Es
Iff has jumps in the normal component across element faces (i.e. does not have a global divergence), this would introduce some factors n·f in the face integral terms.
For the exact eld, if we take the divergence of (1.1) we see that −k2
∇·(nEs) =
∇·f. In addition k2xˆ·Es=ik∇ G
e( ˆx×Es) which just species continuity of the normal component of Es across R. So the above equality just uses the residuals of these terms to estimate the gradient error.
Using the fact that n=I on R, the Cauchy–Schwarz inequality, (4.14) and estimates for the interpolant we obtain
Acknowledgments and disclaimer
Eort of Peter Monk sponsored by the Air Force Oce of Scientic Research, Air Force Mate-rials Command, USAF, under grant number F49620-96-1-0039. The US Government is authorized to reproduce and distribute reprints for governmental purposes notwithstanding any copyright nota-tion thereon. The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the ocial policies or endorsements, either expressed or implied, of the Air Force Oce of Scientic Research or the US Government.
References
[1] T. Abboud, Electromagnetic waves in periodic media, in: R. Kleinman et al. (Ed.), Proc. 2nd Int. Conf. on Mathematical and Numerical Aspects of Wave Propagation, SIAM, 1993.
[2] T. Abboud, J.C. Nedelec, Electromagnetic waves in an inhomogeneous medium, J. Math. Anal. Appl. 164 (1992) 40–58.
[3] F. Assous, P. Degond, E. Heintze, P.A. Raviart, J. Segree, On a nite-element method for solving the three-dimensional Maxwell equations, J. Comput. Phys. 109 (1993) 222.
[4] A. Bayliss, C.I. Goldstein, E. Turkel, On accuracy conditions for the numerical computation of waves, J. Comput. Phys. 59 (1985) 396–494.
[5] R. Beck, P. Deuhard, R. Hiptmair, R.H.W Hoppe, B. Wohlmuth, Adaptive multilevel methods for edge element discretizations of Maxwell’s equations. Preprint.
[6] W.E. Boyse, D.R. Lynch, K.D. Paulsen, G.N. Minerbo, Node based nite element modelling of Maxwell’s equations, IEEE Trans. Antennas Propagat. 40 (1992) 642–651.
[7] S.C. Brenner, L.R. Scott, The Mathematical Theory of Finite Element Methods, Springer, Berlin, 1994.
[8] D. Colton, R. Kress, Inverse Acoustic Electromagnetic Scattering Theory, 2nd ed., Applied Mathematical Sciences, vol. 93, Springer, New York, 1998.
[9] L. Crouzet, G. Meurant, Solving 3D harmonic Maxwell equations with nite elements, Lagrange multipliers iterative methods, in: M. Krizek et al., (Eds.), Finite Element Methods, Marcel-Dekker, New York, 1994, pp. 173–182. [10] F. Dubois, Discrete vector potential representation of a divergence free vector eld in three dimensional domains:
numerical analysis of a model problem, SIAM J. Numer. Anal. 27 (1990) 1103–1142.
[11] C. Hazard, M. Lenoir, On the solution of time-harmonic scattering problems for Maxwell’s equations, SIAM J. Math. Anal. 27 (1996) 1597–1630.
[12] H. Ihlenburg, I. Babuska, Finite element solution of the Helmholtz equation with high wavenumber Part I: the h-version of the FEM. Comput. Math. Appl. 30 (1995) 9–37.
[13] J.-M. Jin, The nite element method in electromagnetics, Wiley, New York, 1993.
[14] J.B. Keller, D. Givoli, Exact non-reecting boundary conditions, J. Comput. Phys. 82 (1989) 172–192.
[15] A. Kirsch, P. Monk, Corrigendum to a nite element/spectral method for approximating the time-harmonic Maxwell system in R3, (SIAM J. Appl. Math., 55 (1995) 1324–1344).
[16] A. Kirsch, P. Monk, An analysis of the coupling of nite element Nystrom methods in acoustic scattering, IMA J. Numer. Anal. 14 (1994) 523–544.
[17] A. Kirsch, P. Monk, A nite element/spectral method for approximating the time harmonic Maxwell system inR3, SIAM J. Appl. Math. 55 (1995) 1324–1344.
[18] M. Krizek, P. Neittaanmaki, On time-harmonic Maxwell equations with nonhomogeneous conductivities: solvability and FE-approximation, Appl. Math. 34 (1989) 480–499.
[19] R. Leis, Initial Boundary Value Problems in Mathematical Physics, Wiley, New York, 1988.
[20] V. Levillain, Coupling integral equation methods and nite volume elements for the resolution of time harmonic Maxwell’s equations in three dimensional heterogeneous medium, Technical Report 222, Centre de Mathematiques Appliquees, Ecole Polytechnique, 1990.
[22] G. Mur, The nite-element modeling of three-dimensional time-domain electromagnetic elds in strongly inhomogeneous media, IEEE Trans. Magn. 28 (1992) 1130–1133.
[23] G. Mur, A.T. de Hoop, A nite-element method for computing three-dimensional electromagnetic elds in inhomogeneous media, IEEE Trans. Magn. MAG-21 (1985) 2188–2191.
[24] J.C. Nedelec, Mixed nite elements inR3, Numer. Math. 35 (1980) 315–341.
[25] J.C. Nedelec, A new family of mixed nite elements inR3, Numer. Math. 50 (1986) 57–81.
