Directory UMM :Data Elmu:jurnal:J-a:Journal of Computational And Applied Mathematics:Vol100.Issue2.1998:
Journal of Computational and Applied Mathematics 100 (1998) 173–190
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 brie
y recall existence and uniqueness 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
c 1998 Elsevier Science B.V. All rights reserved.
in the mesh.
Keywords: Maxwell equations; Residual error indicators; Finite elements
1. Introduction
Suppose a bounded, anisotropic and inhomogeneous body is illuminated by a time-harmonic incident electromagnetic wave. The incident eld will interact with the inhomogeneity to produce
a scattered eld. The total eld, which is the sum of the incident and scattered elds, satis es
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 satis ed by the elds, we need some de nitions. 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 de nite while is positive semide nite. For simplicity, in this paper, we shall assume that all the coecients are smooth functions
∗
E-mail: [email protected]
c 1998 Elsevier Science B.V. All rights reserved.
0377-0427/98/$ – see front matter
PII: S 0 3 7 7 - 0 4 2 7 ( 9 8 ) 0 0 1 8 7 - 3
174
P. Monk / Journal of Computational and Applied Mathematics 100 (1998) 173–190
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) = ”0 I;
(x) = 0 I
and (x) = 0
if |x| ¿ b where I is the identity matrix.
We wish to compute time harmonic solutions with frequency !. We de ne the wave-number k
and functions (x) and n(x) by
k=
√
”0 0 !;
−1
(x) = 0 (x);
n(x) =
”−1
0
i(x)
:
”(x) +
!
The known incident electric eld is denoted by E i and is assumed to satisfy the homogeneous
Maxwell system
∇ × ∇ × E i − k 2 E i = 0 in R3 :
A typical choice for E i is the plane wave
E i = (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 E s the scattered electric eld, then E satis es the
Maxwell system:
∇ × ∇ × E − k 2 nE = 0 in R3 :
(1.1a)
The incident, scattered and total elds are related by
E = E i + Es
in R3
(1.1b)
P. Monk / Journal of Computational and Applied Mathematics 100 (1998) 173–190
175
and the following condition at in nity is needed to complete the problem
lim ((∇ × E s ) × x − ikrE s ) = 0
(1.1c)
r→∞
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 , we de ne u() to be the solution of
∇ × ∇ × u − k 2u = 0
xˆ × u =
on
in R3 \ BR ;
(1.2a)
(1.2b)
R;
lim ((∇ × u) × x − ikru) = 0:
(1.2c)
r→∞
This problem has a unique solution for any ∈ H −1=2 (Div; R ) where the surface space H −1=2
3
(Div; R ) is the space of tangential vector elds on R in H −1=2 ( R ) with surface divergence in
H −1=2 ( R ). For the simple case of a spherical arti cial boundary considered here a series expansion
for u is well known (see for example [8]). Using this expansion we can see that
xˆ ×
1
(∇ × u)| R ∈ H −1=2 (Div;
ik
R)
and hence we can de ne the operator Ge : H −1=2 (Div;
Ge = xˆ ×
R)
→ H −1=2 (Div;
R)
by
1
(∇ × u())| R :
ik
ˆ
Since we shall need the expansion later, we summarize it now (see [17] for details). Let {Ynm (x)},
m = −n; · · · ; n be an orthonormal sequence of spherical harmonics of order n on the unit sphere,
′
normalized such that hYnm ; Ynm′ i = mm′ nn′ . The basis functions for tangential elds on R are then
Unm = √
1
∇ Ynm
n(n + 1)
and
Vnm = xˆ × Unm ;
m = −n; : : : ; n;
n ∈ N:
(1.3)
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).
Any tangential vector eld ∈ H −1=2 (Div; R ) can be written in the form
=
n
∞ X
X
{an; m Unm + bn; m Vnm }
(1.4)
n=1 m=−n
with bounded norm
kk2H −1=2 (Div;
R)
=
∞ X
n
X
p
n=1 m=−n
2
1 + n(n + 1)|an; m | + √
1
|bn; m |2 ¡∞:
1 + n(n + 1)
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P. Monk / Journal of Computational and Applied Mathematics 100 (1998) 173–190
Given ∈ H −1=2 (Div;
s
Ge : =xˆ × H =
R)
we can show that
∞ X
n
X
n=1 m=−n
bn; m m an; m n m
U +
−ikR
V :
n n
ikR n
(1.5)
where
n = kR
′
(h(1)
n ) (kR)
+ 1:
h(1)
n (kR)
(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 de ne
(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
normal to BR ). Then using (1.1b) in (1.1a) we have
R ).
Let xˆ = x=r (this is the unit outer
∇ × ∇ × E s − k 2 nE s = f
where f = −(∇ × ∇ × E i − k 2 nE i ) (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 ; )
=(∇ × ∇ × E s ; ) − k 2 (nE s ; )
=(∇ × E s ; ∇ × ) − k 2 (nE s ; ) + hxˆ × ∇ × E s ; (xˆ × ) × xi:
ˆ
(1.7)
Using Ge and the fact that E s satis es the homogeneous Maxwell equations (1.2a) outside BR and
the radiation condition (1.2c) we know that
xˆ × ∇ × E s = ikGe (xˆ × E s ):
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
n
3
3 o
H (curl; BR ) = u ∈ L2 (BR ) | ∇ × u ∈ L2 (BR )
then we seek E s ∈ H (curl; BR ) such that
(∇ × E s ; ∇ × ) − k 2 (nE s ; )
+ikhGe (xˆ × E s ); (xˆ × ) × xi
ˆ = ( f ; )
;
(1.8)
for all ∈ H (curl; BR ). Note that if a function u ∈ H (curl; BR ), then (xˆ × u) × xˆ is in the dual space
of H −1=2 (Div; R ) so (1.8) is well de ned (see [17] for details).
Eq. (1.8) can be used to understand many of the current nite element methods for approximating the Maxwell’s equations. For example the coupled boundary element – nite element scheme
approximates Ge using boundary element techniques and E s using nite elements [13, 20, 2]. The
P. Monk / Journal of Computational and Applied Mathematics 100 (1998) 173–190
177
Keller–Givolli scheme approximates Ge using a series expansion [14]. The approach of [11] is somewhat di erent and replaces the term Ge (xˆ × E s ) 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 −k 2 (nE s ; ). Second the bilinear form
(∇ × E s ; ∇ × ) is not coercive on H (curl; BR ) and nally n is a complex symmetric matrix function. 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 di erentiable. More recently Hazard [11] has proved uniqueness 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 di erentiable (and satis es 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 coecients to include less smooth coecients (piecewise analytic for example) and variable .
Once uniqueness is veri ed, 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 de ne the spaces
Y : = ∇p: p ∈ H 1 (BR ) ;
Y + : = u ∈ H (curl; BR ) : − k 2 (nu; ∇q ) + ikhGe (xˆ × u); ∇ qi = 0
∀q ∈ H 1 (BR )
= {u ∈ H (curl; BR ): ∇ · (nu) = 0 in BR and
k 2 xˆ · u = ik∇ · Ge (xˆ × u) on
and show that
H (curl; BR ) = Y ⊕ Y + :
R
:
(2.1)
(2.2)
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P. Monk / Journal of Computational and Applied Mathematics 100 (1998) 173–190
First, we determine p ∈ H 1 (BR )=R with
1
a(∇p; ∇q) = b(∇q) for all q ∈ H˜ (BR );
(2.3)
which is exactly the problem of determining p ∈ H 1 (BR )=R such that
− k 2 (n∇p; ∇q) +
ik
hGe (xˆ × ∇p); ∇ qi = ( f ; ∇q)
R
(2.4)
for all q ∈ H 1 (BR )=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
@q
x:
ˆ
∇ q+
R
@xˆ
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 = aˆ1 + ˆ1 where
aˆ1 (u; C) = (∇ × u; ∇ × C) + (nu; C) + ikhGe2 (xˆ × u); C⊤ i ;
ˆ1 (u; C) = −(k 2 + 1) (nu; C) + ikhGe1 (xˆ × u); C⊤ i :
where
Ge =
n
∞ X
X
n=0 m=−n
"
bnm m anm (n − ˜n ) m
Vn
u +
−ikR
n n
ikR
#
n
∞ X
1 X
+
anm ˜n Vnm
ikR n=0 m=−n
= : Ge1 + Ge2 :
(2.6)
(h(1) )′ (iR)
+ 1:
˜n = kR n(1)
hn (iR)
(2.7)
and
Now, aˆ1 is coercive over Y + and ˆ1 de nes 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 (BR ))3 ; there exists a unique solution to (1:8).
At this stage we can apply nite element methods to discretize (1.8) and prove optimal error
estimates [17].
P. Monk / Journal of Computational and Applied Mathematics 100 (1998) 173–190
179
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 modi cation 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 de ned
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 (H 2 (BR ))3 where H 2 (BR ) is the Sobolev space of twice di erentiable functions in the L2 sense (a
weaker space, but not (H 1 (BR ))3 , is possible). The interpolation operator rh has the following error
estimate:
ku − rh ukH (curl; BR ) : =
q
ku − rh uk2L2 (BR ) + k∇ × ( − rh u)k2L2 (BR )
6 ChkukH 2 (BR ) :
(3.1)
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P. Monk / Journal of Computational and Applied Mathematics 100 (1998) 173–190
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 Ehs ∈ Vh such that
(∇ × Ehs ; ∇ × h ) − k 2 (nEhs ; h )
+ik hGe (xˆ × Ehs ); (xˆ × h ) × xi
ˆ = ( f ; h )
(3.2)
for all h ∈ Vh and we can prove the following optimal estimate [17]:
Theorem 3.1. The discrete solution Ehs ∈ Vh is well de ned for all h suciently small and
kE s − Ehs kH( curl; BR ) 6 ChkE s kH 2 (BR ) :
4. A Posteriori error estimates
In practice we would like to have a posteriori error estimates to help with selective mesh re nement. 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 that hk be
suciently small for the interpolant to have reasonable accuracy (although even this is not sucient
for accuracy [4, 12]). However selective mesh re nement to deal with changes in parameters or
geometric features may still be needed to guarantee accuracy.
Let Fint denote the set of all faces in the mesh interior to BR , and let F denote the set of all faces
on R . For each face f let Kf be the element of largest diameter having f as a face. In addition,
let hK be the diameter of element K and hf be the diameter of a face f in the mesh. Now we shall
prove:
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P. Monk / Journal of Computational and Applied Mathematics 100 (1998) 173–190
Theorem 4.1. Under the conditions of this paper there is a constant C independent of E s , Ehs and
h such that
kE s − Ehs kL2 (BR )
6C
(
X
K
+
X
f∈Fint
+
+
h2K kf
X
f∈F
s
hKf k nf × ∇ × Eh
f
nEhs k2L2 (K)
!1=2
1=2
k2L2 (f)
!1=2
·f −k ∇·
X
h Kf k n f ·
(nEhs ) f
X
hKf kk 2 xˆ · Ehs − ik∇ · Ge (xˆ × E s )kL2 (f)
F
1=2
hKf knf × ∇ × Ehs − ikGe (xˆ × Ehs )k2L2 (f)
h2K k∇
Fint
+
+k
2
X
K
+
−
∇ × ∇ × Ehs
2
(nEhs )k2L2 (K)
k2L2 (f)
!1=2
!1=2
where [·]f denotes the jump in the quantity across the face f; and nf is a normal to face f
(chosen outward on R ).
Remark
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 de nes a suitable adjoint solution driven
by the actual error in the solution. Let z ∈ H (curl; BR ) satisfy
a( ; z) = (n ; E s − Ehs );
∀ ∈ H (curl; BR ):
(4.1)
Our rst lemma of this section con rms that z exists uniquely and gives some of its properties.
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P. Monk / Journal of Computational and Applied Mathematics 100 (1998) 173–190
Lemma 4.2. There exists a unique solution z to (4.1). Furthermore z=u+∇s where u ∈ H (curl; BR )
and s ∈ H 1 (BR )=R where
∀p ∈ H 1 (BR )=R;
a(∇p; ∇s) = (n∇p; E s − Ehs );
and
a( ; u) = (n ; (E s − Ehs )) − a( ; ∇s) ∀ ∈ H (curl; BR ):
Furthermore s satis es the following a priori estimate:
k∇skL2 (BR ) 6 CkE s − Ehs kL2 (BR ) :
(4.2)
In addition, if the coecients in and are smooth functions of position
kukH 2 (BR ) 6 CkE s − Ehs kL2 (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
kuk2H 2 (K) 6 CkE s − Ehs k2L2 (BR )
to hold.
To prove this lemma we rst need to study the adjoint of the operator Ge denoted Ge∗ . To do this
we need to recall that the space
H −1=2 (Curl;
n
−1=2
(
R ) = u ∈ (H
is the dual space to H −1=2 (Div; ;
R)
3
ˆ · u = 0 and ∇ R × u ∈ H −1=2 (
R )) | x
in the L2 (
R)
inner product. We have the following lemma:
Lemma 4.3. The operator Ge∗ is a bounded operator from H −1=2 (Curl;
(Curl; R ) is given by
=
∞ X
n
X
n; m Unm + n; m Vnm
n=1 m=−n
then
Ge∗ =
∞ X
n
X
n=1 m=−n
n; m
(2)
ikR m
n
U m;
−
V
n;
m
n
ikR n
(2)
n
where
(2)
n = kR
′
(h(2)
n ) (kR)
+1
(2)
hn (kR)
o
R)
R)
into itself. If ∈ H −1=2
183
P. Monk / Journal of Computational and Applied Mathematics 100 (1998) 173–190
and h(2)
is the spherical Hankel function of second type. Furthermore Ge∗ may be characterized
n
as follows. Let z satisfy
∇ × ∇ × z − k 2 z = 0 in R3 \
R ;
(4.4)
zT = on
(4.5)
R;
(∇ × z) × x + ikrzT → 0 as r → ∞
(4.6)
where zT = (xˆ × z) × xˆ then Ge∗ = (1=ik)(∇ × u)T on
R.
Proof of Lemma 4.3. Ge∗ is de ned 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
sphere):
h; Ge∗ i
=hGe ; i
=
Z
2
=R
R
n
∞ X
X
ikR m
n m
−bn; m
Un + an; m
V
n
ikR n
n=1 m=−n
∞ X
n
X
n=1 m=−n
=
Z
R
−bn; m n; m
n
∞ X
X
!
n
∞ X
X
n; m Unm + n; m Vnm
n=1 m=−n
dA
ikR
n
+ an; m bn; m
n
ikR
an; m Un; m +
bn; m Vnm
n=1 m=−n
!
!
n
∞ X
X
n m
ikr
n; m Vnm − n; m
dA:
U
ikR n
n
n=1 m=−n
−1=2
(Curl;
This proves the series de nition of Ge∗ since (2)
n = n . But if ∈ H
k kH −1=2 (Curl;
!
v
u∞ n
uX X
| n; m |2
t
√
=
)
R
n=1 m=−n
n(n + 1)
+
p
n(n + 1)| n; m
|2
R ),
¡∞
and, using the fact that there are constants c1 and c2 such that c1 n 6 |n | 6 c2 n [16], we have
kGe∗ kH −1=2 (Curl2 ;
R)
v
u∞ n
uX X
=t
n=1 m=−n
2
p
|(2)
k 2 R2
n |
√
| n; m |2 + n(n + 1) (2) | n; m |2
2
2
k R n(n + 1)
|n |2
v
u∞ n
uX X p
6 Ct
n(n + 1)| n; m |2 + √
n=1 m=−n
6 Ck k2H −1=2 (Curl;
R)
1
n; m |2
n(n + 1)
!
:
This proves the claimed boundedness of the operator.
The proof of the representation of Ge∗ in terms of the solution of a scattering problem can also
be done directly by a series argument. Let the volume basis functions Mnm and Nnm be de ned by
Mnm (x) = ∇ × xn(2) (k|x|)Ynm (x);
ˆ
and
Nnm (x) =
1
∇ × Mnm ;
ik
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P. Monk / Journal of Computational and Applied Mathematics 100 (1998) 173–190
then following [8] the solution z of (4.4) and (4.6) can be written as
z=
n
∞ X
X
(
n; m Mnm + n; m Nnm ) :
n=1 m=−n
Since [16], on
R,
p
m
(xˆ × Mnm ) × xˆ = −h(2)
n (kR) n(n + 1)Vn ;
(xˆ × Nnm ) × xˆ =
(4.7)
p
1 (2)
m
′
(hn (kR) + kR(h(2)
n ) (kR)) n(n + 1)Un ;
ikR
(4.8)
we deduce from (4.5) that
1
n; m ;
n(n + 1)h(2)
n (kR)
ikR
1
n; m :
n; m = √
(2)
′
n(n + 1) hn (kR) + kR(h(2)
n ) (kR)
n; m = − √
Then
∞ X
n
X
1
∇×z = −
ik
n=1 m=−n
√
ikR
1
n; m Mnm
(2)
′
n(n + 1) hn (kR) + kR(h(2)
n ) (kR)
1
n; m Nmn
+√
n(n + 1)h(2)
n (kR)
restricting to
R
!
and using (4.7) and (4.8) again proves the representation of Ge∗ .
Proof of Lemma 4.2. Using the de nition of z (4.1) z ∈ H (curl; BR ) satis es
(∇ × ; ∇ × z) − k 2 (n ; z) + ik ¡ Ge (xˆ × ; (xˆ × ) × xˆ ¿ =(n ; E s − Ehs ) ∈ H (curl; BR ):
Hence via the de nition of adjoint operator Ge∗ , and the fact that is real symmetric, we can see
that z is the weak solution of the Maxwell system:
∇ × ∇ × z − k 2 nz = n (E s − Ehs )
(∇ × z)T = ikGe∗ (zT )
on
in BR ;
R:
Hence, using the characterization of Ge∗ in Lemma 4.3, we see that z = w|BR where w is the weak
solution of the scattering problem:
∇ × ∇ × w − k 2 nw = g ∈ R3 ;
lim ((∇ × w) × x + ikrw) = 0;
r→∞
where
g(x) =
n(E s − Ehs ) if x ∈ BR ;
0
otherwise:
P. Monk / Journal of Computational and Applied Mathematics 100 (1998) 173–190
185
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 veri ed 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 ) satis es
∇ × ∇ × u − k 2 nu
= F in R3 ;
lim ((∇ × u) × x + ikru) = 0;
r→∞
where F ∈ L2 (BR ) is de ned by
(n ; F) = (n ; (E s − Ehs )) − a( ; ∇s);
∀ ∈ H (curl; BR )
and is extended to R3 by zero. Note that if we choose = ∇ for any ∈ H 1 (BR )=R we have
(n∇; F) = 0 for any ∈ H 1 (BR )=R so that xˆ · F = 0 on R and so ∇ · (nF) = 0 in all of R3 . Thus
u satis es a standard scattering problem and because nF is divergence free (see [19, p. 168]):
kukH 2 (BR ) 6 CkFkL2 (BR ) 6 CkE s − Ehs kL2 (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 ; (E s − Ehs ));
∀ ∈ H (curl; BR ):
(4.10)
Furthermore
z = u + ∇s
where s ∈ H 1 (BR )=R’s and z ∈ H (curl; BR ),
Thus, choosing = E s − Ehs in (4.10) and using the fact that E s − Ehs satis es the nite element
“orthogonality” relation
a(E s − Ehs ; h ) = 0;
∀ h ∈ Sh ;
we have
(n(E s − Ehs ); (E s − Ehs )) = a(E s − Ehs ; u) + a(E s − Ehs ; ∇s)
= a(E s − Ehs ; u − uh ) + a(E s − 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
E s satis es (1.8), and nally integrating by parts on each element we have
a(E s − Ehs ; u − uh ) = (∇ × (E s − Ehs ); ∇ × (u − uh )) − k 2 (n(E s − Ehs ); (u − uh ))
ˆ
+ik hGe (xˆ × (E s − Ehs )); (xˆ × (u − uh )) × xi
= ( f ; u − uh ) − (∇ × Ehs ; ∇ × (u − uh )) + k 2 (nEhs ; (u − uh ))
ˆ
−ik hGe (xˆ × Ehs ); (xˆ × (u − uh )) × xi
186
P. Monk / Journal of Computational and Applied Mathematics 100 (1998) 173–190
=
(
X
( f − ∇ × ∇ × Ehs + k 2 nEhs ; u − uh )K
K∈ h
− hnK ∇ × Ehs ; nK × (u − uh )i@K
+ik
hGe (xˆ × Ehs ); (xˆ × (u
− uh )) × xi
ˆ
)
(4.12)
where (·; ·)K is the (L2 (K))3 inner product, h·; ·i@K is the duality pairing between H −1=2 (Div; R ) and
H −1=2 Div; R )′ on @K, and nK is the outward normal to K.
If nf denotes a normal to a face f (chosen outward on R ) and if [·]f denotes the jump of the
quantity across a face f, we may rewrite (4.12) as
a(E s − Ehs ; u − uh ) =
(
X
K∈ h
+
( f − ∇ × ∇ × Ehs + k 2 nEhs ; u − uh )K
X D
nf × ∇ × Ehs
f∈Fint
+
X
f∈F
f
; (nf × (u − uh )) × nf
E
f
)
ˆ f :
hxˆ × ∇ × Ehs − ikGe (xˆ × Ehs ); (xˆ × (u − uh )) × xi
Next we choose uh =rh u (this is possible since u ∈ H 2 (
)). Then using the Cauchy–Schwarz inequality and the error estimate for u − rh u from (3.1) together with a similar estimate for the interpolant
on the faces of the mesh, we get
|a(E s − Ehs ; u − uh )|
6
(
X
K∈ h
+
k f − ∇ × ∇ × Ehs + k 2 nEhs kL2 (K) ku − uh kL2 (K)
X
k nf × ∇ × Ehs
X
kxˆ × ∇ × Ehs
ikGe (xˆ × Ehs )kL2 (f) k(xˆ × (u
f∈Fint
+
f∈F
6C
(
+
X
K∈ h
−
f
kL2 (f) knf × (u − uh )) × nf kL2 (f)
− uh )) × xk
ˆ L2 (f)
X
k nf × ∇ × Ehs
X
kxˆ × ∇ × Ehs
ikGe (xˆ × Ehs )kL2 (f) hf kukH 1 (f)
f∈F
)
k f − ∇ × ∇ × Ehs + k 2 nEhs kL2 (K) hK kukH 2 (K)
f∈Fint
+
−
f
kL2 (f) hf kukH 1 (f)
)
:
(4.13)
187
P. Monk / Journal of Computational and Applied Mathematics 100 (1998) 173–190
Next we use the following estimate which is valid for any q ∈ H 1 (Kf ) with constant C independent
of the mesh parameter hK (for a regular mesh)
kqk2L2 (f)
1
6 C kqkL2 (Kf ) k∇qkL2 (Kf ) +
kqk2L2 (Kf )
hK f
!
(4.14)
where Kf is a tetrahedron with f as one face. With this estimate we have
kukH 1 (f) 6 ChK−1=2
kukH 2 (Kf ) :
f
Using this estimate in (4.13) followed by the a priori estimate (4.9) provides the following estimate
for |a(E s − Ehs ; u − uh )|:
s
|a(E −
Ehs ; u
− uh )| 6 C
X
K∈ h
+
X
f∈Fint
h2K k f
−
∇ × ∇ × Ehs
s
hKf k nf × ∇ × Eh
f
+k
2
nEhs k2L2 (K)
!1=2
1=2
k2L2 (f)
1=2
X
s
s
+
hKf kxˆ × ∇ × Eh − ikGe (xˆ × Eh )kL2 (f)
kE s − Ehs kL2 (BR ) :
f∈F
(4.15)
The next step is to estimate the error in a(E s −Ehs ; ∇(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
a(E s − Ehs ; ∇(s − sh ))
= − k 2 (n∇E s − Ehs ; ∇(s − sh )) + ikhGe (xˆ × (E s − Ehs )); ∇(s − sh )i
=−
+
X
(∇ · f + k 2 ∇ · (nEhs ); s − sh )K −
X
ikhGe (xˆ × Ehs ); ∇(s − sh )if :
K∈ h
f∈F
X
K∈ h
k 2 hnK · (nEhs ); s − sh i@K
We take sh ∈ Sh to be the Scott–Zhang interpolant of s [7] which is de ned on functions in H 1 (BR ).
We have also assumed that f has a well-de ned global divergence so there is no jump in the normal
component of f across faces in the mesh ( f = 0 in the neighborhood of R ).
188
P. Monk / Journal of Computational and Applied Mathematics 100 (1998) 173–190
Next we integrate by parts on the boundary
integrals over the face of each element:
a(E s − Ehs ; ∇(s − sh ))
=−
−
X
K∈ h
(∇ · f + k 2 ∇ · (nEhs ); s − sh )K −
X
f∈
R
2
k xˆ ·
Ehs
−∇ ·
Ge (xˆ × Ehs ); s
and rewrite the face integrals by grouping the
R
X
f∈
− sh
int
f
k 2 h[f · (nEhs )]; s − sh if
:
If f 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 −k 2 ∇ · (nE s ) = ∇ · f . In addition
k 2 xˆ · E s = ik∇ Ge (xˆ × E s ) which just speci es continuity of the normal component of E s 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
|a(E s − Ehs ; ∇(s − sh ))|
6
X
k∇ · f + k 2 ∇ · (nEhs )kL2 (K) ks − sh kL2 (K) +
K∈ h
+
X
f∈
6
R
2
kk xˆ ·
Ehs
− ik∇ ·
Ge (xˆ × Ehs )kL2 (f) ks
X
f∈
int
k 2 k[nf · (nEhs )]kL2 (f) ks − sh kL2 (f)
− sh kL2 (f)
X
k∇ · f + k 2 ∇ · (nEhs )kL2 (K) ks − sh kL2 (K)
+C
X
ks − sh kL2 (Kf ) + hK1=2f ks − sh kH 1 (Kf )
k[nf · (nEhs )]kL2 (f) hK−1=2
f
X
kk 2 xˆ · Ehs − ik∇ · Ge (xˆ × Ehs )kL2 (f) hK−1=2
ks − sh kL2 (Kf ) + hK1=2f ks − sh kH 1 (Kf )
f
K∈ h
f∈
+C
f∈
int
R
6 CkE s − Ehs k
+
X
+
X
f∈
f∈
int
X
K∈ h
h2K k∇ · f + k 2 ∇ · (nEhs )k2L2 (K)
!1=2
1=2
hKf k[nf · (nEhs )]k2L2 (f)
1=2
s 2
2
s
;
hKf kk xˆ · Eh − ik∇ · Ge (xˆ × Eh )kL2 (f)
R
where Kf is a tetrahedron with f as one face and we have used the a priori estimate (4.2) to
estimate kskH 1 (
) . Combining the above estimate and (4.15) in (4.11) completes the proof.
P. Monk / Journal of Computational and Applied Mathematics 100 (1998) 173–190
189
Acknowledgments and disclaimer
E ort of Peter Monk sponsored by the Air Force Oce of Scienti c Research, Air Force Materials 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 notation 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 Scienti c 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 threedimensional 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. Deu
hard, 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-re
ecting 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 in R3 ,
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.
[21] P. Monk, A mixed method for approximating Maxwell’s equations, SIAM J. Numer. Anal. 28 (1991) 1610–1634.
190
P. Monk / Journal of Computational and Applied Mathematics 100 (1998) 173–190
[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 in R3 , Numer. Math. 35 (1980) 315–341.
[25] J.C. Nedelec, A new family of mixed nite elements in R3 , Numer. Math. 50 (1986) 57–81.
[26] R. Potthast, Integral equation methods in electromagnetic scattering from anisotropic media. Math. Methods in Appl.
Sci., to appear.
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 brie
y recall existence and uniqueness 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
c 1998 Elsevier Science B.V. All rights reserved.
in the mesh.
Keywords: Maxwell equations; Residual error indicators; Finite elements
1. Introduction
Suppose a bounded, anisotropic and inhomogeneous body is illuminated by a time-harmonic incident electromagnetic wave. The incident eld will interact with the inhomogeneity to produce
a scattered eld. The total eld, which is the sum of the incident and scattered elds, satis es
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 satis ed by the elds, we need some de nitions. 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 de nite while is positive semide nite. For simplicity, in this paper, we shall assume that all the coecients are smooth functions
∗
E-mail: [email protected]
c 1998 Elsevier Science B.V. All rights reserved.
0377-0427/98/$ – see front matter
PII: S 0 3 7 7 - 0 4 2 7 ( 9 8 ) 0 0 1 8 7 - 3
174
P. Monk / Journal of Computational and Applied Mathematics 100 (1998) 173–190
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) = ”0 I;
(x) = 0 I
and (x) = 0
if |x| ¿ b where I is the identity matrix.
We wish to compute time harmonic solutions with frequency !. We de ne the wave-number k
and functions (x) and n(x) by
k=
√
”0 0 !;
−1
(x) = 0 (x);
n(x) =
”−1
0
i(x)
:
”(x) +
!
The known incident electric eld is denoted by E i and is assumed to satisfy the homogeneous
Maxwell system
∇ × ∇ × E i − k 2 E i = 0 in R3 :
A typical choice for E i is the plane wave
E i = (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 E s the scattered electric eld, then E satis es the
Maxwell system:
∇ × ∇ × E − k 2 nE = 0 in R3 :
(1.1a)
The incident, scattered and total elds are related by
E = E i + Es
in R3
(1.1b)
P. Monk / Journal of Computational and Applied Mathematics 100 (1998) 173–190
175
and the following condition at in nity is needed to complete the problem
lim ((∇ × E s ) × x − ikrE s ) = 0
(1.1c)
r→∞
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 , we de ne u() to be the solution of
∇ × ∇ × u − k 2u = 0
xˆ × u =
on
in R3 \ BR ;
(1.2a)
(1.2b)
R;
lim ((∇ × u) × x − ikru) = 0:
(1.2c)
r→∞
This problem has a unique solution for any ∈ H −1=2 (Div; R ) where the surface space H −1=2
3
(Div; R ) is the space of tangential vector elds on R in H −1=2 ( R ) with surface divergence in
H −1=2 ( R ). For the simple case of a spherical arti cial boundary considered here a series expansion
for u is well known (see for example [8]). Using this expansion we can see that
xˆ ×
1
(∇ × u)| R ∈ H −1=2 (Div;
ik
R)
and hence we can de ne the operator Ge : H −1=2 (Div;
Ge = xˆ ×
R)
→ H −1=2 (Div;
R)
by
1
(∇ × u())| R :
ik
ˆ
Since we shall need the expansion later, we summarize it now (see [17] for details). Let {Ynm (x)},
m = −n; · · · ; n be an orthonormal sequence of spherical harmonics of order n on the unit sphere,
′
normalized such that hYnm ; Ynm′ i = mm′ nn′ . The basis functions for tangential elds on R are then
Unm = √
1
∇ Ynm
n(n + 1)
and
Vnm = xˆ × Unm ;
m = −n; : : : ; n;
n ∈ N:
(1.3)
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).
Any tangential vector eld ∈ H −1=2 (Div; R ) can be written in the form
=
n
∞ X
X
{an; m Unm + bn; m Vnm }
(1.4)
n=1 m=−n
with bounded norm
kk2H −1=2 (Div;
R)
=
∞ X
n
X
p
n=1 m=−n
2
1 + n(n + 1)|an; m | + √
1
|bn; m |2 ¡∞:
1 + n(n + 1)
176
P. Monk / Journal of Computational and Applied Mathematics 100 (1998) 173–190
Given ∈ H −1=2 (Div;
s
Ge : =xˆ × H =
R)
we can show that
∞ X
n
X
n=1 m=−n
bn; m m an; m n m
U +
−ikR
V :
n n
ikR n
(1.5)
where
n = kR
′
(h(1)
n ) (kR)
+ 1:
h(1)
n (kR)
(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 de ne
(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
normal to BR ). Then using (1.1b) in (1.1a) we have
R ).
Let xˆ = x=r (this is the unit outer
∇ × ∇ × E s − k 2 nE s = f
where f = −(∇ × ∇ × E i − k 2 nE i ) (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 ; )
=(∇ × ∇ × E s ; ) − k 2 (nE s ; )
=(∇ × E s ; ∇ × ) − k 2 (nE s ; ) + hxˆ × ∇ × E s ; (xˆ × ) × xi:
ˆ
(1.7)
Using Ge and the fact that E s satis es the homogeneous Maxwell equations (1.2a) outside BR and
the radiation condition (1.2c) we know that
xˆ × ∇ × E s = ikGe (xˆ × E s ):
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
n
3
3 o
H (curl; BR ) = u ∈ L2 (BR ) | ∇ × u ∈ L2 (BR )
then we seek E s ∈ H (curl; BR ) such that
(∇ × E s ; ∇ × ) − k 2 (nE s ; )
+ikhGe (xˆ × E s ); (xˆ × ) × xi
ˆ = ( f ; )
;
(1.8)
for all ∈ H (curl; BR ). Note that if a function u ∈ H (curl; BR ), then (xˆ × u) × xˆ is in the dual space
of H −1=2 (Div; R ) so (1.8) is well de ned (see [17] for details).
Eq. (1.8) can be used to understand many of the current nite element methods for approximating the Maxwell’s equations. For example the coupled boundary element – nite element scheme
approximates Ge using boundary element techniques and E s using nite elements [13, 20, 2]. The
P. Monk / Journal of Computational and Applied Mathematics 100 (1998) 173–190
177
Keller–Givolli scheme approximates Ge using a series expansion [14]. The approach of [11] is somewhat di erent and replaces the term Ge (xˆ × E s ) 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 −k 2 (nE s ; ). Second the bilinear form
(∇ × E s ; ∇ × ) is not coercive on H (curl; BR ) and nally n is a complex symmetric matrix function. 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 di erentiable. More recently Hazard [11] has proved uniqueness 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 di erentiable (and satis es 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 coecients to include less smooth coecients (piecewise analytic for example) and variable .
Once uniqueness is veri ed, 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 de ne the spaces
Y : = ∇p: p ∈ H 1 (BR ) ;
Y + : = u ∈ H (curl; BR ) : − k 2 (nu; ∇q ) + ikhGe (xˆ × u); ∇ qi = 0
∀q ∈ H 1 (BR )
= {u ∈ H (curl; BR ): ∇ · (nu) = 0 in BR and
k 2 xˆ · u = ik∇ · Ge (xˆ × u) on
and show that
H (curl; BR ) = Y ⊕ Y + :
R
:
(2.1)
(2.2)
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P. Monk / Journal of Computational and Applied Mathematics 100 (1998) 173–190
First, we determine p ∈ H 1 (BR )=R with
1
a(∇p; ∇q) = b(∇q) for all q ∈ H˜ (BR );
(2.3)
which is exactly the problem of determining p ∈ H 1 (BR )=R such that
− k 2 (n∇p; ∇q) +
ik
hGe (xˆ × ∇p); ∇ qi = ( f ; ∇q)
R
(2.4)
for all q ∈ H 1 (BR )=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
@q
x:
ˆ
∇ q+
R
@xˆ
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 = aˆ1 + ˆ1 where
aˆ1 (u; C) = (∇ × u; ∇ × C) + (nu; C) + ikhGe2 (xˆ × u); C⊤ i ;
ˆ1 (u; C) = −(k 2 + 1) (nu; C) + ikhGe1 (xˆ × u); C⊤ i :
where
Ge =
n
∞ X
X
n=0 m=−n
"
bnm m anm (n − ˜n ) m
Vn
u +
−ikR
n n
ikR
#
n
∞ X
1 X
+
anm ˜n Vnm
ikR n=0 m=−n
= : Ge1 + Ge2 :
(2.6)
(h(1) )′ (iR)
+ 1:
˜n = kR n(1)
hn (iR)
(2.7)
and
Now, aˆ1 is coercive over Y + and ˆ1 de nes 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 (BR ))3 ; there exists a unique solution to (1:8).
At this stage we can apply nite element methods to discretize (1.8) and prove optimal error
estimates [17].
P. Monk / Journal of Computational and Applied Mathematics 100 (1998) 173–190
179
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 modi cation 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 de ned
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 (H 2 (BR ))3 where H 2 (BR ) is the Sobolev space of twice di erentiable functions in the L2 sense (a
weaker space, but not (H 1 (BR ))3 , is possible). The interpolation operator rh has the following error
estimate:
ku − rh ukH (curl; BR ) : =
q
ku − rh uk2L2 (BR ) + k∇ × ( − rh u)k2L2 (BR )
6 ChkukH 2 (BR ) :
(3.1)
180
P. Monk / Journal of Computational and Applied Mathematics 100 (1998) 173–190
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 Ehs ∈ Vh such that
(∇ × Ehs ; ∇ × h ) − k 2 (nEhs ; h )
+ik hGe (xˆ × Ehs ); (xˆ × h ) × xi
ˆ = ( f ; h )
(3.2)
for all h ∈ Vh and we can prove the following optimal estimate [17]:
Theorem 3.1. The discrete solution Ehs ∈ Vh is well de ned for all h suciently small and
kE s − Ehs kH( curl; BR ) 6 ChkE s kH 2 (BR ) :
4. A Posteriori error estimates
In practice we would like to have a posteriori error estimates to help with selective mesh re nement. 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 that hk be
suciently small for the interpolant to have reasonable accuracy (although even this is not sucient
for accuracy [4, 12]). However selective mesh re nement to deal with changes in parameters or
geometric features may still be needed to guarantee accuracy.
Let Fint denote the set of all faces in the mesh interior to BR , and let F denote the set of all faces
on R . For each face f let Kf be the element of largest diameter having f as a face. In addition,
let hK be the diameter of element K and hf be the diameter of a face f in the mesh. Now we shall
prove:
181
P. Monk / Journal of Computational and Applied Mathematics 100 (1998) 173–190
Theorem 4.1. Under the conditions of this paper there is a constant C independent of E s , Ehs and
h such that
kE s − Ehs kL2 (BR )
6C
(
X
K
+
X
f∈Fint
+
+
h2K kf
X
f∈F
s
hKf k nf × ∇ × Eh
f
nEhs k2L2 (K)
!1=2
1=2
k2L2 (f)
!1=2
·f −k ∇·
X
h Kf k n f ·
(nEhs ) f
X
hKf kk 2 xˆ · Ehs − ik∇ · Ge (xˆ × E s )kL2 (f)
F
1=2
hKf knf × ∇ × Ehs − ikGe (xˆ × Ehs )k2L2 (f)
h2K k∇
Fint
+
+k
2
X
K
+
−
∇ × ∇ × Ehs
2
(nEhs )k2L2 (K)
k2L2 (f)
!1=2
!1=2
where [·]f denotes the jump in the quantity across the face f; and nf is a normal to face f
(chosen outward on R ).
Remark
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 de nes a suitable adjoint solution driven
by the actual error in the solution. Let z ∈ H (curl; BR ) satisfy
a( ; z) = (n ; E s − Ehs );
∀ ∈ H (curl; BR ):
(4.1)
Our rst lemma of this section con rms that z exists uniquely and gives some of its properties.
182
P. Monk / Journal of Computational and Applied Mathematics 100 (1998) 173–190
Lemma 4.2. There exists a unique solution z to (4.1). Furthermore z=u+∇s where u ∈ H (curl; BR )
and s ∈ H 1 (BR )=R where
∀p ∈ H 1 (BR )=R;
a(∇p; ∇s) = (n∇p; E s − Ehs );
and
a( ; u) = (n ; (E s − Ehs )) − a( ; ∇s) ∀ ∈ H (curl; BR ):
Furthermore s satis es the following a priori estimate:
k∇skL2 (BR ) 6 CkE s − Ehs kL2 (BR ) :
(4.2)
In addition, if the coecients in and are smooth functions of position
kukH 2 (BR ) 6 CkE s − Ehs kL2 (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
kuk2H 2 (K) 6 CkE s − Ehs k2L2 (BR )
to hold.
To prove this lemma we rst need to study the adjoint of the operator Ge denoted Ge∗ . To do this
we need to recall that the space
H −1=2 (Curl;
n
−1=2
(
R ) = u ∈ (H
is the dual space to H −1=2 (Div; ;
R)
3
ˆ · u = 0 and ∇ R × u ∈ H −1=2 (
R )) | x
in the L2 (
R)
inner product. We have the following lemma:
Lemma 4.3. The operator Ge∗ is a bounded operator from H −1=2 (Curl;
(Curl; R ) is given by
=
∞ X
n
X
n; m Unm + n; m Vnm
n=1 m=−n
then
Ge∗ =
∞ X
n
X
n=1 m=−n
n; m
(2)
ikR m
n
U m;
−
V
n;
m
n
ikR n
(2)
n
where
(2)
n = kR
′
(h(2)
n ) (kR)
+1
(2)
hn (kR)
o
R)
R)
into itself. If ∈ H −1=2
183
P. Monk / Journal of Computational and Applied Mathematics 100 (1998) 173–190
and h(2)
is the spherical Hankel function of second type. Furthermore Ge∗ may be characterized
n
as follows. Let z satisfy
∇ × ∇ × z − k 2 z = 0 in R3 \
R ;
(4.4)
zT = on
(4.5)
R;
(∇ × z) × x + ikrzT → 0 as r → ∞
(4.6)
where zT = (xˆ × z) × xˆ then Ge∗ = (1=ik)(∇ × u)T on
R.
Proof of Lemma 4.3. Ge∗ is de ned 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
sphere):
h; Ge∗ i
=hGe ; i
=
Z
2
=R
R
n
∞ X
X
ikR m
n m
−bn; m
Un + an; m
V
n
ikR n
n=1 m=−n
∞ X
n
X
n=1 m=−n
=
Z
R
−bn; m n; m
n
∞ X
X
!
n
∞ X
X
n; m Unm + n; m Vnm
n=1 m=−n
dA
ikR
n
+ an; m bn; m
n
ikR
an; m Un; m +
bn; m Vnm
n=1 m=−n
!
!
n
∞ X
X
n m
ikr
n; m Vnm − n; m
dA:
U
ikR n
n
n=1 m=−n
−1=2
(Curl;
This proves the series de nition of Ge∗ since (2)
n = n . But if ∈ H
k kH −1=2 (Curl;
!
v
u∞ n
uX X
| n; m |2
t
√
=
)
R
n=1 m=−n
n(n + 1)
+
p
n(n + 1)| n; m
|2
R ),
¡∞
and, using the fact that there are constants c1 and c2 such that c1 n 6 |n | 6 c2 n [16], we have
kGe∗ kH −1=2 (Curl2 ;
R)
v
u∞ n
uX X
=t
n=1 m=−n
2
p
|(2)
k 2 R2
n |
√
| n; m |2 + n(n + 1) (2) | n; m |2
2
2
k R n(n + 1)
|n |2
v
u∞ n
uX X p
6 Ct
n(n + 1)| n; m |2 + √
n=1 m=−n
6 Ck k2H −1=2 (Curl;
R)
1
n; m |2
n(n + 1)
!
:
This proves the claimed boundedness of the operator.
The proof of the representation of Ge∗ in terms of the solution of a scattering problem can also
be done directly by a series argument. Let the volume basis functions Mnm and Nnm be de ned by
Mnm (x) = ∇ × xn(2) (k|x|)Ynm (x);
ˆ
and
Nnm (x) =
1
∇ × Mnm ;
ik
184
P. Monk / Journal of Computational and Applied Mathematics 100 (1998) 173–190
then following [8] the solution z of (4.4) and (4.6) can be written as
z=
n
∞ X
X
(
n; m Mnm + n; m Nnm ) :
n=1 m=−n
Since [16], on
R,
p
m
(xˆ × Mnm ) × xˆ = −h(2)
n (kR) n(n + 1)Vn ;
(xˆ × Nnm ) × xˆ =
(4.7)
p
1 (2)
m
′
(hn (kR) + kR(h(2)
n ) (kR)) n(n + 1)Un ;
ikR
(4.8)
we deduce from (4.5) that
1
n; m ;
n(n + 1)h(2)
n (kR)
ikR
1
n; m :
n; m = √
(2)
′
n(n + 1) hn (kR) + kR(h(2)
n ) (kR)
n; m = − √
Then
∞ X
n
X
1
∇×z = −
ik
n=1 m=−n
√
ikR
1
n; m Mnm
(2)
′
n(n + 1) hn (kR) + kR(h(2)
n ) (kR)
1
n; m Nmn
+√
n(n + 1)h(2)
n (kR)
restricting to
R
!
and using (4.7) and (4.8) again proves the representation of Ge∗ .
Proof of Lemma 4.2. Using the de nition of z (4.1) z ∈ H (curl; BR ) satis es
(∇ × ; ∇ × z) − k 2 (n ; z) + ik ¡ Ge (xˆ × ; (xˆ × ) × xˆ ¿ =(n ; E s − Ehs ) ∈ H (curl; BR ):
Hence via the de nition of adjoint operator Ge∗ , and the fact that is real symmetric, we can see
that z is the weak solution of the Maxwell system:
∇ × ∇ × z − k 2 nz = n (E s − Ehs )
(∇ × z)T = ikGe∗ (zT )
on
in BR ;
R:
Hence, using the characterization of Ge∗ in Lemma 4.3, we see that z = w|BR where w is the weak
solution of the scattering problem:
∇ × ∇ × w − k 2 nw = g ∈ R3 ;
lim ((∇ × w) × x + ikrw) = 0;
r→∞
where
g(x) =
n(E s − Ehs ) if x ∈ BR ;
0
otherwise:
P. Monk / Journal of Computational and Applied Mathematics 100 (1998) 173–190
185
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 veri ed 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 ) satis es
∇ × ∇ × u − k 2 nu
= F in R3 ;
lim ((∇ × u) × x + ikru) = 0;
r→∞
where F ∈ L2 (BR ) is de ned by
(n ; F) = (n ; (E s − Ehs )) − a( ; ∇s);
∀ ∈ H (curl; BR )
and is extended to R3 by zero. Note that if we choose = ∇ for any ∈ H 1 (BR )=R we have
(n∇; F) = 0 for any ∈ H 1 (BR )=R so that xˆ · F = 0 on R and so ∇ · (nF) = 0 in all of R3 . Thus
u satis es a standard scattering problem and because nF is divergence free (see [19, p. 168]):
kukH 2 (BR ) 6 CkFkL2 (BR ) 6 CkE s − Ehs kL2 (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 ; (E s − Ehs ));
∀ ∈ H (curl; BR ):
(4.10)
Furthermore
z = u + ∇s
where s ∈ H 1 (BR )=R’s and z ∈ H (curl; BR ),
Thus, choosing = E s − Ehs in (4.10) and using the fact that E s − Ehs satis es the nite element
“orthogonality” relation
a(E s − Ehs ; h ) = 0;
∀ h ∈ Sh ;
we have
(n(E s − Ehs ); (E s − Ehs )) = a(E s − Ehs ; u) + a(E s − Ehs ; ∇s)
= a(E s − Ehs ; u − uh ) + a(E s − 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
E s satis es (1.8), and nally integrating by parts on each element we have
a(E s − Ehs ; u − uh ) = (∇ × (E s − Ehs ); ∇ × (u − uh )) − k 2 (n(E s − Ehs ); (u − uh ))
ˆ
+ik hGe (xˆ × (E s − Ehs )); (xˆ × (u − uh )) × xi
= ( f ; u − uh ) − (∇ × Ehs ; ∇ × (u − uh )) + k 2 (nEhs ; (u − uh ))
ˆ
−ik hGe (xˆ × Ehs ); (xˆ × (u − uh )) × xi
186
P. Monk / Journal of Computational and Applied Mathematics 100 (1998) 173–190
=
(
X
( f − ∇ × ∇ × Ehs + k 2 nEhs ; u − uh )K
K∈ h
− hnK ∇ × Ehs ; nK × (u − uh )i@K
+ik
hGe (xˆ × Ehs ); (xˆ × (u
− uh )) × xi
ˆ
)
(4.12)
where (·; ·)K is the (L2 (K))3 inner product, h·; ·i@K is the duality pairing between H −1=2 (Div; R ) and
H −1=2 Div; R )′ on @K, and nK is the outward normal to K.
If nf denotes a normal to a face f (chosen outward on R ) and if [·]f denotes the jump of the
quantity across a face f, we may rewrite (4.12) as
a(E s − Ehs ; u − uh ) =
(
X
K∈ h
+
( f − ∇ × ∇ × Ehs + k 2 nEhs ; u − uh )K
X D
nf × ∇ × Ehs
f∈Fint
+
X
f∈F
f
; (nf × (u − uh )) × nf
E
f
)
ˆ f :
hxˆ × ∇ × Ehs − ikGe (xˆ × Ehs ); (xˆ × (u − uh )) × xi
Next we choose uh =rh u (this is possible since u ∈ H 2 (
)). Then using the Cauchy–Schwarz inequality and the error estimate for u − rh u from (3.1) together with a similar estimate for the interpolant
on the faces of the mesh, we get
|a(E s − Ehs ; u − uh )|
6
(
X
K∈ h
+
k f − ∇ × ∇ × Ehs + k 2 nEhs kL2 (K) ku − uh kL2 (K)
X
k nf × ∇ × Ehs
X
kxˆ × ∇ × Ehs
ikGe (xˆ × Ehs )kL2 (f) k(xˆ × (u
f∈Fint
+
f∈F
6C
(
+
X
K∈ h
−
f
kL2 (f) knf × (u − uh )) × nf kL2 (f)
− uh )) × xk
ˆ L2 (f)
X
k nf × ∇ × Ehs
X
kxˆ × ∇ × Ehs
ikGe (xˆ × Ehs )kL2 (f) hf kukH 1 (f)
f∈F
)
k f − ∇ × ∇ × Ehs + k 2 nEhs kL2 (K) hK kukH 2 (K)
f∈Fint
+
−
f
kL2 (f) hf kukH 1 (f)
)
:
(4.13)
187
P. Monk / Journal of Computational and Applied Mathematics 100 (1998) 173–190
Next we use the following estimate which is valid for any q ∈ H 1 (Kf ) with constant C independent
of the mesh parameter hK (for a regular mesh)
kqk2L2 (f)
1
6 C kqkL2 (Kf ) k∇qkL2 (Kf ) +
kqk2L2 (Kf )
hK f
!
(4.14)
where Kf is a tetrahedron with f as one face. With this estimate we have
kukH 1 (f) 6 ChK−1=2
kukH 2 (Kf ) :
f
Using this estimate in (4.13) followed by the a priori estimate (4.9) provides the following estimate
for |a(E s − Ehs ; u − uh )|:
s
|a(E −
Ehs ; u
− uh )| 6 C
X
K∈ h
+
X
f∈Fint
h2K k f
−
∇ × ∇ × Ehs
s
hKf k nf × ∇ × Eh
f
+k
2
nEhs k2L2 (K)
!1=2
1=2
k2L2 (f)
1=2
X
s
s
+
hKf kxˆ × ∇ × Eh − ikGe (xˆ × Eh )kL2 (f)
kE s − Ehs kL2 (BR ) :
f∈F
(4.15)
The next step is to estimate the error in a(E s −Ehs ; ∇(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
a(E s − Ehs ; ∇(s − sh ))
= − k 2 (n∇E s − Ehs ; ∇(s − sh )) + ikhGe (xˆ × (E s − Ehs )); ∇(s − sh )i
=−
+
X
(∇ · f + k 2 ∇ · (nEhs ); s − sh )K −
X
ikhGe (xˆ × Ehs ); ∇(s − sh )if :
K∈ h
f∈F
X
K∈ h
k 2 hnK · (nEhs ); s − sh i@K
We take sh ∈ Sh to be the Scott–Zhang interpolant of s [7] which is de ned on functions in H 1 (BR ).
We have also assumed that f has a well-de ned global divergence so there is no jump in the normal
component of f across faces in the mesh ( f = 0 in the neighborhood of R ).
188
P. Monk / Journal of Computational and Applied Mathematics 100 (1998) 173–190
Next we integrate by parts on the boundary
integrals over the face of each element:
a(E s − Ehs ; ∇(s − sh ))
=−
−
X
K∈ h
(∇ · f + k 2 ∇ · (nEhs ); s − sh )K −
X
f∈
R
2
k xˆ ·
Ehs
−∇ ·
Ge (xˆ × Ehs ); s
and rewrite the face integrals by grouping the
R
X
f∈
− sh
int
f
k 2 h[f · (nEhs )]; s − sh if
:
If f 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 −k 2 ∇ · (nE s ) = ∇ · f . In addition
k 2 xˆ · E s = ik∇ Ge (xˆ × E s ) which just speci es continuity of the normal component of E s 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
|a(E s − Ehs ; ∇(s − sh ))|
6
X
k∇ · f + k 2 ∇ · (nEhs )kL2 (K) ks − sh kL2 (K) +
K∈ h
+
X
f∈
6
R
2
kk xˆ ·
Ehs
− ik∇ ·
Ge (xˆ × Ehs )kL2 (f) ks
X
f∈
int
k 2 k[nf · (nEhs )]kL2 (f) ks − sh kL2 (f)
− sh kL2 (f)
X
k∇ · f + k 2 ∇ · (nEhs )kL2 (K) ks − sh kL2 (K)
+C
X
ks − sh kL2 (Kf ) + hK1=2f ks − sh kH 1 (Kf )
k[nf · (nEhs )]kL2 (f) hK−1=2
f
X
kk 2 xˆ · Ehs − ik∇ · Ge (xˆ × Ehs )kL2 (f) hK−1=2
ks − sh kL2 (Kf ) + hK1=2f ks − sh kH 1 (Kf )
f
K∈ h
f∈
+C
f∈
int
R
6 CkE s − Ehs k
+
X
+
X
f∈
f∈
int
X
K∈ h
h2K k∇ · f + k 2 ∇ · (nEhs )k2L2 (K)
!1=2
1=2
hKf k[nf · (nEhs )]k2L2 (f)
1=2
s 2
2
s
;
hKf kk xˆ · Eh − ik∇ · Ge (xˆ × Eh )kL2 (f)
R
where Kf is a tetrahedron with f as one face and we have used the a priori estimate (4.2) to
estimate kskH 1 (
) . Combining the above estimate and (4.15) in (4.11) completes the proof.
P. Monk / Journal of Computational and Applied Mathematics 100 (1998) 173–190
189
Acknowledgments and disclaimer
E ort of Peter Monk sponsored by the Air Force Oce of Scienti c Research, Air Force Materials 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 notation 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 Scienti c 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 threedimensional 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. Deu
hard, 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-re
ecting 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 in R3 ,
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.
[21] P. Monk, A mixed method for approximating Maxwell’s equations, SIAM J. Numer. Anal. 28 (1991) 1610–1634.
190
P. Monk / Journal of Computational and Applied Mathematics 100 (1998) 173–190
[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 in R3 , Numer. Math. 35 (1980) 315–341.
[25] J.C. Nedelec, A new family of mixed nite elements in R3 , Numer. Math. 50 (1986) 57–81.
[26] R. Potthast, Integral equation methods in electromagnetic scattering from anisotropic media. Math. Methods in Appl.
Sci., to appear.