ON MAXWELL EQUATIONS WITH THE TRANSPARENT ...

Maxwell Equations with the Transparent Boundary Condition 285
propagation, see the review papers Givoli [5], Tsynkov [11], Hagstrom [7] and the references
therein. The purpose of this paper is to study the transparent boundary condition for Maxwell
scattering problems.
For any s ∈ C such that Re (s) > 0, we let E L
= L (E) and H L
= L (H) be respectively
the Laplace transform of E and H in time
E L
(x, s) =
∫ ∞
0
e −st E(x, t)dt, H L
(x, s) =
∫ ∞
0
e −st H(x, t)dt.
Since L (∂ t E) = sE L
−E 0 and L (∂ t H) = sH L
−H 0 , by taking the Laplace transform of (1.1)
and (1.2) we get
ε(sE L
− E 0 ) − ∇ × H L
= J L
in R 3 \ ¯D, (1.5)
µ(sH L
− H 0 ) + ∇ × E L
= 0 in R 3 \ ¯D, (1.6)
where J L
= L (J). Because J, E 0 , H 0 are supported inside B R = {x ∈ R 2 : |x| < R}, we know
that E L
satisfies the time-harmonic Maxwell equation outside B R
∇ × ∇ × E − k 2 E L
= 0
in R 3 \¯D,
where the wave number k = i √ εµs so that Im(k) = √ εµs 1 > 0. Let G e : H −1/2 (Div; Γ R ) →
H −1/2 (Div; Γ R ) be the Dirichlet to Neumann operator
By using (1.6) we have
G e (ˆx × E L
) = 1
ik ˆx × (∇ × E L ) = − 1 √ εµ
1
s ˆx × (∇ × E L ).
G e (ˆx × E L
) =
√ µ
ε ˆx × H L
on Γ R . (1.7)
For ˆx × E L
| ΓR = ∑ ∞ ∑ n
n=1 m=−n a mnU m n (ˆx) + b mnVn m (ˆx), we know that (cf., e.g., in Monk [9]
and also the discussion in Section 2)
G e (ˆx × E L
) =
∞∑
n∑
n=1 m=−n
−ikRb mn h (1)
n (kR)
z n (1) (kR)
U m n + a mnz n (1) (kR)
ikRh (1)
n (kR) Vm n ,
where U m n ,Vn m are the vector spherical harmonics, h (1)
n (z) is the spherical Hankel function of
the first order of order n, and z n (1) (z) = h (1)
n (z) + zh (1)′
n (z).
By taking the inverse Laplace transform of (1.7) we obtain the following transparent boundary
condition for the electromagnetic scattering problems
√ µ
ε ˆx × H = (L −1 ◦ G e ◦ L )(ˆx × E| ΓR ) on Γ R , (1.8)
where
(L −1 ◦ G e ◦ L )(ˆx × E| ΓR )
[ (
∞∑ n∑ √εµsRh (1)
= L −1 n (i √ ) ]
εµsR)
n=1 m=−n z n (1) (i √ ∗ b mn (R, t) U m n
εµsR)
(
−
[L −1 z n (1) (i √ ) ]
εµsR)
√ (1) εµsRh n (i √ ∗ a mn (R, t) Vn m , (1.9)
εµsR)