Well-conditioned boundary integral formulations for the ... - Njit

and
Λ (1)
n (K) = [Kj n (K)] ′ [Kh (1)
n (K)] ′ (64)
Λ (2)
n (K) = −K 2 j n (K)h (1)
n (K). (65)
In the formulas above j n and h (1)
n denote the spherical Bessel and Hankel functions of the first
kind, respectively. From the spectral properties of the operators T K presented above, the spectral
properties of the operator n×S K can be derived immediately. Indeed, it follows from the definition
(7) (with k = K) that
(n × S K )( −−→ curl S 2Yn m ) = Λ(2) n (K)
iK ∇ S 2Y n m , (66)
and, using the definition of the Laplace-Beltrami operator ∆ S 2 = div S 2∇ S 2 and the fact that the
spherical harmonic Yn m is an eigenfunction of both the operator ∆ S 2, with eigenvalue −n(n + 1),
as well as the single layer acoustic operator corresponding to the wavenumber K [44]
∫
(S K Y n (m) )(x) = G K (x − y)Yn m (y)dσ(y) = iKj n (K)h (1)
n (K)Yn m (x), |x| = 1 (67)
S 2
we obtain
(n × S K )(∇ S 2Yn m ) = 1
iK (Λ(1) n (K) + n(n + 1)j n (K)h (1)
n (K)) −−→ curl S 2Yn m = S n
(1) (K) −−→ curl S 2Yn m . (68)
Consequently, the choice of the regularizing operators R = η κ n × S iκ , κ > 0 advocated in [15]
leads to boundary integral operators with the following spectral properties
( ( )
I 1
2 − K k + ηκT k ◦ (n × S iκ ))
∇ S 2Yn m =
2 + λ n(k) + ηκΛ (2)
n (k)S n
(1) (iκ) ∇ S 2Yn
m
( ) I −−→
2 − K k + ηκT k ◦ (n × S iκ ) curlS 2Yn m =
= A (1)
n (k, κ, η)∇ S 2Yn
m ( 1
2 − λ n(k) − ηΛ (1)
n (k)Λ (2)
n (iκ)
) −−→ curlS 2Yn
m
= A (2)
n (k, κ, η) −−→ curl S 2Yn m . (69)
On the other hand, the choice of the regularizing operators R = −ξT iκ , κ > 0, ξ > 0 introduced
in [30] leads to boundary integral operators with the following spectral properties
( )
( )
I 1
2 − K k − ξT k ◦ T iκ ∇ S 2Yn m =
2 + λ n(k) − ξΛ (2)
n (k)Λ (1)
n (iκ) ∇ S 2Yn
m
( I
2 − K k − ξT k ◦ T iκ
) −−→ curlS 2Y m n =
= B n
(1) (k, κ, ξ)∇ S 2Yn
m ( 1
2 − λ n(k) − ξΛ (1)
n (k)Λ (2)
n (iκ)
) −−→ curlS 2Yn
m
= B n
(2) (k, κ, ξ) −−→ curl S 2Yn
m
. (70)
We note that the operators ( I
2 − K k + ηκT k ◦ (n × S iκ ) ) and ( I
2 − K )
(
k − ξT k ◦ T iκ are diagonal in
∇
the orthonormal basis S 2 Yn
√ m −−→
)
curl
, S 2 Yn
√ m of the space L 2 (T M(S 2 )), having the same
n(n+1) n(n+1)
1≤n,−n≤m≤n
eigenvalues (A (1)
n (k, κ, η), A (2)
n (k, κ, η) and (B n (1) (k, κ, ξ), B n
(2) (k, κ, ξ)) in that basis. We investigate
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