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

that the heuristic bounds are true.
(9.3.31)–(9.3.32))
We use the following asymptotic expansions [1] (Formulas
J ν (ν) = aν −1/3 + O(ν −5/3 ) Y ν (ν) = − √ 3aν −1/3 + O(ν −5/3 )
2
which are valid as ν → ∞, and where a = 1/3
the following estimate is valid
3 2/3 Γ(2/3)
b ν (k) ∼ 2 √ 2a 2 k 1/3 , ν ∼ k, k → ∞.
, we obtain that for large enough values of k
The plot of k −1/3 max k≤ν b ν (k) in Figure 1 seems to be consistent with the estimate just derived
above, given that 2 √ 2a 2 ≈ 0.56592. Furthermore, we use the result in Formula 9.3 in [1] which is
valid for fixed k and ν → ∞
J ν (k)H (1)
ν (k) ∼ 1
2πν
( ) ek 2ν
− i 1
2ν πν = O(ν−1 )
and we get that for a fixed and large enough k there exists a constant C(k) such that for all ν large
enough we have
b ν (k) ≤ C(k).
The next result is a coercivity result that establishes that given the choice ξ = k 1/3 in the
definition of the regularizing operators R = −ξT ik , the real parts of the eigenvalues B n (1) (k, k, k 1/3 )
and B n
(2) (k, k, k 1/3 ) are bounded from below by strictly positive constants for all values of n for
large enough values of the wavenumber k. More precisely, we have that
Theorem 3.8 There exists a constant C 9 and a wavenumber k ′ such that for all k ≥ k ′ we have
that
min{R(B n (1) (k, k, k 1/3 )), R(B n (2) (k, k, k 1/3 ))} ≥ C 9 for all n ≥ 1.
Proof. We take into account the estimate (i) established in Lemma 3.2 to obtain
R(B n
(1) (k, k, k 1/3 )) ≥ − πk
2 J n+1/2 ′ (k)Y n+1/2(k) + C 1π 2 k 1/3
4
On the other hand, the estimate (57) can be written explicitly in the form
R(d n+1/2 (k, k)) = − πk
2 J ′ n+1/2 (k)Y n+1/2(k) + kπ 2 J 2 n+1/2 (k) ≥ 1 2
J 2 n+1/2 (k)√ n 2 + k 2 − |b (1)
n (k, k 1/3 )|.
for all 1 ≤ n
and sufficiently large k. Since √ n 2 + k 2 ≥ k and C 1 πk 1/3 ≥ 2 for sufficiently large k, it follows
from the previous two estimates that
R(B (1)
n (k, k, k 1/3 )) ≥ R(d n+1/2 (k, k)) − |b (1)
n (k, k 1/3 )| ≥ 1 2 − |b(1) n (k, k 1/3 )| for all n ≥ 1
and k sufficiently large. The result of the theorem now follows if we take into account estimates (81)
and the result established in Theorem 3.4.
(
∇
Given the fact that S 2 Yn
√ m −−→
)
curl
, S 2 Yn
√ m constitute an orthonormal basis for the
n(n+1) n(n+1)
1≤n,−n≤m≤n
space L 2 (T M(S 2 )) (and hence H m−1/2
div
(S 2 ) for all 0 ≤ m), we derive from the results in Theorem 3.3,
equation (82), and Theorem 3.8 the following
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