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

problems. In addition, the far field errors incurred by our implementation are of the same order as
those resulting from the classical Combined Field Integral Equation counterparts. Thus, the important
gains in computational complexity make the Regularized Combined Field Integral Equations
discussed in this text a viable method of solution to PEC scattering problems.
Acknowledgments
Yassine Boubendir gratefully acknowledge support from NSF through contract DMS-1016405.
Catalin Turc gratefully acknowledge support from NSF through contract DMS-1312169.
6 Appendix
We present in this section a proof of the technical Lemma 3.2
Lemma 6.1 There exist constants C j > 0, j = 1, . . . , 4 and a number ˜k 0 > 0 such that
(i) C 1 k −2 (n 2 + k 2 ) 1/2 ≤ iJ ′ n+1/2 (ik)(H(1) n+1/2 )′ (ik) ≤ C 2 k −2 (n 2 + k 2 ) 1/2
(ii) 1 4 (n2 + k 2 ) −1/2 ≤ −S (1)
n (ik) ≤ C 3 (n 2 + k 2 ) −1/2 + C 4 k −2 ,
(iii)|J ′ n+1/2 (ik)H(1) n+1/2 (ik)| ≤ C 4k −1 , |J n+1/2 (ik)(H (1)
n+1/2 )′ (ik)| ≤ C 4 k −1
for all k > ˜k 0 and all n ≥ 0.
Proof. (i) We begin by using the representation of the functions J n+1/2 (ik) and H (1)
terms of the Bessel and Hankel functions of the third kind
iJ ′ n+1/2 (ik)(H(1) n+1/2 )′ (ik) = − 2 π I′ n+1/2 (k)K′ n+1/2 (k)
n+1/2
(ik) in
where I ′ ν(k) > 0 and K ′ ν(k) < 0 for ν ≥ 0. Using the uniform asymptotic expansions as ν → ∞
(Formulas 9.7.9 and 9.7.10 in [1])
where µ = √ 1 + z 2 + ln
I ν(νz)
′ 1 (1 + z 2 ) 1 4
∼ √ e νµ (1 + O(ν −1 ))
2πν z
√ π
K ν(νz) ′ (1 + z 2 ) 1 4
∼ −√ e −νµ (1 + O(ν −1 )) (94)
2ν z
z
1+ √ 1+z 2 , we get that there exists a constant N 0 > 0 such that
1
4 k−2 ((n + 1/2) 2 + k 2 ) 1/2 ≤ −I ′ n+1/2 (k)K′ n+1/2 (k) ≤ k−2 ((n + 1/2) 2 + k 2 ) 1/2 ,
for all n > N 0 , k > 0. Given that we can choose the constant N 0 above to further satify (n +
1/2) 2 + k 2 < 2(n 2 + k 2 ) for all n > N 0 and all k > 0, we obtain the following estimate
1
4 k−2 (n 2 + k 2 ) 1/2 ≤ −I ′ n+1/2 (k)K′ n+1/2 (k) ≤ √ 2k −2 (n 2 + k 2 ) 1/2 , for all n > N 0 , k > 0. (95)
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