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

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The hyper-singular integrals in the definition of K k and T k should be interpreted in the sense of Cauchy principal value integrals. Regularized Combined Field Integral Equations for the solution of electromagnetic scattering problems with perfectly electrically conducting boundary conditions seek representations of the electromagnetic fields of the form ∫ E s (x) = curl G k (x − y)a(y)dσ(y) Γ + i ∫ k curl curl G k (x − y)(Ra)(y)dσ(y), (8) Γ H s (x) = 1 ik curl E(x), x ∈ R3 \ D, (9) where R denotes a tangential operator to be specified in what follows. Owing to the trace properties (5), equations (8) and (9) define outgoing solutions to the Maxwell equations with perfectconductor boundary conditions provided the tangential density a is a solution to the integral equation a 2 − K ka + T k (Ra) = −n × E i . (10) If the operator R is selected such that, modulo a compact operator, the composition of T k and R plus I/2 is a bounded invertible operator with a bounded inverse in the appropriate functional space setting, then the representations (8)-(9) lead to Regularized Combined Field Integral Equations (CFIER). Operators R with the aforementioned property are referred to as right regularizing operators for the operators T k . We note that the classical Combined Field Integral Equations assume the choice R = ξ n × I, ξ ∈ R [29, 35], and thus those operators R are not regularizing operators for T k . If R is chosen as a right regularizing operator for T k , the integral operator on the left hand side of (10) is a Fredholm operator, and, thus, the unique solvability of equation (10) is equivalent to the injectivity of the left-hand-side operator. Several operators R with the aforementioned property have been proposed and analyzed in the literature [15]. The aim of this paper is to present a general strategy to produce such regularizing operators R that encompasses many of the regularizing operators previously considered. The starting point of constructing suitable regularizing operators R is the Calderón’s identity = I 4 − K2 k [36]. The regularizing operators R should thus resemble the electric field operator T k . If the choice R = T k were made, the ensuing CFIER operators would not be injective since for certain wavenumbers k the operators I 2 − K k are not injective [29, 36]. In order to ensure the injectivity of the resulting CFIER operators, one strategy is to modify the wavenumber in the Tk 2 definition of the regularizing operator R = T k . Specifically, we propose a general regularizing operator R of the form R = η n × S K + ζ TK 1 div Γ (11) where η and ζ are complex numbers and K is a complex wavenumber such that IK > 0. We first establish that given the choice in equation (11), and for general closed and smooth manifolds Γ, the composition T k ◦ R can be represented as a compact perturbation of an invertible diagonal matrix operator. The main idea in the proof is to use the Helmholtz decomposition of tangential vector fields on the smooth surface Γ in conjunction with the regularity properties of integral operators with pseudo-homogeneous kernels [48] in the context of the Sobolev spaces H m− 1 2 div (Γ) of H m− 1 2 (T M(Γ)) tangential vector fields that admit an H m− 1 2 divergence [36]. 6
In what follows we use the notations and relations [44]: −−→ curl Γ φ = ∇ Γ φ × n (12) curl Γ a = div Γ (a × n) (13) −−→ ∆ Γ φ = div Γ ∇ Γ φ = −curl ΓcurlΓ φ (14) where a is a tangential vector field and where φ is a scalar function defined on Γ. A few relevant properties of the Helmholtz decomposition of tangential vector fields in Sobolev spaces are recounted in what follows. For a given smooth tangential vector field a ∈ H m−1/2 div (Γ) we have the Helmholtz decomposition [36, 44] a = ∇ Γ φ + −−→ curl Γ ψ + ω, (15) where ω is a harmonic vector field (i.e., its divergence and curl vanish), and where, using the rightinverse [54] ∆ −1 Γ : Hs (Γ) → H s+2 (Γ) of the Laplace-Beltrami operator ∆ Γ , the functions φ and ψ in the Helmholtz decomposition (15) are given by φ = ∆ −1 Γ div Γ a and ψ = −∆ −1 Γ curl Γ a. (Note that for simply connected surfaces Γ we necessarily have ω = 0.) Clearly for a tangential vector field a ∈ H m−1/2 div (Γ) we have φ ∈ H m+3/2 (Γ) and ψ ∈ H m+1/2 (Γ) [36]; the corresponding projection operators onto the spaces of gradients, rotationals and harmonic fields will be denoted by Π ∇Γ = ∇ Γ ∆ −1 Γ Π−−→ curlΓ = − −−→ curl Γ ∆ −1 Γ div Γ : H m−1/2 div (Γ) → H m−1/2 div (Γ), (16) curl Γ : H m−1/2 div (Γ) → H m−1/2 .div (Γ), and (17) Π ⋂ −−→ ∇Γ curlΓ = I − Π ∇Γ − Π−−→ curlΓ : H m−1/2 div (Γ) → H m−1/2 div (Γ). (18) We will also make use of a more detailed version of the Helmholtz decomposition (15) that we will review in what follows. For a smooth closed two-dimensional manifold Γ the Laplace Beltrami operator ∆ Γ admits a complete and countable sequence of eigenfunctions which form an orthonormal basis in L 2 (Γ) [44], denoted by {Y n } 0≤n such that −∆ Γ Y n = γ n Y n , γ n > 0 for 0 < n. (19) These eigenfunctions of the Laplace Beltrami operator turn out to be the building block for a complete system of eigenfunctions of the vector Laplace-Beltrami operator (or Hodge Laplace operator) −→ ∆Γ = ∇ Γ div Γ − −−→ curl Γ curl Γ . Indeed, the system {∇ Γ Y n , −−→ curl Γ Y n } 1≤n forms a system of orthogonal nontrivial eigenvectors for −→ ∆ Γ with the same eigenvalues γ n − −→ ∆ Γ ∇ Γ Y n = γ n ∇ Γ Y n (20) − −→ −−→ −−→ ∆ ΓcurlΓ Y n = γ ncurlΓ Y n . (21) However, the system {∇ Γ Y n , −−→ curl Γ Y n } 1≤n may not constitute a complete basis for the space of tangential vector fields that are square integrable, as, depending on the topology of Γ, the operator −→ ∆Γ can have a nontrivial but nevertheless finite-dimensional null-space that consists of tangential vector fields whose tangential divergence and tangential rotational are simultaneously zero. Thus, if we denote by {u p , p = 1, . . . , N} an orthonormal basis to the null space of −→ ∆ Γ in the case when this is non-trivial (e.g. the genus of Γ is not zero, i.e. Γ is not simply connected, in which case N equals the first Betti number of the manifold Γ which is two times the number of “holes” in Γ), the 7
Page 1 and 2: Well-conditioned boundary integral
Page 3 and 4: order effect of the pseudodifferent
Page 5: scattering problems. 2 Regularized
Page 9 and 10: Since n × S k : H s (T M(Γ)) →
Page 11 and 12: for all a ∈ H −1/2 div (Γ), a
Page 13 and 14: Combining the estimate above with t
Page 15 and 16: We make use of the following vector
Page 17 and 18: Since γ n > 0 and RK ≥ 0 and IK
Page 19 and 20: where the operator Kk T is defined
Page 21 and 22: next the behavior of the eigenvalue
Page 23 and 24: 4 √ m 2 0 + k2 ≤ k 4/3 . It fol
Page 25 and 26: 0.62 0.6 k −1/3 max k b (k) 0.58
Page 27 and 28: Corollary 3.9 Let B = I 2 − K k
Page 29 and 30: 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 20 40
Page 31 and 32: 0.515 0.51 Coercivity constant PSB
Page 33 and 34: enclosing the scatterer; (2) the nu
Page 35 and 36: 2. Compute the surface derivatives
Page 37 and 38: D N C k A k,Sik/2 B k,2,k,0.4H 2/3
Page 39 and 40: 30 25 20 15 10 5 20 40 60 80 100 12
Page 41 and 42: Using the asymptotic expansion (For
Page 43 and 44: References [1] Abramowitz, M., and
Page 45 and 46: [32] Darbas, M., Generalized combin

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