A generalized Calderón formula for open-arc diffraction problems: theoretical considerations

Abstract

We deal with the general problem of scattering by open-arcs in two-dimensional space. We show that this problem can be solved by means of certain second-kind integral equations of the formÑS[ϕ] = f , whereÑ andS are first-kind integral operators whose composition gives rise to a generalized Calderón formula of the formÑS =J τ 0 +K in a weighted, periodized Sobolev space. (HereJ τ 0 is a continuous and continuously invertible operator andK is a compact operator.) TheÑS formulation provides, for the first time, a second-kind integral equation for the open-arc scattering problem with Neumann boundary conditions. Numerical experiments show that, for both the Dirichlet and Neumann boundary conditions, our second-kind integral equations have spectra that are bounded away from zero and infinity as k → ∞; to the authors' knowledge these are the first integral equations for these problems that possess this desirable property. This situation is in stark contrast with that arising from the related classical open-surface hypersingular and single-layer operators N and S, whose composition NS maps, for example, the function φ = 1 into a function that is not even square integrable. Our proofs rely on three main elements: 1) Algebraic manipulations enabled by the presence of integral weights; 2) Use of the classical result of continuity of the Cesàro operator; and 3) Explicit characterization of the point spectrum ofJ τ 0 , which, interestingly, can be decomposed into the union of a countable set and an open set, both of which are tightly clustered around − 1 4 . As shown in a separate contribution, the new approach can be used to construct simple spectrally-accurate numerical solvers and, when used in conjunction with Krylov-subspace iterative solvers such as GMRES, it gives rise to dramatic reductions of iteration numbers vs. those required by other approaches.

High-Performance Computational Electromagnetics in Frequency-Domain and Time-Domain

“…Meaningful progress concerning well conditioned integral algorithms took place as a result of work sponsored by this contract, including rigorous mathematical theory and powerful numerical algorithms with applicability in a number of fields of science and engineering [12][13][14][15][16][17][18][19][20][21]. A variety of problems and configurations were thus considered, including problems of scattering by open surfaces [12][13][14]; improved integral methods for closed surfaces [15]; problems concerning propagation and scattering by penetrable scatterers [16]; studies of absorption properties of conducting materials containing asperities [17,18]; as well as new methods for evaluation of Laplace eigenfunctions on general domains and under challenging boundary conditions [19,20].…”

“…A variety of problems and configurations were thus considered, including problems of scattering by open surfaces [12][13][14]; improved integral methods for closed surfaces [15]; problems concerning propagation and scattering by penetrable scatterers [16]; studies of absorption properties of conducting materials containing asperities [17,18]; as well as new methods for evaluation of Laplace eigenfunctions on general domains and under challenging boundary conditions [19,20]. A rigorous convergence proof for the original methods [22] was provided in [21].…”

“…This problem has many applications in optics, from the very small to the very large (from nano-scale optical devices up and including the discovery of planets) as well as applications in stealth, antenna design, electronic and photonic devices, etc. For the first time a regularized integral equation yielding a Fredholm equation of the second kind for opensurface problems was put forth in the contributions [12][13][14], including rigorous analysis of the equations and efficient implementations in two-and three-dimensional space. In practice, the second-kind character of the equation enables solution by iterative solvers in very small numbers of iterations and it thus enables effective solution of the problem.…”

“…Complete proofs of the well conditioned character of the aforementioned integral equation formulation for open surfaces was provided in [13,14]. A numerical analysis of the related closed surface solvers [22], in turn, was for the first time put forth in [21].…”

Regularized integral equation methods for elastic scattering problems in three dimensions

“…We are not aware of previous applications of spectral regularization methods to integral equations for the elastic Neumann problem.Elastic versions of the Calderón formulas for open surfaces are not known at present. In view of the acoustic open-surface Calderön relations [13,14,28] and the related study [16] for the 2D open-arc elastic case, we consider "weighted" versions S w and N w of the single-layer and hyper-singular operators which,…”