## 1. Introduction

[2] Problems of diffraction by *infinitely thin open surfaces* play central roles in a wide range of problems in science and engineering, with important applications to antenna and radar design, electronics, optics, etc. Owing to difficulties inherent in rigorous mathematical treatment of open surface scatterers, such problems are often treated by means of physically approximate models [*Keller*, 1962; *Karam and Fung*, 1983; *Lu and Ando*, 2012] whose accuracy is limited by the physical approximations made. Like other wave scattering problems, on the other hand, open surface problems can be rigorously treated—either by means of numerical methods that rely on approximation of Maxwell's Equations over volumetric domains (on the basis of, e.g., finite difference or finite element methods) or by methods based on boundary integral equations. As a result of the singular character of the electromagnetic fields in the vicinity of open edges, open-surface scattering configurations present major difficulties for both volumetric and boundary integral methods. Boundary integral approaches, which require discretization of domains of lower dimensionality than those involved in volumetric methods, can generally be treated efficiently, even for high-frequencies, by means of accelerated iterative scattering solvers [*Bleszynski et al.*, 1996; *Bruno and Kunyansky*, 2001; *Rokhlin*, 1993]. Unfortunately, integral methods for open surfaces do not give rise, at least in their classical formulations, to Fredholm integral operators of the second-kind, and can therefore prove to be computationally expensive—as the eigenvalues of the resulting equations accumulate at zero and/or infinity and, thus, iterative solution of these equations often requires large numbers of iterations.

[3] This paper presents new Fredholm integral equations of *second-kind* and associated numerical algorithms for problems of scattering by two-dimensional open arcs Γ (i.e., infinite cylinders of cross-section Γ) under either transverse electric (TE) or transverse magnetic (TM) incident fields. The new second-kind Fredholm equations (which result from composition of appropriately modified versions **S**_{ω} and **N**_{ω} of the classical single-layer and hypersingular integral operators **S** and **N**) provide, for the first time, a generalization of the classical closed-surface Calderón formulas to the open-arc case. In particular, the new formulations possess highly favorable spectral properties: their eigenvalues are highly clustered, and they remain bounded away from zero and infinity, even for problems of very high frequency. When used in conjunction with spectrally accurate discretization rules and Krylov-subspace linear algebra solvers such as GMRES, the new open-arc formulations produce results of high accuracy in small numbers of iterations—for low and high frequencies alike.

[4] The new second-kind formulation for the TM problem is particularly beneficial, as it gives rise to order-of-magnitude improvements in computing times over the corresponding weighted hypersingular formulation. Such gains do not occur in the TE case: although the new second-kind TE equation requires fewer iterations than the corresponding weighted first kind formulation, the total computational cost of the second-kind equation is generally higher in the TE case—since the application of the first-kind operator can be significantly less expensive than the application of the composite second-kind operator.

[5] The difficulties that arise as integral equations are used to treat open surface scattering problems are of course well known, and many contributions have been devoted to their treatment; early work in these regards can be found for instance in *Meixner* [1949], *Maue* [1949], and *Mittra et al.* [1973]. Like the present work, *Povzner and Suharevskiĭ* [1960] and *Christiansen and Nédélec* [2000] seek to tackle these problems by means of generalizations of the classical Calderón relations to the case of open surfaces. The first of these contributions establishes that the combination **NS** can be expressed in the form **I** + **T**_{K}, where the kernel **K**(*x*, *y*) of the operator **T**_{K} has local singularity of at most *O* . This early result however does not take into account the singular edge behavior; the resulting operator **T**_{K} is not compact (in fact it gives rise to extreme singularities at the edge (S. Lintner and O. Bruno, A generalized Calderón formula for open-arc diffraction problems: Theoretical considerations, submitted to *Proceedings of the Royal Society of Edinburgh*, 2012; available at http://arxiv.org/abs/1204.3699)) and **I** + **T**_{K} is therefore not a second-kind operator in any numerically meaningful functional space. When used in conjunction with boundary elements that vanish on the edges (by means of well-chosen projections) however, the combination **NS** does give rise to reduction of iteration numbers, as demonstrated in *Christiansen and Nédélec* [2000] through numerical examples for low frequency problems. This contribution does not include details on accuracy, and it does not utilize integral weights to resolve the solution's edge singularity. A related but different method was introduced in *Antoine et al.* [2005] which exhibits, once again, low iteration numbers for low-frequency problems, but which does not resolve the singular edge behavior and for which no accuracy studies have been presented. Finally, high-order integration rules for the single-layer and hypersingular operators adapted to open arcs were introduced in a Galerkin framework in *Stephan and Wendland* [1984], *Hsiao et al.* [1991], and *Stephan and Tran* [1998]. These methods have thus far only been applied for simple geometries and at low frequencies, and limited information is available on the actual convergence properties and performance of their computational implementations.

[6] A second class of methods include those proposed in *Atkinson and Sloan* [1991], *Mönch* [1996], and *Jiang and Rokhlin* [2004]. The contribution by *Atkinson and Sloan* [1991], some aspects of which are incorporated in our method, treats the Dirichlet problem for Laplace's equation by means of second kind equations. The basis of this approach lies in the observation that the cosine basis has the dual positive effect of diagonalizing the logarithmic potential for a straight arc and removing the singular edge behavior—so that the inverse of the logarithmic potential can be easily computed and used as a preconditioner to produce a second kind operator for a general arc. The approach by *Mönch* [1996], which also uses a cosine basis, treats the Neumann problem for the non-zero frequency Helmholtz equation with spectral accuracy by means of first kind equations. The contribution by *Jiang and Rokhlin* [2004], finally, treats, just like *Atkinson and Sloan* [1991], the Laplace problem by means of second-kind equations resulting from inversion of the straight arc logarithmic potential; like *Mönch* [1996], further, it produces spectral accuracy through use of the cosine transforms. The second-kind integral approach developed in *Atkinson and Sloan* [1991] and later revisited in *Jiang and Rokhlin* [2004] seems essentially limited to the specific problem for which it was proposed: neither an extension to the Neumann problem nor to the full three dimensional problem seem straightforward. And, more importantly, this approach does not lead to adequately preconditioned equations for non-zero frequencies: a simple experiment conducted in section 3.4 shows that a direct generalization of the algorithm *Atkinson and Sloan* [1991] to the Helmholtz problem generally requires significantly *more* linear algebra iterations than are necessary if the operator **S**_{ω} alone is used.

[7] The remainder of this paper is organized as follows: after recalling in section 2 the classical boundary integral formulations for the TE and TM open-arc scattering problems, in sections 3.1 and 3.2 we present our new weighted operators **S**_{ω} and **N**_{ω} as well as certain periodized counterparts and (which are obtained by considering sinusoidal changes of variables for source and observation points). Our main result, the generalized Calderón formula, is presented in section 3.3. Section 3.4 then presents results of numerical evaluation of eigenvalues for a non-trivial open arc problem, illustrating the spectral properties of previous open-arc operators as well as the second kind operators introduced in this paper. Theoretical considerations concerning the new second-kind equations are presented in section 4, including a succinct but complete proof of the open-arc Calderón formulae; a more detailed theoretical discussion, including full mathematical technicalities, can be found in Lintner and Bruno (submitted manuscript, 2012). The high-order quadrature rules we use for evaluation of the new integral operators are described in section 5. Numerical results, finally, are presented in section 6 for a wide range of frequencies and for various geometries (including a brief study of resonant open cavities) demonstrating the uniformly well conditioned character of the integral formulations proposed in this paper.