Catalin Turc

A high-order integral algorithm for highly singular PDE solutions in Lipschitz domains

We present a new algorithm, based on integral equation formulations, for the solution of constant-coefficient elliptic partial differential equations (PDE) in closed two-dimensional domains with non-smooth boundaries; we focus on cases in which the integral-equation solutions as well as physically meaningful quantities (such as, stresses, electric/magnetic fields, etc.) tend to infinity at singular boundary points (corners). While, for simplicity, we restrict our discussion to integral equations associated with the Neumann problem for the Laplace equation, the proposed methodology applies to integral equations arising from other types of PDEs, including the Helmholtz, Maxwell, and linear elasticity equations. Our numerical results demonstrate excellent convergence as discretizations are refined, even around singular points at which solutions tend to infinity. We demonstrate the efficacy of this algorithm through applications to solution of Neumann problems for the Laplace operator over a variety of domains—including domains containing extremely sharp concave and convex corners, with angles as small as π/100 and as large as 199π/100.

Electromagnetic integral equations requiring small numbers of Krylov-subspace iterations

We present a new class of integral equations for the solution of problems of scattering of electromagnetic fields by perfectly conducting bodies. Like the classical Combined Field Integral Equation (CFIE), our formulation results from a representation of the scattered field as a combination of magnetic- and electric-dipole distributions on the surface of the scatterer. In contrast with the classical equations, however, the electric-dipole operator we use contains a regularizing operator; we call the resulting equations Regularized Combined Field Integral Equations (CFIE-R). Unlike the CFIE, the CFIE-R are Fredholm equations which, we show, are uniquely solvable; our selection of coupling parameters, further, yields CFIE-R operators with excellent spectral distributions-with closely clustered eigenvalues-so that small numbers of iterations suffice to solve the corresponding equations by means of Krylov subspace iterative solvers such as GMRES. The regularizing operators are constructed on the basis of the single layer operator, and can thus be incorporated easily within any existing surface integral equation implementation for the solution of the classical CFIE. We present one such methodology: a high-order Nystrom approach based on use of partitions of unity and trapezoidal-rule integration. A variety of numerical results demonstrate very significant gains in computational costs that can result from the new formulations, for a given accuracy, over those arising from previous approaches.

Well conditioned boundary integral equations for two-dimensional sound-hard scattering problems in domains with corners

We present several well-posed, well-conditioned direct and indirect integral equation formulations for the solution of two-dimensional acoustic scattering problems with Neumann boundary conditions in domains with corners. We focus mainly on Direct Regularized Combined Field Integral Equations (DCFIE-R) formulations whose name reflects that (1) they consist of combinations of direct boundary integral equations of the second-kind and first-kind integral equations which are preconditioned on the left by coercive boundary single-layer operators, and (2) their unknowns are physical quantities, i.e the total field on the boundary of the scatterer. The DCFIE-R equations are shown to be uniquely solvable in appropriate function spaces under certain assumptions on the coupling parameter. Using Calderon’s identities and the fact that the unknowns are bounded in the neighborhood of the corners, the integral operators that enter the DCFIE-R formulations are recast in a form that involves integral operators that are expressed by convergent integrals only. The polynomially-graded mesh quadrature introduced by Kress [30] enables the high-order resolution of the weak singularities of the kernels of the integral operators and the singularities in the derivatives of the unknowns in the vicinity of the corners. This approach is shown to lead to an efficient, high-order Nystrom method capable of producing solutions of sound-hard scattering problems in domains with corners which require small numbers of Krylov subspace iterations throughout the frequency spectrum. We present a variety of numerical results that support our claims.

Superalgebraically convergent smoothly windowed lattice sums for doubly periodic Green functions in three-dimensional space

This work, part I in a two-part series, presents: (i) a simple and highly efficient algorithm for evaluation of quasi-periodic Green functions, as well as (ii) an associated boundary-integral equation method for the numerical solution of problems of scattering of waves by doubly periodic arrays of scatterers in three-dimensional space. Except for certain ‘Wood frequencies’ at which the quasi-periodic Green function ceases to exist, the proposed approach, which is based on smooth windowing functions, gives rise to tapered lattice sums which converge superalgebraically fast to the Green function—that is, faster than any power of the number of terms used. This is in sharp contrast to the extremely slow convergence exhibited by the lattice sums in the absence of smooth windowing. (The Wood-frequency problem is treated in part II.) This paper establishes rigorously the superalgebraic convergence of the windowed lattice sums. A variety of numerical results demonstrate the practical efficiency of the proposed approach.

Windowed Green Function Method for Layered-Media Scattering

This paper introduces a new windowed Green function (WGF) method for the numerical integral-equation solution of problems of electromagnetic scattering by obstacles in the presence of dielectric or conducting half-planes. The WGF method, which is based on the use of smooth windowing functions and integral kernels that can be expressed directly in terms of the free-space Green function, does not require evaluation of expensive Sommerfeld integrals. The proposed approach is fast, accurate, flexible, and easy to implement. In particular, straightforward modifications of existing (accelerated or unaccelerated) integral-equation solvers suffice to incorporate the WGF capability. The method relies on a certain integral equation posed on the union of the boundary of the obstacle and a small flat section of the interface between the penetrable media. Our analysis and numerical experiments demonstrate that both the near- and far-field errors resulting from the proposed approach decrease faster than any negative power of the window size. In the examples considered in this paper the proposed method is up to thousands of times faster, for a given accuracy, than a corresponding method based on use of Sommerfeld integrals.

Regularized Combined Field Integral Equations for Acoustic Transmission Problems

We present a new class of well-conditioned integral equations for the solution of two and three dimensional scattering problems by homogeneous penetrable scatterers. Our novel boundary integral equations result from using regularizing operators which are suitable approximations of the admittance operators that map the transmission boundary conditions to the exterior and, respectively, interior Cauchy data on the interface between the media. We refer to these regularized boundary integral equations as generalized combined source integral equations (GCSIE). The admittance operators can be expressed in terms of Dirichlet-to-Neumann operators and their inverses. We construct our regularizing operators in terms of simple boundary layer operators with complex wavenumbers. The choice of complex wavenumbers in the definition of the regularizing operators ensures the unique solvability of the GCSIE. The GCSIE are shown to be integral equations of the second kind for regular enough interfaces of material discontinu...

Well-conditioned boundary integral equation formulations for the solution of high-frequency electromagnetic scattering problems

Abstract We present several versions of Regularized Combined Field Integral Equation (CFIER) formulations for the solution of three dimensional frequency domain electromagnetic scattering problems with Perfectly Electric Conducting (PEC) boundary conditions. Just as in the Combined Field Integral Equations (CFIE), we seek the scattered fields in the form of a combined magnetic and electric dipole layer potentials that involves a composition of the latter type of boundary layers with regularizing operators. The regularizing operators are of two types: (1) modified versions of electric field integral operators with complex wavenumbers, and (2) principal symbols of those operators in the sense of pseudodifferential operators. We show that the boundary integral operators that enter these CFIER formulations are Fredholm of the second kind, and invertible with bounded inverses in the classical trace spaces of electromagnetic scattering problems. We present a spectral analysis of CFIER operators with regularizing operators that have purely imaginary wavenumbers for spherical geometries—we refer to these operators as Calderon–Ikawa CFIER. Under certain assumptions on the coupling constants and the absolute values of the imaginary wavenumbers of the regularizing operators, we show that the ensuing Calderon–Ikawa CFIER operators are coercive for spherical geometries. These properties allow us to derive wavenumber explicit bounds on the condition numbers of Calderon–Ikawa CFIER operators. When regularizing operators with complex wavenumbers with non-zero real parts are used—we refer to these operators as Calderon-Complex CFIER, we show numerical evidence that those complex wavenumbers can be selected in a manner that leads to CFIER formulations whose condition numbers can be bounded independently of frequency for spherical geometries. In addition, the Calderon-Complex CFIER operators possess excellent spectral properties in the high-frequency regime for both convex and non-convex scatterers. We provide numerical evidence that our solvers based on fast, high-order Nystrom discretization of these equations converge in very small numbers of GMRES iterations, and the iteration counts are virtually independent of frequency for several smooth scatterers with slowly varying curvatures.

Well-posed boundary integral equation formulations and Nyström discretizations for the solution of Helmholtz transmission problems in two-dimensional Lipschitz domains

We present a comparison between the performance of solvers based on Nystrom discretizations of several well-posed boundary integral equation formulations of Helmholtz transmission problems in two-dimensional Lipschitz domains. Specifically, we focus on the following four classes of boundary integral formulations of Helmholtz transmission problems (1) the classical first kind integral equations for transmission problems, (2) the classical second kind integral equations for transmission problems, (3) the {\em single} integral equation formulations, and (4) certain direct counterparts of recently introduced Generalized Combined Source Integral Equations. The former two formulations were the only formulations whose well-posedness in Lipschitz domains was rigorously established. We establish the well-posedness of the latter two formulations in appropriate functional spaces of boundary traces of solutions of transmission Helmholtz problems in Lipschitz domains. We give ample numerical evidence that Nystrom solvers based on formulations (3) and (4) are computationally more advantageous than solvers based on the classical formulations (1) and (2), especially in the case of high-contrast transmission problems at high frequencies.

Three-dimensional quasi-periodic shifted Green function throughout the spectrum, including Wood anomalies

This work, part II in a series, presents an efficient method for evaluation of wave scattering by doubly periodic diffraction gratings at or near what are commonly called ‘Wood anomaly frequencies’. At these frequencies, there is a grazing Rayleigh wave, and the quasi-periodic Green function ceases to exist. We present a modification of the Green function by adding two types of terms to its lattice sum. The first type are transversely shifted Green functions with coefficients that annihilate the growth in the original lattice sum and yield algebraic convergence. The second type are quasi-periodic plane wave solutions of the Helmholtz equation which reinstate certain necessary grazing modes without leading to blow-up at Wood anomalies. Using the new quasi-periodic Green function, we establish, for the first time, that the Dirichlet problem of scattering by a smooth doubly periodic scattering surface at a Wood frequency is uniquely solvable. We also present an efficient high-order numerical method based on this new Green function for scattering by doubly periodic surfaces at and around Wood frequencies. We believe this is the first solver able to handle Wood frequencies for doubly periodic scattering problems in three dimensions. We demonstrate the method by applying it to acoustic scattering.

Harmonic density interpolation methods for high-order evaluation of Laplace layer potentials in 2D and 3D

Abstract We present an effective harmonic density interpolation method for the numerical evaluation of singular and nearly singular Laplace boundary integral operators and layer potentials in two and three spatial dimensions. The method relies on the use of Greens third identity and local Taylor-like interpolations of density functions in terms of harmonic polynomials. The proposed technique effectively regularizes the singularities present in boundary integral operators and layer potentials, and recasts the latter in terms of integrands that are bounded or even more regular, depending on the order of the density interpolation. The resulting boundary integrals can then be easily, accurately, and inexpensively evaluated by means of standard quadrature rules. A variety of numerical examples demonstrate the effectiveness of the technique when used in conjunction with the classical trapezoidal rule (to integrate over smooth curves) in two-dimensions, and with a Chebyshev-type quadrature rule (to integrate over surfaces given as unions of non-overlapping quadrilateral patches) in three-dimensions.