arXiv subjects

Catalin Turc

Publications and source records attributed to Catalin Turc.

At least 19 recordsLinked to original sources

arXiv2026

Maxwell \`a la Helmholtz: Direct boundary integral equations for 3D scattering by perfect electric conductors via Helmholtz operators

This paper is the direct-formulation companion to [Burbano-Gallegos, P\'erez-Arancibia, and Turc, ESAIM: M2AN, 60(1):273--315, 2026], which developed indirect combined-field-only boundary integral equations (BIEs) for time-harmonic electromagnetic scattering by smooth perfectly electrically conducting (PEC) obstacles, relying entirely on Helmholtz boundary integral operators. Here we exploit the same equivalence between the Maxwell PEC scattering problem and a pair of vector Helmholtz boundary value problems -- one for the electric field and one for the magnetic field -- to derive direct BIE formulations whose unknowns are the Dirichlet and Neumann traces of the total fields, decomposed into their normal and tangential surface components. These unknowns carry direct physical meaning: in particular, the magnetic-field formulation yields the surface electric currents as part of its solution. The mixed regularity of the two field-trace components requires introducing a tailored product H\"older space, a distinctive feature absent from the indirect approach. We prove that the resulting Direct Electric and Magnetic Combined-Field-Only Integral Equations (D-ECFOIE and D-MCFOIE) are uniquely solvable at all frequencies, and introduce Calder\'on-type regularizations (RD-ECFOIE and RD-MCFOIE) that render them of the Fredholm second kind. We further examine the low-frequency breakdown affecting the electric-field formulation and introduce a modified equation that enforces the physical charge-conservation constraints, which restores numerical accuracy and well-conditioned linear systems for frequencies arbitrarily close to zero. Numerical experiments, performed using a high-order Nystr\"om solver based on the Density Interpolation Method and implemented in the Julia package Inti.jl, validate the accuracy and robustness of the proposed formulations across a range of geometries and frequencies.

math.NA↗

arXiv2025

Maxwell \`a la Helmholtz: Electromagnetic scattering by 3D perfect electric conductors via Helmholtz integral operators

This paper introduces a novel class of indirect boundary integral equation (BIE) formulations for the solution of electromagnetic scattering problems involving smooth perfectly electric conductors (PECs) in three-dimensions. These combined-field-type BIE formulations rely exclusively on classical Helmholtz boundary operators, resulting in provably well-posed, frequency-robust, Fredholm second-kind BIEs. Notably, we prove that the proposed formulations are free from spurious resonances, while retaining the versatility of Helmholtz integral operators. The approach is based on the equivalence between the Maxwell PEC scattering problem and two independent vector Helmholtz boundary value problems for the electric and magnetic fields, with boundary conditions defined in terms of the Dirichlet and Neumann traces of the corresponding vector Helmholtz solutions. While certain aspects of this equivalence (for the electric field) have been previously exploited in the so-called field-only BIE formulations, we here rigorously establish and generalize the equivalence between Maxwell and Helmholtz problems for both fields. Finally, a variety of numerical examples highlights the robustness and accuracy of the proposed approach when combined with Density Interpolation-based Nystr\"om methods and fast linear algebra solvers, implemented in the open-source Julia package Inti$.$jl.

math.NA↗

arXiv2023

Stability estimates of Nystr\"om discretizations of Helmholtz decomposition boundary integral equation formulations for the solution of Navier scattering problems in two dimensions with Dirichlet boundary conditions

Helmholtz decompositions of elastic fields is a common approach for the solution of Navier scattering problems. Used in the context of Boundary Integral Equations (BIE), this approach affords solutions of Navier problems via the simpler Helmholtz boundary integral operators (BIOs). Approximations of Helmholtz Dirichlet-to-Neumann (DtN) can be employed within a regularizing combined field strategy to deliver BIE formulations of the second kind for the solution of Navier scattering problems in two dimensions with Dirichlet boundary conditions, at least in the case of smooth boundaries. Unlike the case of scattering and transmission Helmholtz problems, the approximations of the DtN maps we use in the Helmholtz decomposition BIE in the Navier case require incorporation of lower order terms in their pseudodifferential asymptotic expansions. The presence of these lower order terms in the Navier regularized BIE formulations complicates the stability analysis of their Nystr\"om discretizations in the framework of global trigonometric interpolation and the Kussmaul-Martensen kernel singularity splitting strategy. The main difficulty stems from compositions of pseudodifferential operators of opposite orders, whose Nystr\"om discretization must be performed with care via pseudodifferential expansions beyond the principal symbol. The error analysis is significantly simpler in the case of arclength boundary parametrizations and considerably more involved in the case of general smooth parametrizations which are typically encountered in the description of one dimensional closed curves.

math.NA↗

arXiv2023

High-Order Nyström/Convolution-Quadrature Solution of Time-Domain Scattering from Closed and Open Lipschitz Boundaries with Dirichlet and Neumann Boundary Conditions

We investigate high-order Convolution Quadratures methods for the solution of the wave equation in unbounded domains in two dimensions that rely on Nyström discretizations for the solution of the ensemble of associated Laplace domain modified Helmholtz problems. We consider two classes of CQ discretizations, one based on linear multistep methods and the other based on Runge-Kutta methods, in conjunction with Nyström discretizations based on Alpert and QBX quadratures of Boundary Integral Equation (BIE) formulations of the Laplace domain Helmholtz problems with complex wavenumbers. We present a variety of accuracy tests that showcase the high-order in time convergence (up to and including fifth order) that the Nyström CQ discretizations are capable of delivering for a variety of two dimensional scatterers and types of boundary conditions.

math.NA↗

arXiv2023

Combined field-only boundary integral equations for PEC electromagnetic scattering problem in spherical geometries

We analyze the well posedness of certain field-only boundary integral equations (BIE) for frequency domain electromagnetic scattering from perfectly conducting spheres. Starting from the observations that (1) the three components of the scattered electric field $\mathbf{E}^s(\mathbf{x})$ and (2) scalar quantity $\mathbf{E}^s(\mathbf{x})\cdot\mathbf{x}$ are radiative solutions of the Helmholtz equation, novel boundary integral equation formulations of electromagnetic scattering from perfectly conducting obstacles can be derived using Green's identities applied to the aforementioned quantities and the boundary conditions on the surface of the scatterer. The unknowns of these formulations are the normal derivatives of the three components of the scattered electric field and the normal component of the scattered electric field on the surface of the scatterer, and thus these formulations are referred to as field-only BIE. In this paper we use the Combined Field methodology of Burton and Miller within the field-only BIE approach and we derive new boundary integral formulations that feature only Helmholtz boundary integral operators, which we subsequently show to be well posed for all positive frequencies in the case of spherical scatterers. Relying on the spectral properties of Helmholtz boundary integral operators in spherical geometries, we show that the combined field-only boundary integral operators are diagonalizable in the case of spherical geometries and their eigenvalues are non zero for all frequencies. Furthermore, we show that for spherical geometries one of the field-only integral formulations considered in this paper exhibits eigenvalues clustering at one -- a property similar to second kind integral equations.

math.NA↗

arXiv2022

Nystrom discretizations of boundary integral equations for the solution of 2D elastic scattering problems

We present three high-order Nystrom discretization strategies of various boundary integral equation formulations of the impenetrable time-harmonic Navier equations in two dimensions. One class of such formulations is based on the four classical Boundary Integral Operators (BIOs) associated with the Green's function of the Navier operator. We consider two types of Nystrom discretizations of these operators, one that relies on Kussmaul-Martensen logarithmic splittings and the other on Alpert quadratures. In addition, we consider an alternative formulation of Navier scattering problems based on Helmholtz decompositions of the elastic fields, which can be solved via a system of boundary integral equations that feature integral operators associated with the Helmholtz equation. Owing to the fact that some of the BIOs that are featured in those formulations are non-standard, we use Quadrature by Expansion (QBX) methods for their high order Nystrom discretization. Alternatively, we use Maue integration by parts techniques to recast those non-standard operators in terms of single and double layer Helmholtz BIOs whose Nystrom discretizations is amenable to the Kussmaul-Martensen methodology. We present a variety of numerical results concerning the high order accuracy that our Nystrom discretization elastic scattering solvers achieve for both smooth and Lipschitz boundaries. We also present extensive comparisons regarding the iterative behavior of solvers based on different integral equations in the high frequency regime. Finally, we illustrate how some of the Nystrom discretizations we considered can be incorporated seamlessly into the Convolution Quadrature (CQ) methodology to deliver high-order solutions of the time domain elastic scattering problems.

math.NA↗

arXiv2022

Boundary integral equation methods for the solution of scattering and transmission 2D elastodynamic problems

We introduce and analyze various Regularized Combined Field Integral Equations (CFIER) formulations of time-harmonic Navier equations in media with piece-wise constant material properties. These formulations can be derived systematically starting from suitable coercive approximations of Dirichlet-to-Neumann operators (DtN), and we present a periodic pseudodifferential calculus framework within which the well posedness of CIER formulations can be established. We also use the DtN approximations to derive and analyze Optimized Schwarz (OS) methods for the solution of elastodynamics transmission problems. The pseudodifferential calculus we develop in this paper relies on careful singularity splittings of the kernels of Navier boundary integral operators which is also the basis of high-order Nyström quadratures for their discretizations. Based on these high-order discretizations we investigate the rate of convergence of iterative solvers applied to CFIER and OS formulations of scattering and transmission problems. We present a variety of numerical results that illustrate that the CFIER methodology leads to important computational savings over the classical CFIE one, whenever iterative solvers are used for the solution of the ensuing discretized boundary integral equations. Finally, we show that the OS methods are competitive in the high-frequency high-contrast regime.

math.NA↗

arXiv2019

Planewave density interpolation methods for the EFIE on simple and composite surfaces

This paper presents an extension of the recently introduced planewave density interpolation (PWDI) method to the electric field integral equation (EFIE) formulation of problems of scattering and radiation by perfect electric conducting (PEC) objects. Relying on Kirchhoff integral formula and local interpolation of surface current densities that regularize the kernel singularities, the PWDI method enables off- and on-surface EFIE operators to be re-expressed in terms of integrands that are globally bounded (or even more regular) over the whole domain of integration, regardless of the magnitude of the distance between target and source points. Surface integrals resulting from the application of the method-of-moments (MoM) using Rao-Wilton-Glisson (RWG) basis functions, can then be directly and easily evaluated by means of elementary quadrature rules irrespective of the singularity location. The proposed technique can be applied to simple and composite surfaces comprising two or more simply-connected overlapping components. The use of composite surfaces can significantly simplify the geometric treatment of complex structures, as the PWDI method enables the use of separate non-conformal meshes for the discretization of each of the surface components that make up the composite surface. A variety of examples, including multi-scale and intricate structures, demonstrate the effectiveness of the proposed methodology.

physics.comp-ph↗

arXiv2019

Planewave density interpolation methods for 3D Helmholtz boundary integral equations

This paper introduces planewave density interpolation methods for the regularization of weakly singular, strongly singular, hypersingular and nearly singular integral kernels present in 3D Helmholtz surface layer potentials and associated integral operators. Relying on Green's third identity and pointwise interpolation of density functions in the form of planewaves, these methods allow layer potentials and integral operators to be expressed in terms of integrand functions that remain smooth (at least bounded) regardless the location of the target point relative to the surface sources. Common challenging integrals that arise in both Nystr\"om and boundary element discretization of boundary integral equation, can then be numerically evaluated by standard quadrature rules that are irrespective of the kernel singularity. Closed-form and purely numerical planewave density interpolation procedures are presented in this paper, which are used in conjunction with Chebyshev-based Nystr\"om and Galerkin boundary element methods. A variety of numerical examples---including problems of acoustic scattering involving multiple touching and even intersecting obstacles, demonstrate the capabilities of the proposed technique.

math.NA↗

arXiv2018

Sweeping preconditioners for the iterative solution of quasiperiodic Helmholtz transmission problems in layered media

We present a sweeping preconditioner for quasi-optimal Domain Decomposition Methods (DDM) applied to Helmholtz transmission problems in periodic layered media. Quasi-optimal DD (QO DD) for Helmholtz equations rely on transmission operators that are approximations of Dirichlet-to-Neumann (DtN) operators. Employing shape perturbation series, we construct approximations of DtN operators corresponding to periodic domains, which we then use as transmission operators in a non-overlapping DD framework. The Robin-to-Robin (RtR) operators that are the building blocks of DDM are expressed via robust boundary integral equation formulations. We use Nystr\"om discretizations of quasi-periodic boundary integral operators to construct high-order approximations of RtR. Based on the premise that the quasi-optimal transmission operators should act like perfect transparent boundary conditions, we construct an approximate LU factorization of the tridiagonal QO DD matrix associated with periodic layered media, which is then used as a double sweep preconditioner. We present a variety of numerical results that showcase the effectiveness of the sweeping preconditioners applied to QO DD for the iterative solution of Helmholtz transmission problems in periodic layered media.

math.NA↗

arXiv2018

Domain decomposition for quasi-periodic scattering by layered media via robust boundary-integral equations at all frequencies

We develop a non-overlapping domain decomposition method (DDM) for scalar wave scattering by periodic layered media. Our approach relies on robust boundary-integral equation formulations of Robin-to-Robin (RtR) maps throughout the frequency spectrum, including cutoff (or Wood) frequencies. We overcome the obstacle of non-convergent quasi-periodic Green functions at these frequencies by incorporating newly introduced shifted Green functions. Using the latter in the definition of quasi-periodic boundary-integral operators leads to rigorously stable computations of RtR operators. We develop Nystr\"om discretizations of the RtR maps that rely on trigonometric interpolation, singularity resolution, and fast convergent windowed quasi-periodic Green functions. We solve the tridiagonal DDM system via recursive Schur complements and establish rigorously that this procedure is always completed successfully. We present a variety of numerical results concerning Wood frequencies in two and three dimensions as well as large numbers of layers.

math.NA↗

arXiv2017

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

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 Green's 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.

math.NA↗

arXiv2017

Domain Decomposition Methods based on quasi-optimal transmission operators for the solution of Helmholtz transmission problems

We present non-overlapping Domain Decomposition Methods (DDM) based on quasi-optimal transmission operators for the solution of Helmholtz transmission problems with piece-wise constant material properties. The quasi-optimal transmission boundary conditions incorporate readily available approximations of Dirichlet-to-Neumann operators. These approximations consist of either complexified hypersingular boundary integral operators for the Helmholtz equation or square root Fourier multipliers with complex wavenumbers. We show that under certain regularity assumptions on the closed interface of material discontinuity, the DDM with quasi-optimal transmission conditions are well-posed. We present a DDM framework based on Robin-to-Robin (RtR) operators that can be computed robustly via boundary integral formulations. More importantly, the use of quasi-optimal transmission operators results in DDM that converge in small numbers of iterations even in the challenging high-contrast, high-frequency regime of Helmholtz transmission problems. Furthermore, the DDM presented in this text require only minor modifications to handle the case of transmission problems in partially coated domains, while still maintaining excellent convergence properties. We also investigate the dependence of the DDM iterative performance on the number of subdomains.

math.NA↗

arXiv2017

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

This work presents an efficient method for evaluation of wave scattering by doubly periodic diffraction gratings at or near "Wood anomaly frequencies". At these frequencies, one or more grazing Rayleigh waves exist, and the lattice sum for the quasi-periodic Green function ceases to exist. We present a modification of this sum by adding two types of terms to it. The first type adds linear combinations of "shifted" Green functions, ensuring that the spatial singularities introduced by these terms are located below the grating and therefore outside of the physical domain. With suitable coefficient choices these terms annihilate the growing contributions in the original lattice sum and yield algebraic convergence. Convergence of arbitrarily high order can be obtained by including sufficiently many shifts. The second type of added terms 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 the this new Green function for the problem of scattering by doubly periodic three-dimensional surfaces at and around Wood frequencies. We believe this is the first solver in existence that is applicable to Wood-frequency doubly periodic scattering problems. We demonstrate the proposed approach by means of applications to problems of acoustic scattering by doubly periodic gratings at various frequencies, including frequencies away from, at, and near Wood anomalies.

math.AP↗

arXiv2017

Multitrace formulations and Domain Decomposition Methods for the solution of Helmholtz transmission problems for bounded composite scatterers

We present Nystr\"om discretizations of multitrace formulations and non-overlapping Domain Decomposition Methods (DDM) for the solution of Helmholtz transmission problems for bounded composite scatterers with piecewise constant material properties. We investigate the performance of DDM with both classical Robin and generalized Robin boundary conditions. The generalized Robin boundary conditions incorporate square root Fourier multiplier approximations of Dirichlet to Neumann operators. While the classical version of DDM is not particularly well suited for Krylov subspace iterative solvers, we show that the associated DDM linear system can be efficiently solved by hierarchical elimination via Schur complements of the Robin data. We show through numerical examples that the latter version of DDM gives rise to small numbers of Krylov subspace iterations that depend mildly on the frequency and number of subdomains.

math.NA↗

arXiv2016

Schur complement Domain Decomposition Methods for the solution of multiple scattering problems

We present a Schur complement Domain Decomposition (DD) algorithm for the solution of frequency domain multiple scattering problems. Just as in the classical DD methods we (1) enclose the ensemble of scatterers in a domain bounded by an artificial boundary, (2) we subdivide this domain into a collection of nonoverlapping subdomains so that the boundaries of the subdomains do not intersect any of the scatterers, and (3) we connect the solutions of the subproblems via Robin boundary conditions matching on the common interfaces between subdomains. We use subdomain Robin-to-Robin maps to recast the DD problem as a sparse linear system whose unknown consists of Robin data on the interfaces between subdomains---two unknowns per interface. The Robin-to-Robin maps are computed in terms of well-conditioned boundary integral operators. Unlike classical DD, we do not reformulate the Domain Decomposition problem in the form a fixed point iteration, but rather we solve the ensuing linear system by Gaussian elimination of the unknowns corresponding to inner interfaces between subdomains via Schur complements. Once all the unknowns corresponding to inner subdomains interfaces have been eliminated, we solve a much smaller linear system involving unknowns on the inner and outer artificial boundary. We present numerical evidence that our Schur complement DD algorithm can produce accurate solutions of very large multiple scattering problems that are out of reach for other existing approaches.

math.NA↗

arXiv2016

Well-conditioned boundary integral equation formulations and Nystr\"om discretizations for the solution of Helmholtz problems with impedance boundary conditions in two-dimensional Lipschitz domains

We present a regularization strategy that leads to well-conditioned boundary integral equation formulations of Helmholtz equations with impedance boundary conditions in two-dimensional Lipschitz domains. We consider both the case of classical impedance boundary conditions, as well as the case of transmission impedance conditions wherein the impedances are certain coercive operators. The latter type of problems is instrumental in the speed up of the convergence of Domain Decomposition Methods for Helmholtz problems. Our regularized formulations use as unknowns the Dirichlet traces of the solution on the boundary of the domain. Taking advantage of the increased regularity of the unknowns in our formulations, we show through a variety of numerical results that a graded-mesh based Nystr\"om discretization of these regularized formulations leads to efficient and accurate solutions of interior and exterior Helmholtz problems with impedance boundary conditions.

math.NA↗

arXiv2015

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

We present a comparison between the performance of solvers based on Nystr\"om 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 Nystr\"om 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.

math.NA↗