arXiv subjects

Carlos Perez-Arancibia

Publications and source records attributed to Carlos Perez-Arancibia.

3 recordsLinked to original sources

arXiv2026

High-order kernel regularization of singular and hypersingular Helmholtz boundary integral operators

This paper extends and analyzes the high-order kernel regularization framework of Beale & Tlupova (arXiv:2510.13639) to all four on-surface boundary integral operators of the Helmholtz Calderon calculus in three dimensions: the single-layer, double-layer, adjoint double-layer, and hypersingular operators. To the best of our knowledge, this work provides the first high-order kernel regularization of the hypersingular operator for both the Helmholtz and Laplace equations in three dimensions. The regularization replaces each singular kernel with a smooth modification constructed from error functions together with a polynomial correction whose coefficients are determined through moment conditions. Alongside the derivation of the regularizing functions, the paper provides a unified error analysis of the combined regularization and quadrature discretization procedure. By coupling the regularization parameter to the mesh size, the two error contributions can be balanced, leading to explicit overall convergence rates that depend jointly on the order of the regularization and the degree of exactness of the surface quadrature rule. A key practical feature of the method is its implementation simplicity. Once the regularizing functions are determined, the numerical task reduces entirely to the evaluation of smooth surface integrals using standard quadrature, without the need for element-local solves, singularity-specific precomputations, or specialized quadrature rules. Although the modified kernel is generally incompatible with kernel-specific fast methods, this limitation is addressed through H-matrix acceleration, applicable in a black-box manner. Numerical examples -- including verification of the predicted convergence rates and solution of sound-soft and sound-hard scattering problems by smooth obstacles -- demonstrate the accuracy and practicality of the proposed methodology.

math.NA↗

arXiv2016

Windowed Green Function Method for Nonuniform Open-Waveguide Problems

This contribution presents a novel Windowed Green Function (WGF) method for the solution of problems of wave propagation, scattering and radiation for structures which include open (dielectric) waveguides, waveguide junctions, as well as launching and/or termination sites and other nonuniformities. Based on use of a "slow-rise" smooth-windowing technique in conjunction with free-space Green functions and associated integral representations, the proposed approach produces numerical solutions with errors that decrease faster than any negative power of the window size. The proposed methodology bypasses some of the most significant challenges associated with waveguide simulation. In particular the WGF approach handles spatially-infinite dielectric waveguide structures without recourse to absorbing boundary conditions, it facilitates proper treatment of complex geometries, and it seamlessly incorporates the open-waveguide character and associated radiation conditions inherent in the problem under consideration. The overall WGF approach is demonstrated in this paper by means of a variety of numerical results for two-dimensional open-waveguide termination, launching and junction problems.

math.NA↗

arXiv2015

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 presence of dielectric or conducting half-planes. The WGF method, which is based on 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) solvers suffice to incorporate the WGF capability. The mathematical basis of the method is simple: 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. 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 the layer-Green-function.

physics.comp-ph↗