General-purpose kernel regularization of boundary integral equations via density interpolation

This paper presents a general high-order kernel regularization technique applicable to all four integral operators of Calder\'on calculus associated with linear elliptic PDEs in two and three spatial dimensions. Like previous density interpolation methods, the proposed technique relies on interpolating the density function around the kernel singularity in terms of solutions of the underlying homogeneous PDE, so as to recast singular and nearly singular integrals in terms of bounded (or more regular) integrands. We present here a simple interpolation strategy which, unlike previous approaches, does not entail explicit computation of high-order derivatives of the density function along the surface. Furthermore, the proposed approach is kernel- and dimension-independent in the sense that the sought density interpolant is constructed as a linear combination of point-source fields, given by the same Green's function used in the integral equation formulation, thus making the procedure applicable, in principle, to any PDE with known Green's function. For the sake of definiteness, we focus here on Nystr\"om methods for the (scalar) Laplace and Helmholtz equations and the (vector) elastostatic and time-harmonic elastodynamic equations. The method's accuracy, flexibility, efficiency, and compatibility with fast solvers are demonstrated by means of a variety of large-scale three-dimensional numerical examples

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

Layer potential quadrature on manifold boundary elements with constant densities for Laplace and Helmholtz kernels in R3\mathbb{R}^3

A method is proposed for evaluation of single and double layer potentials of the Laplace and Helmholtz equations on piecewise smooth manifold boundary elements with constant densities. The method is based on a novel two-term decomposition of the layer potentials, derived by means of differential geometry. The first term is an integral of a differential 2-form which can be reduced to contour integrals using Stokes' theorem, while the second term is related to the element curvature. This decomposition reduces the degree of singularity and the curvature term can be further regularized by a polar coordinate transform. The method can handle singular and nearly singular integrals. Numerical results validating the accuracy of the method are presented for all combinations of single and double layer potentials, for the Laplace and Helmholtz kernels, and for singular and nearly singular integrals

Machine Precision Evaluation of Singular and Nearly Singular Potential Integrals by Use of Gauss Quadrature Formulas for Rational Functions

A new technique for machine precision evaluation of singular and nearly singular potential integrals with 1/R singularities is presented. The numerical quadrature scheme is based on a new rational expression for the integrands, obtained by a cancellation procedure. In particular, by using library routines for Gauss quadrature of rational functions readily available in the literature, this new expression permits the exact numerical integration of singular static potentials associated with polynomial source distributions. The rules to achieve the desired numerical accuracy for singular and nearly singular static and dynamic potential integrals are presented and discussed, and several numerical examples are provide

Quadrature by Expansion: A New Method for the Evaluation of Layer Potentials

Integral equation methods for the solution of partial differential equations, when coupled with suitable fast algorithms, yield geometrically flexible, asymptotically optimal and well-conditioned schemes in either interior or exterior domains. The practical application of these methods, however, requires the accurate evaluation of boundary integrals with singular, weakly singular or nearly singular kernels. Historically, these issues have been handled either by low-order product integration rules (computed semi-analytically), by singularity subtraction/cancellation, by kernel regularization and asymptotic analysis, or by the construction of special purpose "generalized Gaussian quadrature" rules. In this paper, we present a systematic, high-order approach that works for any singularity (including hypersingular kernels), based only on the assumption that the field induced by the integral operator is locally smooth when restricted to either the interior or the exterior. Discontinuities in the field across the boundary are permitted. The scheme, denoted QBX (quadrature by expansion), is easy to implement and compatible with fast hierarchical algorithms such as the fast multipole method. We include accuracy tests for a variety of integral operators in two dimensions on smooth and corner domains

Direct boundary integral equation method for electromagnetic scattering by partly coated dielectric objects

We present a new variational direct boundary integral equation approach for solving the scattering and transmission problem for dielectric objects partially coated with a PEC layer. The main idea is to use the electromagnetic Calderón projector along with transmission conditions for the electromagnetic fields. This leads to a symmetric variational formulation which lends itself to Galerkin discretization by means of divergence-conforming discrete surface currents. A wide array of numerical experiments confirms the efficacy of the new metho