A higher-order singularity subtraction technique for the discretization of singular integral operators on curved surfaces

This note is about promoting singularity subtraction as a helpful tool in the discretization of singular integral operators on curved surfaces. Singular and nearly singular kernels are expanded in series whose terms are integrated on parametrically rectangular regions using high-order product integration, thereby reducing the need for spatial adaptivity and precomputed weights. A simple scheme is presented and an application to the interior Dirichlet Laplace problem on some tori gives around ten digit accurate results using only two expansion terms and a modest programming- and computational effort.Comment: 7 pages, 2 figure

On a Helmholtz transmission problem in planar domains with corners

A particular mix of integral equations and discretization techniques is suggested for the solution of a planar Helmholtz transmission problem with relevance to the study of surface plasmon waves. The transmission problem describes the scattering of a time harmonic transverse magnetic wave from an infinite dielectric cylinder with complex permittivity and sharp edges. Numerical examples illustrate that the resulting scheme is capable of obtaining total magnetic and electric fields to very high accuracy in the entire computational domain.Comment: 28 pages, 8 figure

An explicit kernel-split panel-based Nystr\"om scheme for integral equations on axially symmetric surfaces

A high-order accurate, explicit kernel-split, panel-based, Fourier-Nystr\"om discretization scheme is developed for integral equations associated with the Helmholtz equation in axially symmetric domains. Extensive incorporation of analytic information about singular integral kernels and on-the-fly computation of nearly singular quadrature rules allow for very high achievable accuracy, also in the evaluation of fields close to the boundary of the computational domain.Comment: 30 pages, 5 figures, errata correcte

Determination of normalized electric eigenfields in microwave cavities with sharp edges

The magnetic field integral equation for axially symmetric cavities with perfectly conducting piecewise smooth surfaces is discretized according to a high-order convergent Fourier--Nystr\"om scheme. The resulting solver is used to accurately determine eigenwavenumbers and normalized electric eigenfields in the entire computational domain.Comment: 34 pages, 6 figure

An accurate boundary value problem solver applied to scattering from cylinders with corners

In this paper we consider the classic problems of scattering of waves from perfectly conducting cylinders with piecewise smooth boundaries. The scattering problems are formulated as integral equations and solved using a Nystr\"om scheme where the corners of the cylinders are efficiently handled by a method referred to as Recursively Compressed Inverse Preconditioning (RCIP). This method has been very successful in treating static problems in non-smooth domains and the present paper shows that it works equally well for the Helmholtz equation. In the numerical examples we specialize to scattering of E- and H-waves from a cylinder with one corner. Even at a size kd=1000, where k is the wavenumber and d the diameter, the scheme produces at least 13 digits of accuracy in the electric and magnetic fields everywhere outside the cylinder.Comment: 19 pages, 3 figure

Determination of normalized magnetic eigenfields in microwave cavities

The magnetic field integral equation for axially symmetric cavities with perfectly conducting surfaces is discretized according to a high-order convergent Fourier--Nystr\"om scheme. The resulting solver is used to determine eigenwavenumbers and normalized magnetic eigenfields to very high accuracy in the entire computational domain.Comment: 23 pages, 4 figure

On the polarizability and capacitance of the cube

An efficient integral equation based solver is constructed for the electrostatic problem on domains with cuboidal inclusions. It can be used to compute the polarizability of a dielectric cube in a dielectric background medium at virtually every permittivity ratio for which it exists. For example, polarizabilities accurate to between five and ten digits are obtained (as complex limits) for negative permittivity ratios in minutes on a standard workstation. In passing, the capacitance of the unit cube is determined with unprecedented accuracy. With full rigor, we develop a natural mathematical framework suited for the study of the polarizability of Lipschitz domains. Several aspects of polarizabilities and their representing measures are clarified, including limiting behavior both when approaching the support of the measure and when deforming smooth domains into a non-smooth domain. The success of the mathematical theory is achieved through symmetrization arguments for layer potentials.Comment: 33 pages, 7 figure

Faster convergence and higher accuracy for the Dirichlet-Neumann map

New techniques allow for more efficient boundary integral algorithms to compute the Dirichlet–Neumann map for Laplace’s equation in two-dimensional exterior domains. Novelties include a new post-processor which reduces the need for discretization points with 50%, a new integral equation which reduces the error for resolved geometries with a factor equal to the system size, systematic use of regularization which reduces the error even further, and adaptive mesh generation based on kernel resolution

The effective conductivity of arrays of squares: large random unit cells and extreme contrast ratios

An integral equation based scheme is presented for the fast and accurate computation of effective conductivities of two-component checkerboard-like composites with complicated unit cells at very high contrast ratios. The scheme extends recent work on multi-component checkerboards at medium contrast ratios. General improvement include the simplification of a long-range preconditioner, the use of a banded solver, and a more efficient placement of quadrature points. This, together with a reduction in the number of unknowns, allows for a substantial increase in achievable accuracy as well as in tractable system size. Results, accurate to at least nine digits, are obtained for random checkerboards with over a million squares in the unit cell at contrast ratio 10^6. Furthermore, the scheme is flexible enough to handle complex valued conductivities and, using a homotopy method, purely negative contrast ratios. Examples of the accurate computation of resonant spectra are given.Comment: 28 pages, 11 figures, submitted to J. Comput. Phy