Related papers: Quadrature by Two Expansions: Evaluating Laplace L…
We study scattering by a high aspect ratio particle using boundary integral equation methods. This problem has important applications in nanophotonics problems, including sensing and plasmonic imaging. To illustrate the effect of parity and…
A high-order quadrature scheme is constructed for the evaluation of Laplace single and double layer potentials and their normal derivatives on smooth surfaces in three dimensions. The construction begins with a harmonic approximation of the…
When using boundary integral equation methods, we represent solutions of a linear partial differential equation as layer potentials. It is well-known that the approximation of layer potentials using quadrature rules suffer from poor…
Panel-based, kernel-split quadrature is currently one of the most efficient methods available for accurate evaluation of singular and nearly singular layer potentials in two dimensions. However, it can fail completely for the layer…
We construct and analyze a hierarchical direct solver for linear systems arising from the discretization of boundary integral equations using the Quadrature by Expansion (QBX) method. Our scheme builds on the existing theory of Hierarchical…
We propose a high-precision numerical quadrature framework based on local Fourier extension (LFE) approximations. The method constructs, on each subinterval, a truncated-SVD stabilized local Fourier continuation of the integrand on an…
Panel-based, kernel-split quadrature is currently one of the most efficient methods available for accurate evaluation of singular and nearly singular layer potentials in two dimensions. However, it can fail completely for the layer…
This paper presents a quadrature method for evaluating layer potentials in two dimensions close to periodic boundaries, discretized using the trapezoidal rule. It is an extension of the method of singularity swap quadrature, which recently…
This paper introduces a new functional expansion framework that extends classical ideas beyond the Taylor series. Unlike traditional Taylor expansions based on local polynomial approximations, the proposed approach arises from exact…
The method of Helsing and co-workers evaluates Laplace and related layer potentials generated by a panel (composite) quadrature on a curve, efficiently and with high-order accuracy for arbitrarily close targets. Since it exploits complex…
Many quantum systems admit an explicit analytic Fourier space expansion, besides the usual analytic Schrodinger configuration space representation. We argue that the use of weighted orthonormal polynomial expansions for the physical states…
Layer potentials represent solutions to partial differential equations in an integral equation formulation. When numerically evaluating layer potentials at evaluation points close to the domain boundary, specialized quadrature techniques…
There has been growing interest in high-order tensor methods for nonconvex optimization, with adaptive regularization, as they possess better/optimal worst-case evaluation complexity globally and faster convergence asymptotically. These…
Well-conditioned boundary integral methods for the solution of elliptic boundary value problems (BVPs) are powerful tools for static and dynamic physical simulations. When there are many close-to-touching boundaries (eg, in complex fluids)…
Convolution quadrature (CQ) methods have enjoyed tremendous interest in recent years as an efficient tool for solving time-domain wave problems in unbounded domains via boundary integral equation techniques. In this paper we consider CQ…
In boundary element methods (BEM) in $\mathbb{R}^3$, matrix elements and right hand sides are typically computed via analytical or numerical quadrature of the layer potential multiplied by some function over line, triangle and tetrahedral…
We describe a new, adaptive solver for the two-dimensional Poisson equation in complicated geometries. Using classical potential theory, we represent the solution as the sum of a volume potential and a double layer potential. Rather than…
We propose a patchwise local Fourier extension method for approximating smooth functions on general two dimensional domains with curved boundaries. The domain is embedded into a Cartesian background grid and decomposed into rectangular…
Boundary integral methods are highly suited for problems with complicated geometries, but require special quadrature methods to accurately compute the singular and nearly singular layer potentials that appear in them. This paper presents a…
A quasimodal expansion method (QMEM) is developed to model and understand the scattering properties of arbitrary shaped two-dimensional (2-D) open structures. In contrast with the bounded case which have only discrete spectrum (real in the…