A fast algorithm for particle simulations
Journal of Computational Physics
High-Order Corrected Trapezoidal Quadrature Rules for Singular Functions
SIAM Journal on Numerical Analysis
Hybrid Gauss-Trapezoidal Quadrature Rules
SIAM Journal on Scientific Computing
Spectral methods in MatLab
Nonlinear Optimization, Quadrature, and Interpolation
SIAM Journal on Optimization
Iterative Methods for Sparse Linear Systems
Iterative Methods for Sparse Linear Systems
A kernel-independent adaptive fast multipole algorithm in two and three dimensions
Journal of Computational Physics
A fast direct solver for boundary integral equations in two dimensions
Journal of Computational Physics
Journal of Computational Physics
Integral equation methods for elliptic problems with boundary conditions of mixed type
Journal of Computational Physics
Efficient discretization of Laplace boundary integral equations on polygonal domains
Journal of Computational Physics
Universal quadratures for boundary integral equations on two-dimensional domains with corners
Journal of Computational Physics
A fast direct solver for the integral equations of scattering theory on planar curves with corners
Journal of Computational Physics
Journal of Computational Physics
Boundary integral equations in time-harmonic acoustic scattering
Mathematical and Computer Modelling: An International Journal
Approximation Theory and Approximation Practice (Other Titles in Applied Mathematics)
Approximation Theory and Approximation Practice (Other Titles in Applied Mathematics)
Quadrature by expansion: A new method for the evaluation of layer potentials
Journal of Computational Physics
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Boundary integral equations and Nyström discretization provide a powerful tool for the solution of Laplace and Helmholtz boundary value problems. However, often a weakly-singular kernel arises, in which case specialized quadratures that modify the matrix entries near the diagonal are needed to reach a high accuracy. We describe the construction of four different quadratures which handle logarithmically-singular kernels. Only smooth boundaries are considered, but some of the techniques extend straightforwardly to the case of corners. Three are modifications of the global periodic trapezoid rule, due to Kapur---Rokhlin, to Alpert, and to Kress. The fourth is a modification to a quadrature based on Gauss---Legendre panels due to Kolm---Rokhlin; this formulation allows adaptivity. We compare in numerical experiments the convergence of the four schemes in various settings, including low- and high-frequency planar Helmholtz problems, and 3D axisymmetric Laplace problems. We also find striking differences in performance in an iterative setting. We summarize the relative advantages of the schemes.