GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems
SIAM Journal on Scientific and Statistical Computing
A fast algorithm for particle simulations
Journal of Computational Physics
Quadrature methods for periodic singular and weakly singular Fredholm integral equations
Journal of Scientific Computing
An analysis of quadrature errors in second-kind boundary integral methods
SIAM Journal on Numerical Analysis
High-order and efficient methods for the vorticity formulation of the Euler equations
SIAM Journal on Scientific Computing
Locally corrected multidimensional quadrature rules for singular functions
SIAM Journal on Scientific Computing
On the numerical solution of a hypersingular integral equation in scattering theory
Journal of Computational and Applied Mathematics
High-Order Corrected Trapezoidal Quadrature Rules for Singular Functions
SIAM Journal on Numerical Analysis
Generalized Gaussian Quadratures and Singular Value Decompositions of Integral Operators
SIAM Journal on Scientific Computing
Hybrid Gauss-Trapezoidal Quadrature Rules
SIAM Journal on Scientific Computing
Journal of Computational Physics
Evaluation of Single Layer Potentials over Curved Surfaces
SIAM Journal on Scientific Computing
A high-order 3D boundary integral equation solver for elliptic PDEs in smooth domains
Journal of Computational Physics
On the evaluation of layer potentials close to their sources
Journal of Computational Physics
Journal of Computational Physics
Integral equation methods for elliptic problems with boundary conditions of mixed type
Journal of Computational Physics
NIST Handbook of Mathematical Functions
NIST Handbook of Mathematical Functions
A Nonlinear Optimization Procedure for Generalized Gaussian Quadratures
SIAM Journal on Scientific Computing
Boundary integral equations in time-harmonic acoustic scattering
Mathematical and Computer Modelling: An International Journal
Computers & Mathematics with Applications
Journal of Computational Physics
Advances in Computational Mathematics
Hi-index | 31.45 |
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.