Stress intensity factors and improved convergence estimates at a corner
SIAM Journal on Numerical Analysis
High-order methods for linear functionals of solutions of second kind integral equations
SIAM Journal on Numerical Analysis
SIAM Journal on Numerical Analysis
Locally corrected multidimensional quadrature rules for singular functions
SIAM Journal on Scientific Computing
A Finite-Element Method for Laplace- and Helmholtz-Type Boundary Value Problems with Singularities
SIAM Journal on Numerical Analysis
Journal of Computational Physics
A high-order algorithm for obstacle scattering in three dimensions
Journal of Computational Physics
The kink phenomenon in Fejér and Clenshaw–Curtis quadrature
Numerische Mathematik
Is Gauss Quadrature Better than Clenshaw-Curtis?
SIAM Review
A discrete collocation method for Symm's integral equation on curves with corners
Journal of Computational and Applied Mathematics
A quadrature formula of Clenshaw-Curtis type for the Gegenbauer weight-function
Journal of Computational and Applied Mathematics
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
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We present a new algorithm, based on integral equation formulations, for the solution of constant-coefficient elliptic partial differential equations (PDE) in closed two-dimensional domains with non-smooth boundaries; we focus on cases in which the integral-equation solutions as well as physically meaningful quantities (such as, stresses, electric/magnetic fields, etc.) tend to infinity at singular boundary points (corners). While, for simplicity, we restrict our discussion to integral equations associated with the Neumann problem for the Laplace equation, the proposed methodology applies to integral equations arising from other types of PDEs, including the Helmholtz, Maxwell, and linear elasticity equations. Our numerical results demonstrate excellent convergence as discretizations are refined, even around singular points at which solutions tend to infinity. We demonstrate the efficacy of this algorithm through applications to solution of Neumann problems for the Laplace operator over a variety of domains—including domains containing extremely sharp concave and convex corners, with angles as small as π/100 and as large as 199π/100.