Public Research Seminar by Advanced Materials (AMAT) Thrust, HKUST(GZ) - Euler-Maclaurin Formula for Near-singular Integrals

The Integral Equation Method is one of the most efficient numerical methods for mathematical physics problems. The accuracy and stability of an integral equation solver crucially depend on its underlying quadrature method. Developing efficient quadrature schemes for surface integral operators, especially for near-singular integrals, has been a key challenge in integral equation method. Approximating near-singular integrals using regular quadrature methods is in general inefficient and expensive: adaptive meshes are often required; an excessive number of mesh points must be used in regions of localized rapid variations, leading to high computational cost and algorithmic instability.
We develop a simple near-singular quadrature method based on an error correction approach. Uniform meshes are used. The errors of applying the trapezoidal quadrature rule to nearly singular integrals are corrected to high-order accuracy based on generalized Euler-Maclaurin formulas. This approach results in robust, accurate, and low-cost quadrature rules that are insensitive to the degree of near singularities in the integrand. We embed this near-singular quadrature method in a broader framework to emphasize that simple quadrature rules combined with local corrections can effectively handle a great variety of difficult integrals in a broad range of applications.