Spectrally-accurate quadratures for evaluation of layer potentials close to the boundary for the 2D Stokes and Laplace equations

TL;DR

This paper develops spectrally accurate quadrature methods for evaluating layer potentials near boundaries in 2D Stokes and Laplace equations, crucial for dense particulate flow simulations involving close interfaces.

Contribution

It introduces a globally compensated trapezoid rule quadrature for Laplace potentials and constructs accurate Stokes potential evaluators using Laplace potentials, enhancing near-boundary accuracy.

Findings

Achieves typically 12 digits of accuracy with few discretization nodes.

Extensively tested on vesicle interactions and multiple close-to-touching ellipses.

Provides accessible, documented code for implementation.

Abstract

Dense particulate flow simulations using integral equation methods demand accurate evaluation of Stokes layer potentials on arbitrarily close interfaces. In this paper, we generalize techniques for close evaluation of Laplace double-layer potentials in J. Helsing and R. Ojala, J. Comput. Phys. 227 (2008) 2899-2921. We create a "globally compensated" trapezoid rule quadrature for the Laplace single-layer potential on the interior and exterior of smooth curves. This exploits a complex representation, a product quadrature (in the style of Kress) for the sawtooth function, careful attention to branch cuts, and second-kind barycentric-type formulae for Cauchy integrals and their derivatives. Upon this we build accurate single- and double-layer Stokes potential evaluators by expressing them in terms of Laplace potentials. We test their convergence for vesicle-vesicle interactions, for an extensive set of Laplace and Stokes problems, and when applying the system matrix in a boundary value problem solver in the exterior of multiple close-to-touching ellipses. We achieve typically 12 digits of accuracy using very small numbers of discretization nodes per curve. We provide documented codes for other researchers to use.