# Asymptotic analysis for close evaluation of layer potentials

## TL;DR

This paper develops an asymptotic expansion technique to accurately evaluate layer potentials near boundaries, overcoming errors in standard quadrature methods for boundary-adjacent points in 2D Laplace problems.

## Contribution

It introduces an asymptotic expansion approach that removes boundary layer errors in layer potential evaluations, improving accuracy near boundaries.

## Key findings

- Effectively reduces boundary layer errors in layer potential evaluation.
- Demonstrates improved accuracy for interior and exterior Laplace problems.
- Applicable to problems in fluid-structure interactions and nano-optics.

## Abstract

We study the evaluation of layer potentials close to the domain boundary. Accurate evaluation of layer potentials near boundaries is needed in many applications, including fluid-structure interactions and near-field scattering in nano-optics. When numerically evaluating layer potentials, it is natural to use the same quadrature rule as the one used in the Nystr\"om method to solve the underlying boundary integral equation. However, this method is problematic for evaluation points close to boundaries. For a fixed number of quadrature points, $N$, this method incurs $O(1)$ errors in a boundary layer of thickness $O(1/N)$. Using an asymptotic expansion for the kernel of the layer potential, we remove this $O(1)$ error. We demonstrate the effectiveness of this method for interior and exterior problems for Laplace's equation in two dimensions.