# An adaptive kernel-split quadrature method for parameter-dependent layer potentials

## TL;DR

This paper introduces an adaptive quadrature algorithm that maintains high accuracy for parameter-dependent layer potentials across a wide range of parameters, overcoming limitations of existing methods.

## Contribution

It presents a novel adaptive sampling approach with recursive bisection to improve kernel-split quadrature for parameter-dependent equations.

## Key findings

- Achieves accurate evaluation for a broader range of the parameter 
- Maintains accuracy with a computational cost scaling as log
- Extends the applicability of kernel-split quadrature to complex problems

## Abstract

Panel-based, kernel-split quadrature is currently one of the most efficient methods available for accurate evaluation of singular and nearly singular layer potentials in two dimensions. However, it can fail completely for the layer potentials belonging to the modified Helmholtz, biharmonic and Stokes equations. These equations depend on a parameter, denoted $\alpha$, and kernel-split quadrature loses its accuracy rapidly when this parameter grows beyond a certain threshold. This paper describes an algorithm that remedies this problem, using per-target adaptive sampling of the source geometry. The refinement is carried out through recursive bisection, with a carefully selected rule set. This maintains accuracy for a wide range of the parameter $\alpha$, at an increased cost that scales as $\log\alpha$. Using this algorithm allows kernel-split quadrature to be both accurate and efficient for a much wider range of problems than previously possible.