# Quadrature by Two Expansions: Evaluating Laplace Layer Potentials using Complex Polynomial and Plane Wave Expansions

## TL;DR

This paper introduces QB2X, a novel method combining complex polynomial and plane wave expansions to evaluate Laplace layer potentials with high accuracy near boundaries, improving upon existing QBX techniques.

## Contribution

The paper develops new representations of Laplace layer potentials using both polynomial and plane wave expansions, enhancing boundary evaluation accuracy and analytical understanding.

## Key findings

- Plane wave expansion captures boundary layer features effectively.
- Error bounds relate to Fourier extension accuracy.
- Preliminary results confirm improved potential evaluation.

## Abstract

The recently developed quadrature by expansion (QBX) technique accurately evaluates the layer potentials with singular, weakly or nearly singular, or even hyper singular kernels in the integral equation reformulations of partial differential equations. The idea is to form a local complex polynomial or partial wave expansion centered at a point away from the boundary to avoid the singularity in the integrand, and then extrapolate the expansion at points near or even exactly on the boundary. In this paper, in addition to the local complex Taylor polynomial expansion, we derive new representations of the Laplace layer potentials using both the local complex polynomial and plane wave expansions. Unlike in the QBX, the local complex polynomial expansion in the new quadrature by two expansions (QB2X) method only collects the far-field contributions and its number of expansion terms can be analyzed using tools from the classical fast multipole method. The plane wave type expansion in the QB2X method better captures the layer potential features near the boundary. It is derived by applying the Fourier extension technique to the density and boundary geometry functions and then analytically utilizing the Residue Theorem for complex contour integrals. The internal connections of the layer potential with its density function and curvature on the boundary are explicitly revealed in the plane wave expansion and its error is bounded by the Fourier extension errors. We present preliminary numerical results to demonstrate the accuracy of the QB2X representations and to validate our analysis.