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

TL;DR

This work introduces QB2X, a two-expansion framework for Laplace layer potentials that couples a local complex polynomial expansion (far-field) with a plane wave (Fourier-extended) expansion (near-field) inside leaf boxes of an FMM tree. It provides representations for both double and single layer potentials, including cases with curved boundaries, by leveraging contour integration and residues to derive plane-wave coefficients and by applying Fourier extension for efficient exponential expansions. The authors analyze error sources and demonstrate, through preliminary numerical experiments, that QB2X can achieve high-precision results (up to machine precision in some tests) while offering a clearer dependency on expansion parameters than classical QBX. The method promises improved accuracy, stability, and efficiency and is extensible to other elliptic PDEs, such as Helmholtz and Yukawa problems, within the FMM framework.

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.

Figures (12)

• Figure 1: Different expansions for the leaf boxes in a uniform FMM hierarchical tree structure. Green: complex polynomial expansion; Yellow: one QB2X for the leaf node; Red: two QB2X required, one for the interior and one for the exterior.
• Figure 1: An implementation of QBX. (a): Analytical layer potential. (b) and (c): approximation using $K=5$ and $K=15$ terms, respectively. (d), (e), and (f): $\log_{10}$ errors for $K=5$, $K=15$, and $K=25$.
• Figure 1: A leaf box close to the boundary.
• Figure 1: Approximation errors of QB2X representations for double layer potentials with $K = 40$ for different density functions. The plot legends are $\log_{10} (Error)$.
• Figure 2: (a) and (c): Fourier extension $g(x)=\sum_{p=-P}^P c_p e^{\mathrm{i} p x}$ (solid line) of the given function $f$ (dashed line), $P=30$. (b) and (d): approximation errors on $[-1,1]$.