Quadrature by two expansions: Evaluating Laplace layer potentials using complex polynomial and plane wave expansions

JOURNAL OF COMPUTATIONAL PHYSICS

卷:428

Article

Ding, Lingyun¹  Huang, Jingfang¹  Marzuola, Jeremy L.¹  Tang, Zhuochao¹ 

[1] Univ N Carolina, Dept Math, Chapel Hill, NC 27515 USA

关键词: Layer potential;    Quadrature by expansion;    Partial wave expansion;    Plane wave expansion;    Fourier extension;    Integral equation;   

DOI  :  10.1016/j.jcp.2020.109963

来源: Elsevier

【 摘 要 】

The recently developed quadrature by expansion (QBX) technique [24] 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 type 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 (FMM). The plane wave type expansion in the QB2X method is derived by first applying the Fourier extension technique to the density and polynomial approximation of the boundary geometry, and then analytically evaluating the integral using the Residue Theorem with properly chosen complex contour. The plane wave type expansion accurately captures the high frequency properties of the layer potential that are determined (up to a prescribed accuracy) only by the local features of the density function and boundary geometry, and the nonlinear impact of the boundary on the layer potential becomes explicit. The QB2X technique allows high order numerical discretizations and can be adopted easily in existing FMM based fast integral equation solvers. We present preliminary numerical results to validate our analysis and demonstrate the accuracy and efficiency of the QB2X representations when compared with the classical QBX method. (c) 2020 Elsevier Inc. All rights reserved.

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