Quadrature by two expansions: Evaluating Laplace layer potentials using complex polynomial and plane wave expansions
Quadrature by two expansions: Evaluating Laplace layer potentials using complex polynomial and plane wave expansions
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通过两次展开式求积：使用复数多项式和平面波展开式评估拉普拉斯层势
DOI：
10.1016/j.jcp.2020.109963
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发表时间：
2021
影响因子：
4.1
通讯作者：
Tang, Zhuochao
中科院分区：
文献类型：
--
作者：
Ding, Lingyun;Huang, Jingfang;Marzuola, Jeremy L.;Tang, Zhuochao
最近发展起来的展开求积(QBX)技术[24]精确地计算了偏微分方程积分公式中具有奇异核、弱奇异核、近奇异核甚至超奇异核的层势。其思想是以远离边界的一点为中心形成局部复多项式或分波展开式，以避免被积函数的奇异性，然后在靠近甚至正好位于边界上的点上外推展开式。在本文中，除了局域复泰勒多项式展开式外，我们还利用局域复多项式展开式和平面波展开式导出了拉普拉斯层势的新表示。与QBX方法不同，新的二次展开的局部复多项式展开法(QB2X)只收集远场贡献，其展开项的个数可以用经典的快速多极子方法(FMM)的工具来分析。QB2X方法中的平面波展开式是通过将傅里叶延拓技术应用于边界几何的密度和多项式近似，然后利用适当选择复杂轮廓的留数定理来解析计算积分而得到的。平面波展开式精确地捕捉了仅由密度函数和边界几何的局部特征决定的层势的高频特性(直到规定的精度)，边界对层势的非线性影响变得明显。QB2X技术允许高精度的数值离散，可以很容易地在现有的基于FMM的快速积分方程组求解器中采用。我们给出了初步的数值结果来验证我们的分析，并与经典的QBX方法进行了比较，证明了QB2X表示的准确性和效率。
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.