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.

最近开发的正交展开（QBX）技术[24]可以准确地评估偏微分方程积分方程式中具有奇异，弱或近乎奇异，甚至超奇异核的层势。这个想法是形成一个中心复数多项式或偏波扩展，以远离边界的点为中心，以避免被积物的奇异性，然后外推在边界附近或什至恰好在边界上的点的扩展。在本文中，除了局部复数泰勒多项式展开式之外，我们还使用局部复数多项式展开式和平面波类型展开式导出了Laplace层势的新表示形式。与QBX不同，新正交二次膨胀（QB2X）方法中的局部复多项式展开仅收集远场贡献，并且可以使用经典快速多极子方法（FMM）中的工具来分析其展开项的数量。在QB2X方法平面波型膨胀是通过首先将所述傅立叶扩展技术的密度和边界几何多项式近似，然后分析评估使用残留定理适当选择复杂轮廓的积分得到。平面波类型扩展仅精确地捕获层电势的高频特性，该高频特性仅由密度函数和边界几何的局部特征确定（达到规定的精度），并且边界对层电势的非线性影响变为明确的。QB2X技术允许进行高阶数值离散化，并且可以在现有的基于FMM的快速积分方程求解器中轻松采用。我们提供了初步的数值结果来验证我们的分析，并证明与经典QBX方法相比，QB2X表示的准确性和效率。