This paper presents a quadrature method for evaluating layer potentials in two dimensions close to periodic boundaries, discretized using the trapezoidal rule. It is an extension of the method of singularity swap quadrature, which recently was introduced for boundaries discretized using composite Gauss-Legendre quadrature. The original method builds on swapping the target singularity for its preimage in the complexified space of the curve parametrization, where the source panel is flat. This allows the integral to be efficiently evaluated using an interpolatory quadrature with a monomial basis. In this extension, we use the target preimage to swap the singularity to a point close to the unit circle. This allows us to evaluate the integral using an interpolatory quadrature with complex exponential basis functions. This is well-conditioned, and can be efficiently evaluated using the fast Fourier transform. The resulting method has exponential convergence, and can be used to accurately evaluate layer potentials close to the source geometry. We report experimental results on a simple test geometry, and provide a baseline Julia implementation that can be used for further experimentation.

翻译：本文提出了一种用于二维周期边界附近层势计算的求积方法，采用梯形法则离散化。该方法是对近期提出的基于复合高斯-勒让德求积离散边界的奇点交换求积法的扩展。原方法通过将目标奇点交换到曲线参数化复化空间中的原像，使源面板保持平整，从而允许使用单项式基的插值求积高效计算积分。在本文扩展中，我们利用目标原像将奇点交换到接近单位圆的点，从而允许使用复指数基函数的插值求积计算积分。该方法条件良好，且可通过快速傅里叶变换高效实现。该方法具有指数收敛性，可用于精确计算接近源几何体的层势。我们在简单测试几何体上报告了实验验证结果，并提供了可用于进一步实验的基准Julia实现。