Two-dimensional Stokes flow through a periodic channel is considered. The channel walls need only be Lipschitz continuous, in other words they are allowed to have corners. Boundary integral methods are an attractive tool for numerically solving the Stokes equations, as the partial differential equation can be reformulated into an integral equation that must be solved only over the boundary of the domain. When the boundary is at least C 1 smooth, the boundary integral kernel is a compact operator, and traditional Nystr\"om methods can be used to obtain highly accurate solutions. In the case of Lipschitz continuous boundaries, however, obtaining accurate solutions using the standard Nystr\"om method can require high resolution. We adapt a technique known as recursively compressed inverse preconditioning to accurately solve the Stokes equations without requiring any more resolution than is needed to resolve the boundary. Combined with a periodic fast summation method we construct a method that is O(N log N ) where N is the number of quadrature points on the boundary. We demonstrate the robustness of this method by extending an existing boundary integral method for viscous drops to handle the movement of drops near corners.

中文翻译：

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一种具有分段光滑边界的Stokes流的精确积分方程方法

考虑通过周期性通道的二维斯托克斯流。通道壁只需要是 Lipschitz 连续的，换句话说，它们可以有拐角。边界积分方法是对 Stokes 方程进行数值求解的一种有吸引力的工具，因为偏微分方程可以重新表述为一个积分方程，该方程只能在域的边界上求解。当边界至少为 C 1 平滑时，边界积分核是一个紧算子，可以使用传统的 Nystr\"om 方法获得高精度解。然而，在 Lipschitz 连续边界的情况下，使用标准 Nystr\"om 方法可能需要高分辨率。我们采用一种称为递归压缩逆预处理的技术来准确求解斯托克斯方程，而无需比求解边界所需的更多分辨率。结合周期性快速求和方法，我们构造了一个 O(N log N ) 的方法，其中 N 是边界上的正交点的数量。我们通过扩展粘性液滴的现有边界积分方法来处理靠近角落的液滴运动，证明了该方法的稳健性。