An accurate integral equation method for Stokes flow with piecewise smooth boundaries

TL;DR

This paper introduces an accurate boundary integral method for 2D Stokes flow in channels with Lipschitz continuous, possibly cornered, boundaries, using recursive preconditioning and fast summation to achieve high accuracy efficiently.

Contribution

It develops a novel boundary integral approach with recursive preconditioning for Lipschitz boundaries, enabling accurate and efficient solutions for Stokes flow with corners.

Findings

Achieves O(N log N) computational complexity.

Successfully handles flow near corners and drop movement.

Demonstrates robustness and high accuracy of the method.

Abstract

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.