Solution of Stokes flow in complex nonsmooth 2D geometries via a linear-scaling high-order adaptive integral equation scheme

TL;DR

This paper introduces a fast, high-order adaptive boundary integral method for solving Stokes flow in complex 2D geometries, effectively handling nonsmooth boundaries with high accuracy and efficiency.

Contribution

It develops a novel adaptive panel refinement and quadrature scheme for high-accuracy integral evaluation in complex geometries, including nearly-singular integrals.

Findings

Achieves 9-digit accuracy with fewer than 200K points in complex geometries.

Demonstrates effectiveness in handling nonsmooth, corner-rich geometries.

Provides a scalable, high-order method for 2D Stokes flow problems.

Abstract

We present a fast, high-order accurate and adaptive boundary integral scheme for solving the Stokes equations in complex---possibly nonsmooth---geometries in two dimensions. The key ingredient is a set of panel quadrature rules capable of evaluating weakly-singular, nearly-singular and hyper-singular integrals to high accuracy. Near-singular integral evaluation, in particular, is done using an extension of the scheme developed in J.~Helsing and R.~Ojala, {\it J. Comput. Phys.} {\bf 227} (2008) 2899--2921. The boundary of the given geometry is ``panelized'' automatically to achieve user-prescribed precision. We show that this adaptive panel refinement procedure works well in practice even in the case of complex geometries with large number of corners. In one example, for instance, a model 2D vascular network with 378 corners required less than 200K discretization points to obtain a 9-digit solution accuracy.