Solution of Stokes flow in complex nonsmooth 2D geometries via a linearscaling highorder adaptive integral equation scheme
Abstract
We present a fast, highorder accurate and adaptive boundary integral scheme for solving the Stokes equations in complexpossibly nonsmoothgeometries in two dimensions. We apply the panelbased quadratures of Helsing and coworkers to evaluate to high accuracy the weaklysingular, hypersingular, and supersingular integrals arising in the Nyström discretization, and also the nearsingular integrals needed for flow and traction evaluation close to boundaries. The resulting linear system is solved iteratively via calls to a Stokes fast multipole method. We include an automatic algorithm to "panelize" a given geometry, and choose a panel order, which will efficiently approximate the density (and hence solution) to a userprescribed tolerance. We show that this adaptive panel refinement procedure works well in practice even in the case of complex geometries with large number of corners, or closetotouching smooth curves. In one example, for instance, a model 2D vascular network with 378 corners required less than 200K discretization points to obtain a 9digit solution accuracy.
 Publication:

Journal of Computational Physics
 Pub Date:
 June 2020
 DOI:
 10.1016/j.jcp.2020.109361
 arXiv:
 arXiv:1909.00049
 Bibcode:
 2020JCoPh.41009361W
 Keywords:

 Boundary integral methods;
 Stokes flow;
 Corner singularities;
 Fast algorithms;
 Mathematics  Numerical Analysis
 EPrint:
 24 pages, 8 figures, 1 table