Convergence of the boundary integral method for interfacial Stokes flow

TL;DR

This paper provides the first convergence analysis of boundary integral methods for interfacial Stokes flow, clarifying how numerical filters influence stability and accuracy in simulating elastic capsules, drops, and bubbles.

Contribution

It introduces a rigorous convergence analysis for boundary integral methods in interfacial Stokes flow, highlighting the impact of numerical filters on stability and accuracy.

Findings

Convergence of boundary integral methods is established for interfacial Stokes flow.

Numerical filters play a crucial role in ensuring stability and accuracy.

The analysis applies to simulations of elastic capsules, viscous drops, and bubbles in 2D flows.

Abstract

Boundary integral numerical methods are among the most accurate methods for interfacial Stokes flow, and are widely applied. They have the advantage that only the boundary of the domain must be discretized, which reduces the number of discretization points and allows the treatment of complicated interfaces. Despite their popularity, there is no analysis of the convergence of these methods for interfacial Stokes flow. In practice, the stability of discretizations of the boundary integral formulation can depend sensitively on details of the discretization and on the application of numerical filters. We present a convergence analysis of the boundary integral method for Stokes flow, focusing on a rather general method for computing the evolution of an elastic capsule, viscous drop, or inviscid bubble in 2D strain and shear flows. The analysis clarifies the role of numerical filters in practical computations.