A fast direct solver for integral equations on locally refined boundary discretizations and its application to multiphase flow simulations

TL;DR

This paper introduces a novel fast direct solver for integral equations on locally refined boundary discretizations, significantly improving efficiency in transient particulate Stokes flow simulations by pre-constructing the solver and applying low-rank updates.

Contribution

The authors develop a pre-constructed fast direct solver that efficiently handles local refinements in boundary discretizations for integral equations, enhancing simulation speed.

Findings

Solver accelerates particulate Stokes simulations

Efficient handling of local boundary refinement

Low-rank factorization enables quick updates

Abstract

In transient simulations of particulate Stokes flow, to accurately capture the interaction between the constituent particles and the confining wall, the discretization of the wall often needs to be locally refined in the region approached by the particles. Consequently, standard fast direct solvers lose their efficiency since the linear system changes at each time step. This manuscript presents a new computational approach that avoids this issue by pre-constructing a fast direct solver for the wall ahead of time, computing a low-rank factorization to capture the changes due to the refinement, and solving the problem on the refined discretization via a Woodbury formula. Numerical results illustrate the efficiency of the solver in accelerating particulate Stokes simulations.