# Accurate quadrature and fast summation in boundary integral methods for Stokes flow

**Time: **
Wed 2023-06-14 14.00

**Location: **
F3, Lindstedtsvägen 26 & 28, Stockholm

**Language: **
English

**Subject area: **
Applied and Computational Mathematics, Numerical Analysis

**Doctoral student: **
Joar Bagge
, Numerisk analys, NA

**Opponent: **
Associate Professor Adrianna Gillman, University of Colorado, Boulder, USA

**Supervisor: **
Professor Anna-Karin Tornberg, Numerisk analys, NA; Universitetslektor Katarina Gustavsson, Numerisk analys, NA

QC 2023-05-17

## Abstract

This thesis concerns accurate and efficient numerical methods for the simulation of fluid flow on the microscale, known as Stokes flow or creeping flow. Such flows are important, for example, in understanding the swimming of microorganisms, spreading of dust particles, as well as in developing new nano-materials, and microfluidic devices that can be used for on-the-fly analysis of blood samples, among other things.

Flow on the microscale is dominated by viscous forces, meaning that a fluid such as water or air will behave as a very viscous fluid, like e.g. honey. The equations governing the flow, known as the Stokes equations, are linear PDEs, which permits the use of boundary integral methods (BIMs). In these methods, the PDE is reformulated as a boundary integral equation, thus reducing the dimensionality of the computational problem from three dimensions to two dimensions. The boundary integral formulation is well-conditioned, so that high accuracy can be achieved.

We consider two main challenges related to BIMs. The first challenge is that the integrals in the formulation contain integrands that vary rapidly for evaluation points close to the boundary, and cannot be accurately resolved using a standard method for numerical integration. Therefore, special quadrature methods are needed. We consider two such methods: quadrature by expansion (QBX) and the “line extrapolation/interpolation method” (also known as the Hedgehog method). In particular, we consider these methods applied to simulations involving rigid rodlike particles and surrounding walls.

The second challenge is that discretizing the boundary integral formulation leads to a dense linear system, which requires *O*(*N*^{2}) operations to solve iteratively, where *N* is the number of unknowns. This becomes too expensive for large systems. A fast summation method, such as the Spectral Ewald (SE) method considered in this thesis, reduces the number of operations required, for example to *O*(*N* log *N*). The SE method can also be used for problems with periodic boundary conditions in any number of the spatial directions (arbitrary periodicity).

We also consider an application of these methods to a flow problem involving an inertial spheroid in a parabolic flow profile, and analyze the lateral drift of this spheroidal particle.

The numerical methods studied in this thesis enable fast and accurate computer simulations of e.g. suspensions of rigid particles in three-dimensional Stokes flow, including surrounding walls and arbitrary periodicity.