# A boundary integral equation approach to computing eigenvalues of the Stokes operator

## TL;DR

This paper introduces a boundary integral equation method for efficiently computing eigenvalues and eigenfunctions of the Stokes operator, especially in complex geometries, with theoretical justification and numerical validation.

## Contribution

It provides a new BIE-based framework for Stokes eigenvalue problems, including theoretical analysis and practical numerical implementation for complex domains.

## Key findings

- The BIE approach reduces problem dimension and avoids high-frequency pollution.
- Numerical results confirm the method's accuracy and efficiency.
- Theoretical analysis ensures the uniqueness and validity of the approach.

## Abstract

The eigenvalues and eigenfunctions of the Stokes operator have been the subject of intense analytical investigation and have applications in the study and simulation of the Navier-Stokes equations. As the Stokes operator is a fourth-order operator, computing these eigenvalues and the corresponding eigenfunctions is a challenging task, particularly in complex geometries and at high frequencies. The boundary integral equation (BIE) framework provides robust and scalable eigenvalue computations due to (a) the reduction in the dimension of the problem to be discretized and (b) the absence of high frequency "pollution" when using a Green's function to represent propagating waves. In this paper, we detail the theoretical justification for a BIE approach to the Stokes eigenvalue problem on simply and multiply-connected planar domains, which entails a treatment of the uniqueness theory for oscillatory Stokes equations on exterior domains. Then, using well-established techniques for discretizing BIEs, we present numerical results which confirm the analytical claims of the paper and demonstrate the efficiency of the overall approach.