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The eigenvalues and eigenfunctions of the Stokes operator have been thesubject of intense analytical investigation and have applications in the studyand simulation of the Navier-Stokes equations. As the Stokes operator is afourth-order operator, computing these eigenvalues and the correspondingeigenfunctions is a challenging task, particularly in complex geometries and athigh frequencies. The boundary integral equation (BIE) framework providesrobust and scalable eigenvalue computations due to (a) the reduction in thedimension of the problem to be discretized and (b) the absence of highfrequency "pollution" when using a Green's function to represent propagatingwaves. In this paper, we detail the theoretical justification for a BIEapproach to the Stokes eigenvalue problem on simply and multiply-connectedplanar domains, which entails a treatment of the uniqueness theory foroscillatory Stokes equations on exterior domains. Then, using well-establishedtechniques for discretizing BIEs, we present numerical results which confirmthe analytical claims of the paper and demonstrate the efficiency of theoverall approach.

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