In this work, we provide a fast, spectrally accurate method for the evaluation of boundary integral operators (BIOs) on a suspension of prolate and oblate spheroids. We first derive formulas for the standard layer potential operators for the Laplace equation applied to an expansion of the integral densities in the appropriate spheroidal harmonic basis. These then lead to analytical expressions in solid harmonics that allow spectrally accurate evaluation of near-field particle interactions. Finally, a standard quadrature scheme is used to evaluate smooth, far-field interactions; these are then accelerated using the fast multipole method. Through a number of numerical test cases, we verify the accuracy and efficiency of our BIO evaluation framework for dense, polydisperse suspensions of spheroids. Through the use of standard formulas linking Stokes and Laplace potentials, we show our scheme can be readily applied to problems involving particulate suspension flows. For both Laplace and Stokes, our method allows us to evaluate BIOs for suspensions up to hundreds of particles on a single processor.

Submitted to Journal of Computational Physics, June 2025