Surface discretization considerations for the boundary element method applied to 3D ellipsoidal particles in Stokes flow

The boundary element method has often been used for simulating particle motion in Stokes flow, yet there is a scarcity of quantitative studies examining local errors induced by meshing highly elongated particles. In this paper, we study the eigenvalues and eigenfunctions of the double-layer operator for an ellipsoid in an external linear or quadratic flow. We examine the local and global errors induced by changing the interpolation order of the geometry (flat or curved triangular elements) and the interpolation order of the double-layer density (piecewise-constant or piecewise-linear over each element). Interestingly, we find that increasing the interpolation orders for the geometry and the double layer density does not always guarantee smaller errors. Depending on the nature of the meshing near high curvature regions, the number of high aspect ratio elements, and the fitness of the particle geometry, a piecewise-constant density can exhibit lower errors than piecewise-linear density, and there can be little benefit for using curved triangular elements. Overall, this study provides practical insights on how to appropriately discretize and parameterize 3D boundary element simulations for elongated particles with prolate-like and oblate-like geometries.