  We have performed spectral/spectral-element simulations of a single oblate
spheroid with small geometrical aspect ratio settling in an unbounded ambient
fluid, for a range of Galileo numbers covering the various regimes of motion
(steady vertical, steady oblique, vertical periodic and chaotic). The
high-fidelity data provided includes particle quantities (statistics in the
chaotic case), as well as flow profiles and pressure maps. The reference data
can be used as an additional benchmark for other numerical approaches, where a
careful grid convergence study for a specific target parameter point is often
useful. We further describe an extension of a specific immersed boundary method
(Uhlmann, J. Comput. Phys, 209(2):448--476, 2005) to enable the tracking of
non-spherical particles. Finally, the reference cases are computed with this
immersed boundary method at various spatial and temporal resolutions, and grid
convergence is discussed over the various regimes of spheroidal particle
motion. The cross-validation results can serve as a guideline for the design of
simulations with the aid of similar non-conforming methods, involving
spheroidal particles with Galileo numbers of ${\cal O}(100)$.
