  Thermal noise is expected to be one of the noise sources limiting the
astrophysical reach of Advanced LIGO (once commissioning is complete) and
third-generation detectors. Adopting crystalline materials for thin, reflecting
mirror coatings, rather than the amorphous coatings used in current-generation
detectors, could potentially reduce thermal noise. Understanding and reducing
thermal noise requires accurate theoretical models, but modeling thermal noise
analytically is especially challenging with crystalline materials. Thermal
noise models typically rely on the fluctuation-dissipation theorem, which
relates the power spectral density of the thermal noise to an auxiliary elastic
problem. In this paper, we present results from a new, open-source tool that
numerically solves the auxiliary elastic problem to compute the Brownian
thermal noise for both amorphous and crystalline coatings. We employ
open-source frameworks to solve the auxiliary elastic problem using a
finite-element method, adaptive mesh refinement, and parallel processing that
enables us to use high resolutions capable of resolving the thin reflective
coating. We compare with approximate analytic solutions for amorphous
materials, and we verify that our solutions scale as expected. Finally, we
model the crystalline coating thermal noise in an experiment reported by Cole
and collaborators (2013), comparing our results to a simpler numerical
calculation that treats the coating as an "effectively amorphous" material. We
find that treating the coating as a cubic crystal instead of as an effectively
amorphous material increases the thermal noise by about 3%. Our results are a
step toward better understanding and reducing thermal noise to increase the
reach of future gravitational-wave detectors. (Abstract abbreviated.)
