  Brucite (Mg(OH)$_2$) is a mineral of great interest owing to its various
applications and roles in geological processes. Its structure, behavior under
different conditions, and unique properties have been the subject of numerous
studies and persistent debate. As a stable hydrous phase in subduction zones,
its elastic anisotropy can significantly contribute to the seismological
properties of these regions. We performed ab initio calculations to investigate
brucite's stability, elasticity, and acoustic velocities. We tested several
exchange-correlation functionals and managed to obtain stable phonons for the
P$\bar{3}$ phase with r$^2$SCAN for the first time at all relevant pressures up
to the mantle transition zone. We show that r$^2$SCAN performs very well in
brucite, reproducing the experimental equation of state and several key
structure parameters related to hydrogen positions. The room temperature
elasticity results in P$\bar{3}$ reproduces the experimental results at ambient
pressure. These results, together with the stable phonon dispersion of
P$\bar{3}$ at all relevant pressures, indicate P$\bar{3}$ is the stable
candidate phase not only at elevated pressures but also at ambient conditions.
The success of r$^2$SCAN in brucite, suggests this functional should be
suitable for other challenging layer-structured minerals, e.g., serpentines, of
great geophysical significance.
