  The thermalization process of the holographic entanglement entropy (HEE) of
an annular domain is investigated over the Vaidya-AdS geometry. We numerically
determine the Hubeny-Rangamani-Takayanagi (HRT) surface which may be a
hemi-torus or two disks, depending on the ratio of the inner radius to the
outer radius of the annulus. More importantly, for some fixed ratio of two
radii, it undergoes a phase transition or double phase transitions from a
hemi-torus configuration to a two-disk configuration, or vice versa, during the
thermalization. The occurrence of various phase transitions is determined by
the ratio of two radii of the annulus. The rate of entanglement growth is also
investigated during the thermal quench. The local maximal rate of entanglement
growth occurs in the region with double phase transitions. Finally, if the
quench process is fairly slow which may be controlled by the thickness of null
shell, the region with double phase transitions vanishes.
