The behaviors of a rapidly rotating Bose-Einstein condensate under extreme elongation in a two-dimensional anisotropic harmonic plus quartic trap are investigated. Due to the quartic trap, the system remains stable at high rotating velocity, $\ensuremath{\Omega}\ensuremath{\geqslant}{\ensuremath{\omega}}_{\ensuremath{\perp}}$ (${\ensuremath{\omega}}_{\ensuremath{\perp}}$ is the radial harmonic trap frequency), and vortex lattices form even in the absence of the repulsive
s-wave interaction (
g). When
g is present, the interplay between
g and the quartic trap potential can lead to rich vortex lattice transition states as a function of $\ensuremath{\Omega}$, to which vortex lattices vanish eventually at some higher $\ensuremath{\Omega}$.
We investigate the vortex patterns appearing in a two-dimensional annular Bose-Einstein condensate rotating in a quadratic plus quartic confining potential. We show that in the limit of small anharmonicity the Gross-Pitaevskii energy can be minimized among the lowest-Landau-level wave functions and use this particular form to obtain theoretical results in the spirit of Aftalion et al. [Phys. Rev. A 73, 011601(R) (2006)]. In particular, we show that the vortex pattern is infinite but not uniform. We also compute numerically the complete vortex structure: It is an Abrikosov lattice strongly distorted near the edges of the condensate with multiply quantized vortices appearing at the center of the trap.