The time-dependent Ginzburg-Landau equations have been solved numerically by a finite element analysis for the mesoscopic superconducting samples with cylindrical shape in a uniform axial magnetic field. We obtain the different vortex patterns as a function of the applied field perpendicular to its surface. We find that multi-vortex states are ground state in three-dimensional mesoscopic cylinders. These results show that our approach is an effective and useful to interpret experimental data on vortex states in the mesoscopic superconductors.
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