The spectral element method is used to obtain 3D, time-dependent solutions for a thermocapillary driven liquid bridge with magnetic stabilization, which arises from the float-zone crystal growth process. The methods and implementation of the general, in-house developed fluid flow and heat transfer spectral element solver are discussed and the code is benchmarked. This work compares three-dimensional, time-dependent results to perturbations predicted by linear stability theory for the full-zone problem with Prandtl number of 0.02. Critical points, mode numbers, and azimuthal velocity perturbations are matched for the instabilities. Additionally, the simulations extend the study beyond the initial bifurcation point to find modal competition between two steady modes for the zero magnetic field case. Applying an axial magnetic field damps the perturbations and delays instabilities, providing a quiescent interior region that is conducive to growing defect free, uniform composition crystals. Weak magnetic fields are shown to remove the modal competition that leads to undesirable, time-dependent flow with mode switching.