Three-dimensional (3D) simulation is conducted for the floating-zone (FZ) growth of silicon in an ellipsoid mirror furnace. for the first time, simultaneously considering the time-dependent Marangoni flow, heat transfer, and moving interfaces. The numerical method is based on an efficient multigrid finite-volume method with front tracking. The growth of an 8 mm diameter silicon crystal under microgravity is considered. A half-zone configuration is also used for benchmarking, and the calculated bifurcation points and flow structures are in good agreement with previous results. However. quasi-periodic modes and angular waves are observed at higher Marangoni number. For full zone calculations, including the feeding and growth interfaces. the bifurcation behavior is similar, but the primary bifurcation is found to be subcritical and the 3D one-fold flow mode is dominant. Significant growth rate fluctuations and back melting are found for a typical growth condition as well. The major fluctuation frequency ranges from 0.1 to 0.3 Hz, and the severe back melting may be related to the angular waves.