Sapphire crystals are used in a substrate of the LED devices. Both Kyropoulos (Ky) and Czochralski (CZ) growth process are widely applied techniques for growing high quality sapphire single crystal. A successful growth of the sapphire crystals requires the control of heat and mass transport phenomena. In this study, the growth processes of the sapphire crystal using the resistivity-heated Ky method and the inductively-heated CZ method have been analyzed numerically using finite element method. Based on the simulation results, the melt-crystal interface of the crystal changed from the concave to the flat shape as the Ky process progressed. In case of the CZ method, the high temperature positions moved from the crucible surface to inside the melt and the interface changed to the flat shape when the RPM was increased. Also the interface shape of the grown crystal has been influenced by the formed shoulder shape at the initial stage.