We　propose　the　fully　discrete　traditional　finite　element　and　mixed　finite　element　two-level　schemes　for　solving　the　Cahn-Hilliard　equation　in　the　paper.　We　give　the　stability　and　convergence　of　the　traditional　finite　element　and　mixed　finite　element　two-level　methods.　The　analysis　shows　that　the　two-level　methods　can　get　the　same　convergence　as　the　one-level　methods　provided　that　we　choose　proper　coarse　and　fine　mesh　sizes.　However,　the　two-level　methods　can　save　much　computational　time　compared　with　the　one-level　methods.　Finally,　some　numerical　experiments　are　provided　to　confirm　the　theoretical　analysis.　(C)　2016　International　Association　for　Mathematics　and　Computers　in　Simulation　(IMACS).　Published　by　Elsevier　B.V.　All　rights　reserved.