This　paper　considers　the　H-2-stability　results　for　the　second　order　fully　discrete　schemes　based　on　the　mixed　finite　element　method　for　the　2D　time-dependent　Navier-Stokes　equations　with　the　initial　data　u(0)　is　an　element　of　H-alpha,　where　alpha　=　0,　1　and　2.　A　mixed　finite　element　method　is　used　to　the　spatial　discretization　of　the　Navier-Stokes　equations,　and　the　temporal　treatments　of　the　spatial　discrete　Navier-Stokes　equations　are　the　second　order　semi-implicit,　implicit/explict　and　explicit　schemes.　The　H-2-stability　results　of　the　schemes　are　provided,　where　the　second　order　semi-implicit　and　implicit/explicit　schemes　are　almost　unconditionally　H-2-stable,　the　second　order　explicit　scheme　is　conditionally　H-2-stable　in　the　case　of　alpha　=　2,　and　the　semi-implicit,　implicit/explicit　and　explicit　schemes　are　conditionally　H-2-stable　in　the　case　of　alpha　=　1,　0.　Finally,　some　numerical　tests　are　made　to　verify　the　above　theoretical　results.