Due　to　the　indefiniteness　and　poor　spectral　properties,　the　discretized　linear　algebraic　system　of　the　vector　Laplacian　by　mixed　finite　element　methods　is　hard　to　solve.　A　block　diagonal　preconditioner　has　been　developed　and　shown　to　be　an　effective　preconditioner　by　Arnold　et　al.　(Acta　Numer　15:1-155,　2006).　The　purpose　of　this　paper　is　to　propose　alternative　and　effective　block　diagonal　and　approximate　block　factorization　preconditioners　for　solving　these　saddle　point　systems.　A　variable　V-cycle　multigrid　method　with　the　standard　point-wise　Gauss-Seidel　smoother　is　proved　to　be　a　good　preconditioner　for　the　discrete　vector　Laplacian　operator.　The　major　benefit　of　our　approach　is　that　the　point-wise　Gauss-Seidel　smoother　is　more　algebraic　and　can　be　easily　implemented　as　a　black-box　smoother.　This　multigrid　solver　will　be　further　used　to　build　preconditioners　for　the　saddle　point　systems　of　the　vector　Laplacian.　Furthermore　it　is　shown　that　Maxwell＇s　equations　with　the　divergent　free　constraint　can　be　decoupled　into　one　vector　Laplacian　and　one　scalar　Laplacian　equation.