In　this　paper,　symmetric　space-fractional　partial　differential　equations　(SSFPDE)　with　the　Riesz　fractional　operator　are　considered.　The　SSFPDE　is　obtained　from　the　standard　advection-dispersion　equation　by　replacing　the　first-order　and　second-order　space　derivatives　with　the　Riesz　fractional　derivatives　of　order　2　beta　is　an　element　of　(0,1)　and　2　alpha　is　an　element　of　2　(1,　2],　respectively.　We　prove　that　the　variational　solution　of　the　SSFPDE　exists　and　is　unique.　Using　the　Galerkin　finite　element　method　and　a　backward　difference　technique,　a　fully　discrete　approximating　system　is　obtained,　which　has　a　unique　solution　according　to　the　Lax-Milgram　theorem.　The　stability　and　convergence　of　the　fully　discrete　schemes　are　derived.　Finally,　some　numerical　experiments　are　given　to　confirm　our　theoretical　analysis.　(C)　2010　Elsevier　Inc.　All　rights　reserved.