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　2β　∈　(0,　1)　and　2α　∈　(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.　©　2010　Elsevier　Inc.　All　rights　reserved.