The　fractional　Laplacian,　(-Delta)(s)　,　s　is　an　element　of　(0　,　1)　,　appears　in　a　wide　range　of　physical　systems,　including　Levy　flights,　some　stochastic　interfaces,　and　theoretical　physics　in　connection　to　the　problem　of　stability　of　the　matter.　In　this　paper,　a　matrix　transfer　technique　(MTT)　is　employed　combining　with　spectral/element　method　to　solve　fractional　diffusion　equations　involving　the　fractional　Laplacian.　The　convergence　of　the　MTT　method　is　analyzed　by　the　abstract　operator　theory.　Our　method　can　be　applied　to　solve　various　fractional　equation　involving　fractional　Laplacian　on　some　complex　domains.　Numerical　results　indicate　exponential　convergence　in　the　spatial　discretization　which　is　in　good　agreement　with　the　theoretical　analysis.　(C)　2021　Published　by　Elsevier　B.V.　on　behalf　of　IMACS.