In　this　paper,　we　propose　a　coupled　discontinuous　Galerkin　(DG)　and　continuous　Galerkin　(CG)　scheme　for　solving　the　nonlinear　evolution　Davey-Stewartson　(DS)　system　in　dimensionless　form.　The　DS　system　consists　of　two　coupled　nonlinear　and　complex　structure　partial　differential　equations.　The　wave’s　amplitude　in　the　first　equation　is　solved　by　the　high-efficiency　local　DG　method,　and　the　velocity　in　the　second　equation　is　obtained　by　a　standard　CG　method.　No　matching　conditions　are　needed　for　the　two　finite　element　spaces　since　the　normal　component　of　the　velocity　is　continuous　across　element　boundaries.　The　main　strengths　of　our　approach　are　that　we　combine　the　advantage　of　DG　and　CG　methods,　using　DG　methods　handling　the　nonlinear　Schrödinger　equation　to　obtain　high　parallelizability　and　high-order　formal　accuracy,　using　the　continuous　finite　elements　solving　the　velocity　to　maintain　total　energy　conservation.　We　prove　the　energy-conserving　properties　of　our　scheme　and　error　estimates　in　L2-norm.　However,　the　non-linearity　terms　bring　a　lot　of　trouble　to　the　proof　of　error　estimates.　With　the　help　of　energy-conserving　properties,　we　construct　a　series　of　energy　equations　to　obtain　error　estimates.　Numerical　tests　for　different　types　of　systems　are　presented　to　clarify　the　effectiveness　of　numerical　methods.　©2025　Global-Science　Press.