In　this　paper,　a　type　of　accurate　a　posteriori　error　estimator　is　proposed　for　semilinear　Neumann　problem,　which　provides　an　asymptotic　exact　estimate　for　the　finite　element　approximate　solution.　As　its　applications,　we　design　two　types　of　cascadic　adaptive　finite　element　methods　for　semilinear　Neumann　problem　based　on　the　proposed　a　posteriori　error　estimator.　The　first　scheme　is　based　on　the　Newton　iteration,　which　needs　to　solve　a　linearized　boundary　value　problem　by　some　smoothing　steps　on　each　adaptive　space.　The　second　scheme　is　based　on　the　multilevel　correction　method,　which　contains　some　smoothing　steps　for　a　linearized　boundary　value　problem　on　each　adaptive　space　and　a　solving　step　for　semilinear　Neumann　equation　on　a　low　dimensional　space.　In　addition,　the　proposed　a　posteriori　error　estimator　provides　the　strategy　to　refine　mesh　and　control　the　number　of　smoothing　steps　for　both　of　the　cascadic　adaptive　methods.　Some　numerical　examples　are　presented　to　validate　the　efficiency　of　the　proposed　algorithms　in　this　paper.　(C)　2019　Elsevier　Inc.　All　rights　reserved.