We　present　in　this　paper　a　fully　discrete　two-grid　finite　element　method　to　solve　the　diffusive　Peterlin　viscoelastic　model　on　the　basis　of　studying　the　fully　discrete　standard　finite　element　method.　The　fully　discrete　two-grid　method　involves　solving　small　nonlinear　coarse　spatial　grid　systems　and　linear　fine　spatial　grid　systems　with　the　same　time　step.　Under　the　reasonable　assumptions,　it　is　proven　theoretically　that　the　accuracy　of　fully　discrete　two-grid　method　is　the　same　as　the　fully　discrete　finite　element　method　on　the　fine　spatial　grid.　Numerical　examples　are　given　to　validate　the　accuracy　and　convergence　order　of　the　standard　finite　element　method　and　two-grid　method.　©　2022　Elsevier　Ltd