This　paper　proposes　and　analyzes　some　fully　discrete　(multiphysics)　finite　element　methods　for　a　displacement-pressure　model　which　describes　swelling　dynamics　of　polymer　gels　under　mechanical　constraints.　By　introducing　an　＂elastic　pressure＂　we　first　present　a　reformulation　of　the　original　model.　We　then　propose　a　time-stepping　scheme　which　decouples　the　PDE　system　at　each　time　step　into　two　subproblems,　one　of　which　is　a　Stokes-like　problem　for　the　displacement　vector　field　(of　the　solid　network　of　the　gel)　and　the　other　is　a　diffusion　problem　for　a　＂pseudopressure＂　field　(of　the　solvent　of　the　gel).　To　make　such　a　multiphysics　approach　feasible,　it　is　vital　to　find　admissible　constraints　to　resolve　the　uniqueness　issue　for　the　Stokes-like　problem　and　to　construct　a　good　boundary　condition　for　the　diffusion　equation　so　that　it　also　becomes　uniquely　solvable.　The　key　to　the　first　difficulty　is　to　discover　certain　conserved　quantities　for　the　PDE　solution,　and　the　solution　to　the　second　difficulty　is　to　use　the　Stokes-like　problem　to　generate　a　boundary　condition　for　the　diffusion　problem.　The　time-stepping　scheme　allows　one　to　use　any　convergent　Stokes　solver　(and　its　code)　together　with　any　convergent　diffusion　equation　solver　(and　its　code)　to　solve　the　polymer　gel　model.　In　the　paper,　the　Taylor-Hood　mixed　finite　element　method　combined　with　the　continuous　linear　finite　element　method　are　chosen　as　an　example　to　present　the　ideas　and　to　demonstrate　the　viability　of　the　proposed　multiphysics　approach.　It　is　proved　that,　under　a　mesh　constraint,　both　the　semidiscrete　(in　space)　and　fully　discrete　methods　enjoy　some　discrete　energy　laws　which　mimic　the　differential　energy　law　satisfied　by　the　PDE　solution.　Optimal　order　error　estimates　in　various　norms　are　established　for　both　semidiscrete　and　fully　discrete　methods.　Numerical　experiments　are　also　presented　to　show　the　performance　of　the　proposed　approach　and　methods.