In　this　paper,　we　study　two-grid　mixed　finite　element　algorithms　(MFEAs)　for　the　stationary　incompressible　Stokes　equations　with　damping　and　nonlinear　slip　boundary　conditions.　We　set　up　the　weak　formulation　and　obtain　the　well-posedness.　Then　employing　the　mixed　finite　element　method,　some　optimal　error　estimates　between　the　exact　solution　and　the　approximation　solution　are　derived.　Two　kinds　of　two-grid　MFEAs　based　on　the　Stokes　correction　and　Newton-type　correction　are　established.　Some　error　estimates　for　two-grid　MFEAs　are　obtained.　Several　numerical　results　are　presented　to　confirm　our　theoretical　findings.　©　2019　Elsevier　Ltd