- We consider the standard Taylor-Hood finite element method for the stationary Stokes system on polyhedral domains. We prove local a posteriori error estimates for the maximum error in the gradient of the velocity field. Because the gradient of the velocity field blows up near reentrant corners and edges, such local error control is necessary when pointwise control of the gradient error is desirable. Computational examples confirm the utility of our estimates in adaptive codes. 2012 American Mathematical Society.