  In this work, the dual-weighted residual method is applied to a space-time
formulation of nonstationary Stokes and Navier-Stokes flow. Tensor-product
space-time finite elements are being used to discretize the variational
formulation with discontinuous Galerkin finite elements in time and inf-sup
stable Taylor-Hood finite element pairs in space. To estimate the error in a
quantity of interest and drive adaptive refinement in time and space, we
demonstrate how the dual-weighted residual method for incompressible flow can
be extended to a partition of unity based error localization. We derive the
space-time Newton method for the Navier-Stokes equations and substantiate our
methodology on 2D benchmark problems from computational fluid mechanics.
