  We present an adaptive finite element method for the incompressible
Navier--Stokes equations based on a standard splitting scheme (the incremental
pressure correction scheme). The presented method combines the efficiency and
simplicity of a splitting method with the powerful framework offered by the
finite element method for error analysis and adaptivity. An a posteriori error
estimate is derived which expresses the error in a goal functional of interest
as a sum of contributions from spatial discretization, time discretization and
a term that measures the deviation of the splitting scheme from a pure Galerkin
scheme (the computational error). Numerical examples are presented which
demonstrate the performance of the adaptive algorithm and high quality
efficiency indices. It is further demonstrated that the computational error of
the Navier--Stokes momentum equation is linear in the size of the time step
while the computational error of the continuity equation is quadratic in the
size of the time step.
