  We present a 3D finite element solver for the nonlinear Poisson-Nernst-Planck
(PNP) equations for electrodiffusion, coupled to the Stokes system of fluid
dynamics. The model serves as a building block for the simulation of
macromolecule dynamics inside nanopore sensors.
  We add to existing numerical approaches by deploying goal-oriented adaptive
mesh refinement. To reduce the computation overhead of mesh adaptivity, our
error estimator uses the much cheaper Poisson-Boltzmann equation as a
simplified model, which is justified on heuristic grounds but shown to work
well in practice. To address the nonlinearity in the full PNP-Stokes system,
three different linearization schemes are proposed and investigated, with two
segregated iterative approaches both outperforming a naive application of
Newton's method. Numerical experiments are reported on a real-world nanopore
sensor geometry.
  We also investigate two different models for the interaction of target
molecules with the nanopore sensor through the PNP-Stokes equations. In one
model, the molecule is of finite size and is explicitly built into the
geometry; while in the other, the molecule is located at a single point and
only modeled implicitly -- after solution of the system -- which is
computationally favorable. We compare the resulting force profiles of the
electric and velocity fields acting on the molecule, and conclude that the
point-size model fails to capture important physical effects such as the
dependence of charge selectivity of the sensor on the molecule radius.
