There is considerable interest in understanding the dynamics of complex fluids, including macromolecular solutions, in microfluidic devices. That interest has fueled the development of simulation techniques capable of describing the effect of hydrodynamic interactions in confined complex fluids. In this work, we examine the dynamics of DNA and the concomitant chain migration that arises in a parallel plate slit, at equilibrium and under pressure-driven flow. Results are presented from both the lattice Boltzmann method (LBM) and the Brownian dynamics simulations with fluctuating hydrodynamic interactions (BD-HI). It is found that the results of both methods are consistent with each other. We find that the lattice Boltzmann method is well-suited for long polymer chains as well as semidilute and concentrated DNA solutions, while Brownian dynamics is more efficient in dilute DNA solutions.