<p>We introduce a parallel adaptive space-time discontinuous Galerkin method for the linear transport equation, where the transport vector is determined from Darcy’s law for porous media flow. Given the permeability distribution, in the first step the pressure head and the flux is computed by a mixed approximation of the linear porous media problem. Then, for a given initial pollution distribution the linear transport is approximated by an adaptive DG space-time discretization on a truncated space-time cylinder which turns out to be very efficient since the adaptively refined region is transported with the pollution distribution. The full linear system in space and time is solved with a parallel multigrid method where the stopping criterium for the linear solver is controlled by the convergence of a linear goal functional. Finally we apply this method to solve the inverse problem to reconstruct the initial pollution distribution from measurements of the outflow.</p>