Physical systems encompassing a variety of strongly coupled scales pose major computational challenges in terms of analysis modeling and simulation. In this report we first discuss a multiscale modeling approach for the transport of particles such as DNA in heterogeneous devices (microarrays). The model involves a multiscale elliptic equation coupled with a multiscale advection-diffusion equation. We introduce a numerical method for the solution of the coupled system of equations. We first discuss a new multiscale finite element method for the solution of the elliptic problem. We then explain how it is possible to use an explicit stabilized method (ROCK) for the numerical solution of the stiff system of ordinary differential equations of large dimension, originating from the method of lines discretization of the advection-diffusion equation.