Numerical simulations of multiphysics problems require not only an accurate solution of all the physical phenomena involved, but also an accurate representation of inter-physical couplings. As a natural consequence of the mutual couplings between different physics, most multiphysics problems are also multiscale problems. They can be geometrical, spatial, and temporal multiscales, and can span over several orders of magnitude in terms of the respective characteristics. To accurately simulate such a multiphysics and multiscale problem, it is essential that the multiscale couplings between different physics are properly modeled and simulated. In a numerical simulation, one can use either finer geometrical meshes or higher polynomial orders to resolve a smaller spatial feature, known as the h - and p-refinement, respectively. Unfortunately, since the multiphysics coupling is a dynamic process, the small spatial features of the physics can evolve and propagate in both space and time. In this case, a static refinement does not work well and the h - or p-refinement has to be performed in a dynamic fashion, which results in a numerical system with a not only large but also time-varying size. As a result, the application of h - or p-refinement becomes extremely expensive and impractical to apply.