Bubble and multiscale stabilization of bilinear finite element methods for transient advection-diffusion equations on rectangular grids

In this paper a Petrov-Galerkin type stabilization for a time dependent advection-diffusion equation is considered. We first enrich the bilinear test space with bubble functions and the bilinear trial space with a special combination of bubble and multiscale functions for the steady state advection-diffusion equation. It is known that solving the residual equation obtained by the bubble elimination procedure is as difficult as solving the steady case of the original problem, which makes the enriched methods quite costly for two-dimensional problems. In this study, we instead utilize their cheap and efficient approximations in each rectangular element. Then we suggest a recipe for a proper adaptation of these functions combined with the generalized Euler time integration to the unsteady problem. Some numerical experiments for typical model problems are presented to illustrate the accuracy of our method.