A discontinuous residual-free bubble method for advection-diffusion problems

A discontinuous finite-element method is presented for solving the linear advection-diffusion equation, based on the Residual-Free Bubble (RFB) finite-element formulation. After the macro-scales (usual piecewise-polynomials elements) are separated from the micro-scales (the bubble part), they are computed by a standard Galerkin formulation, while the bubble part is approximated by a discontinuous Galerkin method. The advantage of this approach, as compared to other implementations of the Residual-Free Bubble formulation, is that the macro-scales are computed accurately, at least for the model problem presently considered. Numerical tests are performed to confirm the validity of the proposed approach.