Recovering SUPG using Petrov-Galerkin formulations enriched with adjoint residual-free bubbles

We consider conforming Petrov-Galerkin formulations for the advective and advective-diffusive equations. For the linear hyperbolic equation, the continuous formulation is set up using different spaces and the discretization follows with different 'bubble' enrichments for the test and trial spaces. Boundary conditions for residual-free bubbles are modified to accommodate with the first-order equation case and regular bubbles are used to enrich the other space. Using piecewise linears with these enrichments, the final formulations are shown to be equivalent to the SUPG method, provided the data are assumed to be piecewise constant. Generalization to include diffusion is also presented.