Combining adjoint stabilized methods for the advection-diffusion-reaction problem

Still computational methods for the advection-diffusion-reaction transport equa-
tion are a challenge. Although there exist globally stable methods, oscillations
around sharp layers such as boundary, inner and outflow layers, are typical in
multi-dimensional flows. In this paper a variational formulation that combines two
types of stabilization integrals is proposed, namely an adjoint stabilization and a
gradient adjoint stabilization. The two free parameters are chosen imposing one-
dimensional superconvergence. Then, when applied to multi-dimensional flows, the method presents better local stability than present stabilized methods. Furthermore, in the advective-diffusive limit and for piecewise linear functional spaces, the method recovers the classical SUPG method.