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

Computational methods for the advection-diffusion-reaction transport equation are still 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. Two free parameters are chosen by imposing one-dimensional superconvergence. Then, when applied to multi-dimensional flows, the method presents better local stability than the present stabilized methods. Furthermore, in the advective-diffusive limit and for piecewise linear functional spaces, the method recovers the classical SUPG method.