Quasi-optimality of the SUPG method for the one-dimensional advection-diffusion problem

In this paper we propose quasi-optimal error estimates, in various norms, for the Streamline-Upwind Petrov-Galerkin (SUPG) method applied to the linear one-dimensional advection-diffusion problem. We follow the classical argument due to Babuska and Brezzi, therefore the goal of this work is the proof of the inf-sup and of the continuity conditions for the bilinear stabilized variational form, with respect to suitable norms. These norms are suggested by our previous work, in which we analyze the continuous multi-dimensional advection-diffusion operator. We obtain these results by means of functional spaces interpolation.