$b = \int G$.

In this paper we show the equivalence between the variational multiscale and the residual-free bubbles concepts. 1 Introduction The subject of stabilized finite element methods has existed now for over sixteen years. The basic reason for introducing stabilized methods is that straightforward application of the Galerkin version of the finite element method to certain problems of mathematical physics and engineering yields numerical approximations that are deficient in that they do not inherit the stability properties of the continuous problem. Stabilized methods constitute a systematic methodology for improving stability behavior without compromising accuracy. As such, stabilized methods have provided fundamental solutions to the problem of discrete approximations in several practically important areas. Perhaps the most notable area is fluid dynamics. Convection-diffusion operators have perplexed numerical analysts for decades. Historically they have been treated by methods which hav..