Residual-free bubbles for advection-diffusion problems: the general error analysis

We develop the general a priori error analysis of residual-free bubble finite element approximations to non-self-adjoint elliptic problems of the form \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $(\varepsilon A + C)u = f$\end{document} subject to homogeneous Dirichlet boundary condition, where A is a symmetric second-order elliptic operator, C is a skew-symmetric first-order differential operator, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $\varepsilon$\end{document} is a positive parameter. Optimal-order error bounds are derived in various norms, using piecewise polynomial finite elements of degree \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $k\geq 1$\end{document}.