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 (epsilon A + C)u = f 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 epsilon is a positive parameter. Optimal-order error bounds are derived in various norms, using piecewise polynomial finite elements of degree k greater than or equal to 1.