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 (εA + 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 ε is a positive parameter. Optimal-order error bounds are derived in various norms, using piecewise polynomial finite elements of degree k ≥ 1