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

Summary.

We develop the general a priori error analysis of residual-free bubble finite element approximations to non-self-adjoint elliptic problems of the form $(\varepsilon A + C)u = f$ 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 $\varepsilon$ is a positive parameter. Optimal-order error bounds are derived in various norms, using piecewise polynomial finite elements of degree $k\geq 1$.