An anisotropic functional setting for convection-diffusion problems

Abstract A new functional framework for consistently stabilized discrete approximations to convection-diffusion problems was recently proposed. The key ideas are the evaluation of the residual in an inner product of the type H –1/2 and the realization of this inner product via explicitely computable multilevel decompositions of function spaces. Here we improve such approach, by taking into account the anisotropic nature of the convection-diffusion operator. We derive uniform (in the diffusion parameter) anisotropic estimates for both the exact and the discrete solutions and we study the convergence of the approximation. To this end we develop a functional framework involving anisotropic Sobolev spaces which depend on the velocity field.