Approximate symmetrization and Petrov-Galerkin methods for diffusion-convection problems

Conforming finite element approximations of a given diffusion-convection problem with positive diffusion coefficient, incompressible convective velocity field, and a Dirichlet section of the boundary that includes all the inflow boundary are considered. The concept of symmetrization is introduced and general error bounds for a Petrov-Galerkin method are derived. Approximations based on two symmetric forms are studied, and the problem of interpreting a finite element approximation which attempts to achieve optimality in one of these symmetric forms is addressed. One and two-dimensional numerical example are included.