Optimal Petrov--Galerkin Methods through Approximate Symmetrization

^*Present address: Department of Mathematics, Imperial College, London SW7 2BZ. A technique of approximate symmetrization is used to derivea test space from a given trial space for a Petrov—Galerkinmethod. This is applied to one-dimensional diffusion—convectionproblems to give approximations which are near optimal in anenergy norm. Rigorous and precise error bounds are derived todemonstrate the uniformly good behaviour and near optimalityof the procedure over all values of the mesh Péclet number.