A robust Petrov-Galerkin discretisation of convection-diffusion equations

A Petrov-Galerkin discretization is studied of an ultra-weak variational formulation of the convection-diffusion equation in mixed form. To arrive at an implementable method, the truly optimal test space has to be replaced by its projection onto a finite dimensional test search space. To prevent that this latter space has to be taken increasingly large for vanishing diffusion, a formulation is constructed that is well-posed in the limit case of a pure transport problem. Numerical experiments show approximations that are very close to the best approximations to the solution from the trial space, uniformly in the size of the diffusion term.