A robust Petrov–Galerkin discretisation of convection–diffusion equations

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A Petrov–Galerkin discretisation is studied of an ultra-weak variational formulation of the convection–diffusion equation in a 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.