On convergence of upwinding Petrov-Galerkin methods for convection-diffusion

We consider special upwinding Petrov-Galerkin discretizations for convection-diffusion problems. For the one dimensional case with a standard continuous linear element as the trial space and a special exponential bubble test space, we prove that the Green function associated to the continuous solution can generate the test space. In this case, we find a formula for the exact inverse of the discretization matrix, that is used for establishing new error estimates for other bubble upwinding Petrov-Galerkin discretizations. We introduce a quadratic bubble upwinding method with a special scaling parameter that provides optimal approximation order for the solution in the discrete infinity norm while avoiding exponential test functions. Provided the linear interpolant has standard approximation properties, we prove optimal approximation estimates in standard L2 and H1 norms. The quadratic bubble method is extended to a two dimensional convection diffusion problem. The proposed discretization produces optimal L2 and H1 convergence orders on subdomains that avoid the boundary layers. The tensor idea of using an efficient upwinding Petrov-Galerkin discretization along each stream line direction in combination with a standard discretizations for the orthogonal direction(s) can lead to efficient new discretization methods for multidimensional convection dominated models.