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 $L^2$ and $H^1$ norms. The quadratic bubble method is extended to a two dimensional convection diffusion problem. The proposed discretization produces optimal $L^2$ and $H^1$ 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 new and efficient discretization methods for multidimensional convection dominated models.