On the choice of stabilizing sub-grid for convection–diffusion problem on rectangular grids

A stabilizing sub-grid which consists of a single additional node in each rectangular element is analyzed for solving the convection–diffusion problem, especially in the case of small diffusion. We provide a simple recipe for spotting the location of the additional node that contributes a very good stabilizing effect to the overall numerical method. We further study convergence properties of the method under consideration and prove that the standard Galerkin finite element solution on augmented grid produces a discrete solution that satisfies the same type of a priori error estimates that are typically obtained with the SUPG method. Some numerical experiments that confirm the theoretical findings are also presented.