Recent progress in the development and understanding of SUPG methods with special reference to the compressible Euler and Navier-Stokes equations

The current status of streamline-upwind/Petrov-Galerkin (SUPG) methods for the analysis of flow problems is surveyed in an analytical review. Problem areas addressed include classical Galerkin, upwind, artificial-diffusion, SUPG, discontinuous Galerkin, space-time FEM, and discontinuity-capturing approaches to the scalar advection-diffusion equation; incompressible flows; advective-diffusive systems; and the compressible Euler and Navier-Stokes equations. Graphs and diagrams are provided, and the good stability properties of state-of-the-art SUPG methods are pointed out.