Stability of an upwind Petrov Galerkin discretization of convection diffusion equations
Provides rigorous stability analysis for a numerical method for convection-dominated problems, which is important for computational fluid dynamics and related fields.
The paper proves stability (uniform inf-sup condition) for an upwind Petrov-Galerkin discretization of convection-diffusion equations in the small-viscosity regime, with bounds uniform in mesh width and viscosity up to a logarithmic factor.
We study a numerical method for convection diffusion equations, in the regime of small viscosity. It can be described as an exponentially fitted conforming Petrov-Galerkin method. We identify norms for which we have both continuity and an inf-sup condition, which are uniform in mesh-width and viscosity, up to a logarithm, as long as the viscosity is smaller than the mesh-width or the crosswind diffusion is smaller than the streamline diffusion. The analysis allows for the formation of a boundary layer.