Stability of an upwind Petrov Galerkin discretization of convection diffusion equations

TL;DR

This paper analyzes the stability of an exponentially fitted Petrov-Galerkin method for convection-diffusion equations in the small viscosity regime, establishing uniform stability conditions and accommodating boundary layer formation.

Contribution

It introduces a stability analysis with norms that are uniform in mesh-width and viscosity, extending understanding of Petrov-Galerkin methods for convection-diffusion problems.

Findings

Stability norms are uniform in mesh-width and viscosity, up to a logarithm.

The method remains stable even with boundary layer formation.

Conditions depend on the relation between viscosity, mesh-width, and diffusion parameters.

Abstract

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.