On hp convergence of stabilized finite element methods for the convection–diffusion equation

Abstract

This work analyzes some aspects of the hp convergence of stabilized finite element methods for the convection-diffusion equation when diffusion is small. The methods discussed are classical-residual based stabilization techniques and also projection-based stabilization methods. The theoretical impossibility of obtaining an optimal convergence rate in terms of the polynomial order p for all possible Péclet numbers is explained. The key point turns out to be an inverse estimate that scales as $p^{2}$ . The use of this estimate is not needed in a particular case of ($H^{1}-$)projection-based methods, and therefore the theoretical lack of convergence described does not exist in this case.