We consider a linear convection diffusion equation in two space dimensions and propose to stabilise the Galerkin method with a fourth-order artificial dissipation. We then analyse the effect of this dissipation on the accuracy of the finite element method. We focus our study on stretched meshes in the boundary layers. The main result is that we can choose an appropriate local metric to control the dissipation in stretched regions of the mesh. This metric is defined by a local coordinate system and dissipation coefficients. We compare in this coordinate system, the dissipation distribution of the present finite element scheme with the one obtained for some classical second order finite volume schemes.