Construction of boundary layer elements for singularly perturbed convection-diffusion equations and L(2)-stability analysis

Abstract

It has been demonstrated that the ordinary boundary layer elements play an essential role in the finite element approximations for singularly perturbed problems producing ordinary boundary layers. Here we revise the element so that it has a small compact support and hence the resulting linear system becomes sparse, more precisely, block tridiagonal. We prove the validity of the revised element for some singularly perturbed convection-diffusion equations via numerical simulations and via the H1-approximation error analysis. Furthermore due to the compact structure of the boundary layer we are able to prove the L2-stability analysis of the scheme and derive the L2-error approximations.