Numerical Approximation of two dimensional Singularly Perturbed Problems

Abstract

We demonstrate how one can improve the numerical solution of singularly perturbed problems involving multiple boundary layers by using a combination of analytic and numerical tools. Incorporating the so-called boundary layer elements (BLE), which absorb the singularities due to the boundary layers, into finite element spaces can improve the accuracy of approximate solutions and result in significant simplifications. We discuss here convection-diffusion equations in the case where both ordinary and parabolic boundary layers are present. We also revise the BLE 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 H^1- approximation error analysis. Furthermore due to the compact structure of the BLE we are able to prove the L^2- stability analysis of the scheme and derive the L^2- error approximations.