Error estimates for a fully discrete $varepsilon-$uniform finite element methodon quasi uniform meshes

Abstract

In this article, we analyze a fully discrete $varepsilon-$uniformly convergent finite element method for singularly perturbed convection-diffusion-reaction boundary-value problems, on piecewise-uniform meshes. Here, we choose $L-$splines as basis functions.We will concentrate on the convergence analysis of thefinite element methodwhich employ the discrete$L-$spline basis functions instead of their continuous counterparts.The $L-$splines are approximated on the piecewise-uniform Shishkin mesh inside each element. These approximations are used as basis functions in the frame of Galerkin FEM on a coarse piecewise-uniform meshto discretize the domain. Further, we determinethe amount of error introduced by the discrete $L-$spline basis functions in the overall numerical method, and explore the possibility of recovering the order of convergence that are consistent with the classical order of convergence for the numerical methods usingthe exact $L-$splines.