Error estimates for a fully discrete $\varepsilon-$uniform finite element method on quasi uniform meshes

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, choose $L-$splines as basis functions. We will concentrate the convergence analysis of which employ $L-$spline functions instead their continuous counterparts. The are approximated Shishkin mesh inside each element. These approximations used in frame Galerkin FEM coarse to discretize domain. Further, determine amount error introduced by overall numerical method, and explore possibility recovering order that consistent with classical methods using exact $L-$splines.