Error estimates for a fully discrete $\varepsilon-$uniform finite element method on 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 the finite element method which 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 mesh to discretize the domain. Further, we determine the 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 using the exact $L-$splines.

Keywords

• singularly perturbed convection-diffusion-reaction boundary-value problem
• finite element method
• boundary layers
• piecewise-uniform Shishkin mesh
• uniform convergence

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