Finite Element Analysis of an Exponentially Graded Mesh for Singularly Perturbed Problems

We present the mathematical analysis for the convergence of an h version Finite Element Method (FEM) with piecewise polynomials of degree p >= 1, defined on an exponentially graded mesh. The analysis is presented for a singularly perturbed reaction-diffusion and a convection-diffusion equation in one dimension. We prove convergence of optimal order and independent of the singular perturbation parameter, when the error is measured in the natural energy norm associated with each problem. Numerical results comparing the exponential mesh with the Bakhvalov-Shishkin mesh from the literature are also presented.