A note on the convergence rate of the finite element method for singularly perturbed problems using the Shishkin mesh

摘要

We consider the numerical approximation of singularly perturbed problems by the h version of the finite element method on a piecewise uniform, Shishkin mesh. It is well known that this method yields uniform approximations, with respect to the perturbation parameter, at a quasi-optimal algebraic convergence rate. In this note we show that the known rate is asymptotically sharp (as the meshwidth h→0), by obtaining lower and upper bounds on the error. Numerical results that agree with the analysis are presented, as are comparisons with certain other non-uniform mesh strategies.