A note on the convergence rate of the finite element method for singularly perturbed problems using the Shishkin mesh (Q1399760)

The paper deals with the following singularly perturbed problem \[ -\varepsilon^2 u''(x) + u(x) = f(x), \quad x \in (-1,1), \] \[ u(\pm 1) = 0. \] The author considers the numerical approximation of the above problem by the \(h\) version of the finite element method on a piecewise uniform Shishkin mesh. Lower and upper bounds on the error are obtained and it is shown that the rate is asymptotically sharp.