We present a one-dimensional version of a general mesh adaptation technique developed in [1, 2] which is valid for two and three-dimensional problems. The simplicity of the
one-dimensional case allows to detail all the necessary steps with very simple computations. We
show how the error can be estimated on a piecewise finite element of degree k and how this information
can be used to modify the grid using local mesh operations: element division, node
elimination and node displacement. Finally, we apply the whole strategy to many challenging
singularly perturbed boundary value problems where the one-dimensional setting allows to push
the adaptation method to its limits.