  We propose a new analysis of convergence for a $k$th order ($k\ge 1$) finite
element method, which is applied on Bakhvalov-type meshes to a singularly
perturbed two-point boundary value problem. A novel interpolant is introduced,
which has a simple structure and is easy to generalize. By means of this
interpolant, we prove an optimal order of uniform convergence with respect to
the perturbation parameter.
  Numerical experiments illustrate these theoretical results.
