  In this article we prove that it is possible to construct, using
newest-vertex bisection, meshes that equidistribute the error in $H^1$-norm,
whenever the function to approximate can be decomposed as a sum of a regular
part plus a singular part with singularities around a finite number of points.
This decomposition is usual in regularity results of Partial Differential
Equations (PDE). As a consequence, the meshes turn out to be quasi-optimal, and
convergence rates for adaptive finite element methods (AFEM) using Lagrange
finite elements of any polynomial degree are obtained.
