  Normalization is a fundamental ring-theoretic operation; geometrically it
resolves singularities in codimension one. Existing algorithmic methods for
computing the normalization rely on a common recipe: successively enlarge the
given ring in form an endomorphism ring of a certain (fractional) ideal until
the process becomes stationary. While Vasconcelos' method uses the dual
Jacobian ideal, Grauert-Remmert-type algorithms rely on so-called test ideals.
  For algebraic varieties, one can apply such normalization algorithms
globally, locally, or formal analytically at all points of the variety. In this
paper, we relate the number of iterations for global Grauert-Remmert-type
normalization algorithms to that of its local descendants.
  We complement our results by an explicit study of ADE singularities. This
includes the description of the normalization process in terms of value
semigroups of curves. It turns out that the intermediate steps produce only ADE
singularities and simple space curve singularities from the list of
Fruehbis-Krueger.
