  We present a study of the fully relativistic spherical collapse in presence
of quintessence using on Numerical Relativity, following the method proposed by
the authors in a previous article [arXiv:1409.3476]. We ascertain the validity
of the method by studying the evolution of a spherically symmetric quintessence
inhomogeneity on a de Sitter background and we find that it has an impact on
the local expansion around the centre of coordinates. We then proceed to
compare the results of our method to those of the more largely adopted top-hat
model. We find that quintessence inhomogeneities do build up under the effect
that matter inhomogeneities have on the local space-time yet remain very small
due to the presence of momentum transfer from the over-dense to the background
regions. We expect that these might have an even more important role in
modified theories of gravitation.
