We consider the interaction of multivalent counterions with spherical polyelectrolyte brushes (SPBs). SPBs result if linear polyelectrolyte chains (contour length 60 nm) are densely grafted to colloidal spheres of 116 nm in diameter. When dispersed in water the surface layer, consisting of chains of the strong polyelectrolyte poly(styrene sulfonic acid), will swell. Recent work [Mei, Phys. Rev. Lett. 97, 158301 (2006)] has demonstrated that spherical polyelectrolyte brushes undergo a collapse in the presence of a mixture of monovalent and multivalent counterions. The collapse crossover could be well described by a mean-field approach. Here we demonstrate that the application of a mean-field approach is well founded by simulation results done with molecular dynamics (MD). MD simulations show that over a wide range of multivalent counterion concentration the effects of ion correlation and fluctuations can be neglected. Higher-valent counterions are shown to interact strongly with the polyelectrolyte chains of the SPBs and thus exhibit a much reduced osmotic activity in the system. This reduction is the driving force for the collapse.