  In the context of intra-cluster medium turbulence, it is essential to be able
to split the turbulent velocity field in a compressive and a solenoidal
component. We describe and implement a new method for this aim, i.e.,
performing a Helmholtz-Hodge decomposition, in multi-grid, multi-resolution
descriptions, focusing on (but not being restricted to) the outputs of AMR
cosmological simulations. The method is based on solving elliptic equations for
a scalar and a vector potential, from which the compressive and the solenoidal
velocity fields, respectively, are derived through differentiation. These
equations are addressed using a combination of Fourier (for the base grid) and
iterative (for the refinement grids) methods. We present several idealised
tests for our implementation, reporting typical median errors in the order of
$1\unicode{x2030}$-$1\%$, and with 95-percentile errors below a few percents.
Additionally, we also apply the code to the outcomes of a cosmological
simulation, achieving similar accuracy at all resolutions, even in the case of
highly non-linear velocity fields. We finally take a closer look to the
decomposition of the velocity field around a massive galaxy cluster.
