  The Hodge decomposition provides a very powerful mathematical method for the
analysis of 2D and 3D vector fields. It states roughly that any vector field
can be $L^2$-orthogonally decomposed into a curl-free, divergence-free, and a
harmonic field. The harmonic field itself can be further decomposed into three
components, two of which are closely tied to the topology of the underlying
domain. For practical computations it is desirable to find a discretization
which preserves as many aspects inherent to the smooth theory as possible while
at the same time remains computationally tractable, in particular on
large-sized models. The correctness and convergence of such a discretization
depends strongly on the choice of ansatz spaces defined on the surface or
volumetric mesh to approximate infinite dimensional subspaces. This paper
presents a consistent discretization of Hodge-type decomposition for piecewise
constant vector fields on volumetric meshes. Our approach is based on a careful
interplay between edge-based \textNedelec elements and face-based
Crouzeix-Raviart elements resulting in a very simple formulation. The method is
stable under noisy vector field and mesh resolution, and has a good performance
for large sized models. We give pseudocodes for a possible implementation of
the method together with some insights on how the Hodge decomposition could
answer some central question in computational fluid.
