  Mimetic methods discretize divergence by restricting the Gauss theorem to
mesh cells. Because point clouds lack such geometric entities, construction of
a compatible meshfree divergence remains a challenge. In this work, we define
an abstract Meshfree Mimetic Divergence (MMD) operator on point clouds by
contraction of field and virtual face moments. This MMD satisfies a discrete
divergence theorem, provides a discrete local conservation principle, and is
first-order accurate. We consider two MMD instantiations. The first one assumes
a background mesh and uses generalized moving least squares (GMLS) to obtain
the necessary field and face moments. This MMD instance is appropriate for
settings where a mesh is available but its quality is insufficient for a robust
and accurate mesh-based discretization. The second MMD operator retains the
GMLS field moments but defines virtual face moments using computationally
efficient weighted graph-Laplacian equations. This MMD instance does not
require a background grid and is appropriate for applications where mesh
generation creates a computational bottleneck. It allows one to trade an
expensive mesh generation problem for a scalable algebraic one, without
sacrificing compatibility with the divergence operator. We demonstrate the
approach by using the MMD operator to obtain a virtual finite-volume
discretization of conservation laws on point clouds. Numerical results in the
paper confirm the mimetic properties of the method and show that it behaves
similarly to standard finite volume methods.
