  We explore the recently-proposed Virtual Element Method (VEM) for numerical
solution of boundary value problems on arbitrary polyhedral meshes. More
specifically, we focus on the elasticity equations in three-dimensions and
elaborate upon the key concepts underlying the first-order VEM. While the point
of departure is a conforming Galerkin framework, the distinguishing feature of
VEM is that it does not require an explicit computation of the trial and test
spaces, thereby circumventing a barrier to standard finite element
discretizations on arbitrary grids. At the heart of the method is a particular
kinematic decomposition of element deformation states which, in turn, leads to
a corresponding decomposition of strain energy. By capturing the energy of
linear deformations exactly, one can guarantee satisfaction of the engineering
patch test and optimal convergence of numerical solutions. The decomposition
itself is enabled by local projection maps that appropriately extract the rigid
body motion and constant strain components of the deformation. As we show,
computing these projection maps and subsequently the local stiffness matrices,
in practice, reduces to the computation of purely geometric quantities. In
addition to discussing aspects of implementation of the method, we present
several numerical studies in order to verify convergence of the VEM and
evaluate its performance for various types of meshes.
