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We explore the recently-proposed Virtual Element Method (VEM) for numericalsolution of boundary value problems on arbitrary polyhedral meshes. Morespecifically, we focus on the elasticity equations in three-dimensions andelaborate upon the key concepts underlying the first-order VEM. While the pointof departure is a conforming Galerkin framework, the distinguishing feature ofVEM is that it does not require an explicit computation of the trial and testspaces, thereby circumventing a barrier to standard finite elementdiscretizations on arbitrary grids. At the heart of the method is a particularkinematic decomposition of element deformation states which, in turn, leads toa corresponding decomposition of strain energy. By capturing the energy oflinear deformations exactly, one can guarantee satisfaction of the engineeringpatch test and optimal convergence of numerical solutions. The decompositionitself is enabled by local projection maps that appropriately extract the rigidbody 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. Inaddition to discussing aspects of implementation of the method, we presentseveral numerical studies in order to verify convergence of the VEM andevaluate its performance for various types of meshes.

On the Virtual Element Method for Three-Dimensional Elasticity Problems on Arbitrary Polyhedral Meshes