Abstract The chapter presents Virtual Element Methods for linear elasticity. In particular, displacement based methods stemming from the principle of virtual works for 2D problems, and mixed methods based on the Hellinger-Reissner variational principle for 2D and 3D problems are discussed and detailed. A series of numerical examples for each set of methods are given in order to show the characteristic features of the newly developed methods and to assess their accuracy and convergence properties.

Abstract We present the details of the implementation of the Virtual Element Method (VEM) for nonlinear elliptic and parabolic problems, over polygonal meshes. The VEM implementation of a nonlinear problem does not follow the traditional way of implementation of the nonlinear problems as done in finite element/finite volume or finite difference methods. To facilitate this, suitable projection operators are used for the nonlinear terms that are rewritten in terms of the degrees of freedom associated with the virtual element space and thus are computable. We explicitly present the discretization and include sample codes that will help readers with the basic implementation of the nonlinear problems.