Annotations on the virtual element method for second-order elliptic problems

Abstract

stiffness matrix is rank deficient). In the VEM we fix this issue by adding a stabilization term to the bilinear form. Stabilization terms are required not to upset the exactness property for polynomials and to scale properly with respect to the mesh size and the problem coefficients.1.4.1. The low-order setting for polygonal cells (in 2D) and polyhedral cells (in 3D) can be obtained as a straightforward generalization of the FEM. The high-order setting deserves special care in the treatment of variable coefficients in order not to loose the optimality of the discretization and in the 3D formulation.1.5. The shape functions are virtual in the sense that we never compute them explicitly (not even approximately) inside the elements. Note that the polynomial projection of the approximate solution is readily available from the degrees of freedom and we can use it as the numerical solution.1.6. The major advantage offered by the VEM is in the great flexibility in admitting unstructured polygonal and polyhedral meshes.1.7. The VEM can be seen as an evolution of the MFD method. Background material (and a few historical notes) will be given in the final section of this document.