Virtual Element Methods for Elliptic Problems on Polygonal Meshes

This chapter establishes a connection between the harmonic generalized barycentric coordinates and the lowest order virtual element method, resulting from the fact that the discrete function space is the same for both methods. It discusses how the virtual element methodology can be used to compute approximate solutions to partial differential equations (PDEs) without requiring any evaluation of functions in the local discrete spaces, which are implicitly defined through local boundary-value problems. The chapter discusses the construction of the local function spaces from the perspective of generalized barycentric coordinates, gives a short overview of the nodal virtual element method for elliptic problems, and outlines the connection with similar methods, like the nodal mimetic finite difference method and the boundary element methods (BEM)-based Finite Element Method (FEM). It summarizes how the local bases (associated with each vertex, edge, or face) can be approximated using the BEM approach and their use within the BEM-based FEM.