Virtual element methods for elliptic problems on polygonal meshes

This chapter establishes a connection between the harmonic generalized barycentric coordinates (GBCs) and the lowest order virtual element method, resulting from the fact that the discrete function space is the same for both methods. This connection allows us to look at the high order virtual element spaces as a further generalization of the harmonic GBC in both two and three spatial dimensions. We also discuss how the virtual element methodology can be used to compute approximate solutions to PDEs without requiring any evaluation of functions in the local discrete spaces, which are implicitly defined through local boundary-value problems.