A family of three-dimensional virtual elements with applications to magnetostatics

We consider, as a simple model problem, the application of Virtual Element Methods (VEM) to the linear Magnetostatic three-dimensional problem in the formulation of F. Kikuchi. In doing so, we also introduce new serendipity VEM spaces, where the serendipity reduction is made only on the faces of a general polyhedral decomposition (assuming that internal degrees of freedom could be more easily eliminated by static condensation). These new spaces are meant, more generally, for the combined approximation of $H^1$-conforming (0-forms), $H({\rm {\bf curl}})$-conforming (1-forms), and $H({\rm div})$-conforming (2-forms) functional spaces in three dimensions, and they could surely be useful for other problems and in more general contexts.