A family of three-dimensional virtual elements with applications to magnetostatic
For computational scientists using virtual element methods, this work offers a more efficient discretization for 3D magnetostatics and related problems, though it is incremental in nature.
The paper introduces new serendipity Virtual Element spaces for approximating H^1-, H(curl)-, and H(div)-conforming spaces in 3D, applied to a linear magnetostatic problem. The method reduces degrees of freedom on faces, enabling efficient static condensation.
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 would surely be useful for other problems and in more general contexts.