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

TL;DR

This paper develops a family of three-dimensional virtual element methods, including new serendipity spaces, for magnetostatic problems, enabling efficient approximation of various conforming functional spaces in 3D.

Contribution

Introduction of new serendipity VEM spaces for 3D magnetostatic problems, facilitating combined approximation of multiple conforming spaces with potential broader applications.

Findings

New serendipity VEM spaces on polyhedral decompositions

Application to 3D magnetostatic problems

Potential for use in other conforming space approximations

Abstract

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.