Lowest order Virtual Element approximation of magnetostatic problems

TL;DR

This paper presents a simplified lowest order Virtual Element method for 3D magnetostatic problems, generalizing edge finite elements to arbitrary polyhedral meshes with high accuracy and robustness.

Contribution

It introduces a novel lowest order Serendipity Virtual Element method for magnetostatics, extending edge finite elements to complex polyhedral domains.

Findings

Exhibits high accuracy for a lowest order element

Demonstrates robustness against mesh distortions

Applicable to general polyhedral decompositions

Abstract

We give here a simplified presentation of the lowest order Serendipity Virtual Element method, and show its use for the numerical solution of linear magneto-static problems in three dimensions. The method can be applied to very general decompositions of the computational domain (as is natural for Virtual Element Methods) and uses as unknowns the (constant) tangential component of the magnetic field $\mathbf{H}$ on each edge, and the vertex values of the Lagrange multiplier $p$ (used to enforce the solenoidality of the magnetic induction $\mathbf{B}=\mu\mathbf{H}$). In this respect the method can be seen as the natural generalization of the lowest order Edge Finite Element Method (the so-called "first kind N\'ed\'elec" elements) to polyhedra of almost arbitrary shape, and as we show on some numerical examples it exhibits very good accuracy (for being a lowest order element) and excellent robustness with respect to distortions.