Lowest order Virtual Element approximation of magnetostatic problems

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 Nédélec" 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.

References in corpus (4)

• Approximation by quadrilateral finite elements
• H(div) and H(curl)-conforming VEM
• Serendipity Face and Edge VEM Spaces
• A virtual element method for the acoustic vibration problem

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• Edge-averaged virtual element methods for convection-diffusion and convection-dominated problems
• A family of three-dimensional virtual elements with applications to magnetostatic