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 H on each edge, and the vertex values of the Lagrange multiplier p (used to enforce the solenoidality of the magnetic induction B = µH). 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.

The Virtual Element Method with curved edges

“…and we have therefore the degrees of freedom: (11) -the values at each vertex of P, -(for k ≥ 2) the values at the k − 1 Gauss-Lobatto points of each edge of P, -(for k ≥ 2) the moments of order ≤ k − 2 inside P Remark 1. In previous works we used the bigger spaces (corresponding to k ∆ = k − 1 or k ∆ = k) as starting point for a Serendipity correction (see [12]) that allowed to end up with local VEM spaces smaller than the V k defined in (10).…”

Bricks for the mixed high-order virtual element method: Projectors and differential operators

“…Notice that, according to (4), the gradient component of the decomposition has strictly positive degree. Decomposition (5) comes from a proper linear combination of (7) and (8).…”

“…The Virtual Element Method has been developed successfully for a large range of mathematical and engineering problems, we mention, as sample, the very brief list of papers [10,28,32,18,46,6,33,7], while for the specific topic of implementation aspects related to the VEM we mention [43,25,14,24,4,8,42,35,45,19]. Concernig the mixed PDEs we refer to [41,30,29,37,31] as a sample of VEM papers dealing with such kind of problem, and to [40,34,26,36] as a representative list of papers treating the same topic with different polytopal technologies.…”

A C1 Virtual Element Method on polyhedral meshes

“…Another strategy to get a conforming discrete approximation space in H 2 is to follow the recently born Virtual Element Method (VEM). The VEM is a novel generalization of the finite element method, introduced in [12,8], that allows to use general polygonal/polyhedral meshes and which has been already succesfully applied to a large number of problems (a very brief list being [2,20,38,37,45,10,19,31,15,57,21,24,29,30,52,36,17,32,50,13,6,5,27]). The Virtual Element Method is not restricted to piecewise polynomials but avoids nevertheless the explicit integration of non-polynomial shape functions by a wise choice of the degrees of freedom and an innovative construction of the stiffness matrix.…”