Virtual Element approximations of the Vector Potential Formulation of Magnetostatic problems
SMAI Journal of Computational Mathematics
We consider, as a simple model problem, the application of Virtual Element Methods (VEM) to the linear Magnetostatic three-dimensional problem in the classical Vector Potential formulation. The Vector Potential is treated as a triplet of 0−f orms, approximated by nodal VEM spaces. However this is not done using three classical H 1 -conforming nodal Virtual Elements, and instead we use the Stokes Elements introduced originally in the paper Divergence free Virtual Elements for the Stokes problem
... the Stokes problem on polygonal meshes (ESAIM Math. Model. Numer. Anal. 51 (2017), 509-535) for the treatment of incompressible fluids. L. Beirão da Veiga, F. Brezzi, et al. given j ∈ H(div; Ω) with divj = 0 in Ω, and given µ find H ∈ H(curl; Ω) and B ∈ H(div; Ω) such that: curlH = j and divB = 0, with B = µH in Ω, with the boundary conditions B · n = 0 (or H ∧ n = 0) on ∂Ω. (1.1) Clearly the formulation needs the usual adjustments if Ω is not simply connected (or does not have a simply connected boundary) in order to have uniqueness of the solution, regardless of the numerical method that one has in mind to solve it numerically. We will not deal with these issues here. In some previous papers [12, 13] we dealt with two-dimensional and three-dimensional approximations of the above magneto-static problems using the variational formulation of Kikuchi  . Here, instead, we tackle the discretization of the problem in the (more classical) Vector Potential formulation (see e.g.  and the references therein). Other important contributions to the numerical approximation of Magnetostatic problems can be found, for instance, in [4, 50, 64, 25 ] and the references therein. As far as we know, the vector potential formulation has not yet been tackled with Virtual Elements, and the possible benefits due to the great freedom in the element shapes have not yet been investigated in practice. Here, in particular, we also take advantage from the use of the Virtual Element spaces introduced in  for dealing with Stokes problems (that however are used here in a slightly different way). This choice allows the use (for test and trial functions) of vector-valued fields that have a constant divergence in each element. We think that, together with the generality in the element geometry, this could represent a nice feature (in particular for higher order approximations) when compared to more classical Finite Element formulations. We also point out that here the computed vector potential will have a divergence that is exactly zero. It has to be pointed out from the very beginning that the major interest of applying VEMs (as presented here) to the vector-potential formulation is, in practice, restricted to cases in which the solution is expected to be reasonably smooth, and hence where higher order methods could be more profitable. In particular, they cannot be applied (in the present form) to situations where the computational domain has re-entrant corner, since in that case (see e.g. [47, 30] ) one cannot approximate the solution with vectors belonging to (H 1 ) 3 (as is the case for the VEMs proposed here). The same problem could occur for discontinuous coefficients (see, e.g., [48, 29] ). Needless to say, it would be very interesting to extend to VEMs the tricks that have been developed for FEMs in order to use nodal elements (as for instance in [30, 32, 59, 37, 10] , and the references therein). Similarly, it would also be interesting to extend to VEMs some of the ideas used in FEMs to deal with unbounded domains, as for instance in [24, 31, 63, 58] . All these issues, however, escape the aims of the present paper, where for simplicity we prove error estimates in the case where the solution is regular. For simplicity, we assume that Ω is a convex polyhedron and µ is constant. A layout of the paper is as follows: in Section 2 we will introduce some basic notation, and recall some well known properties of polynomial spaces. Nothing is new there. In Section 3 we will first recall, in Subsection 3.1, the Vector Potential approach to (1.1) and its variational formulation. Then, in Subsections 3.2 and 3.3 we present the local two-dimensional Virtual Element spaces (of nodal type) to be used on the inter-element boundaries. Here we use a simpler (although less powerful) version of the Serendipity spaces of , corresponding, roughly, to the approach that is called lazy choice there. Note that, instead, always with the aim of keeping the presentation as simple as possible, we do not use three-dimensional Serendipity elements to reduce the number of degrees of freedom inside the polyhedrons. Actually, as is well known, in a three-dimensional problem it is more important to reduce the number of degrees of freedom on faces (where static condensation is quite cumbersome to perform), than to reduce the number of degrees of freedom internal to polyhedrons (that can be tackled by static condensation, which is practically done in an almost automatic way by several recent direct solvers).