Virtual Elements for Magnetostatic Problems
by Lourenco Beirao da Veiga (University of Milano-Bicocca, Italy)
The Virtual Element Method (VEM) is a recent technology for the discretization of partial differential equations that follows a similar paradigm to standard Finite Elements, but with important differences. By avoiding the explicit integration of the shape functions that span the discrete space and introducing an innovative construction of the stiffness matrixes, the VEM acquires very interesting properties and advantages with respect to more standard methods. For instance, the VEM easily allows for general polygonal/polyhedral meshes, even non-conforming and with non-convex elements.
In the present talk we investigate the development of some Virtual Element families for a classical magnetostatic problem in two and three dimensions, that can be considered as a first step towards more complex applications. We start by introducing a first set of Virtual Spaces that constitute a complex (both in 2D and 3D) and detail its application for the discretization of the problem. Afterwards, we also present a modified set of spaces that are more efficient in terms of degrees of freedom, and still constitute a complex. We support the theoretical analysis of the method with a set of numerical tests.