On a general implementation of hh- and pp-adaptive curl-conforming finite elements

Edge (or N\'ed\'elec) finite elements are theoretically sound and widely used by the computational electromagnetics community. However, its implementation, specially for high order methods, is not trivial, since it involves many technicalities that are not properly described in the literature. To fill this gap, we provide a comprehensive description of a general implementation of edge elements of first kind within the scientific software project FEMPAR. We cover into detail how to implement arbitrary order (i.e., $p$-adaptive) elements on hexahedral and tetrahedral meshes. First, we set the three classical ingredients of the finite element definition by Ciarlet, both in the reference and the physical space: cell topologies, polynomial spaces and moments. With these ingredients, shape functions are automatically implemented by defining a judiciously chosen polynomial pre-basis that spans the local finite element space combined with a change of basis to automatically obtain a canonical basis with respect to the moments at hand. Next, we discuss global finite element spaces putting emphasis on the construction of global shape functions through oriented meshes, appropriate geometrical mappings, and equivalence classes of moments, in order to preserve the inter-element continuity of tangential components of the magnetic field. Finally, we extend the proposed methodology to generate global curl-conforming spaces on non-conforming hierarchically refined (i.e., $h$-adaptive) meshes with arbitrary order finite elements. Numerical results include experimental convergence rates to test the proposed implementation

Serendipity Face and Edge VEM Spaces

We extend the basic idea of Serendipity Virtual Elements from the previous case (by the same authors) of nodal ($H^1$-conforming) elements, to a more general framework. Then we apply the general strategy to the case of $H(div)$ and $H(curl)$ conforming Virtual Element Methods, in two and three dimensions

Subdivision Directional Fields

We present a novel linear subdivision scheme for face-based tangent directional fields on triangle meshes. Our subdivision scheme is based on a novel coordinate-free representation of directional fields as halfedge-based scalar quantities, bridging the finite-element representation with discrete exterior calculus. By commuting with differential operators, our subdivision is structure-preserving: it reproduces curl-free fields precisely, and reproduces divergence-free fields in the weak sense. Moreover, our subdivision scheme directly extends to directional fields with several vectors per face by working on the branched covering space. Finally, we demonstrate how our scheme can be applied to directional-field design, advection, and robust earth mover's distance computation, for efficient and robust computation