On a general implementation of $h$- and $p$-adaptive curl-conforming finite elements

TL;DR

This paper provides a comprehensive methodology for implementing high-order, $p$-adaptive curl-conforming finite elements, specifically Nédélec edge elements, on complex meshes within the FEMPAR software, addressing technical challenges and demonstrating convergence.

Contribution

It introduces a detailed, general implementation approach for $p$-adaptive edge elements on hexahedral and tetrahedral meshes, filling a gap in the literature.

Findings

Successful implementation of arbitrary order edge elements

Demonstrated convergence rates through numerical experiments

Extended methodology to non-conforming hierarchically refined meshes

Abstract

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.