Algorithmic realization of the solution to the sign conflict problem for hanging nodes on hp-hexahedral Nédélec elements

TL;DR

The work tackles the sign-conflict problem that arises when constructing Nédélec $H(\mathrm{curl})$ elements on locally refined hp-meshes with hanging nodes, especially on non-orientable grids. It develops an algorithmic framework that modifies the constraint matrix and cell orientations to enforce tangential continuity across refined/coarse interfaces in 3D hp-hexahedral meshes. By extending high-order Nédélec implementations in deal.II (via the FE_NedelecSZ extension) to non-orientable meshes, the authors classify hanging faces/edges and apply orientation-aware constraint adjustments, covering both 2D and 3D cases and providing pseudo-code. The approach is validated through time-harmonic Maxwell's equations on two numerical experiments, including a simple waveguide and a laser-written waveguide, showing correct behavior and convergence where prior implementations fail, and the code is released as open-source.

Abstract

In this work, Nédélec elements on locally refined meshes with hanging nodes are considered. A crucial aspect is the orientation of the hanging edges and faces. For non-orientable meshes, no solution or implementation has been available to date. The problem statement and corresponding algorithms are described in great detail. As a model problem, the time-harmonic Maxwell's equations are adopted because Nédélec elements constitute their natural discretization. The algorithms and implementation are demonstrated through two numerical examples on different uniformly and adaptively refined meshes. The implementation is performed within the finite element library deal.II.

Figures (12)

• Figure 2.1: The integrated Legendre polynomials $L_2,~L_3,~L_4$ and $L_5$ are depicted. Integrated Legendre polynomials corresponding to even polynomial degrees are symmetric, while those corresponding to odd polynomial degrees are antisymmetric.
• Figure 2.2: Left: Vertex and face ordering of the two-dimensional reference element. Right: Vertex, edge, and face ordering of the three-dimensional reference element.
• Figure 3.1: Most cells have no hanging faces but a hanging edge.
• Figure 3.2: Comparison of grid orientations. The left-hand side shows the grid without special treatment for hanging edges, while the right-hand side shows the grid with special treatment for hanging edges.
• Figure 4.1: Left: Natural coarse-fine mapping, resulting from ignoring the hanging edges. Right: Canonical coarse-fine mapping.