Partially discontinuous nodal finite elements for $H(\mathrm{curl})$ and $H(\mathrm{div})$

TL;DR

This paper introduces a new class of partially discontinuous nodal finite elements for discretizing $H(\mathrm{curl})$ and $H(\mathrm{div})$ spaces, combining continuous and discontinuous elements within de Rham complexes.

Contribution

It develops a novel construction of well-conditioned nodal bases for these finite elements, enhancing their implementation and stability.

Findings

Spaces can be implemented as a combination of continuous and discontinuous elements

Constructed well-conditioned nodal bases for these elements

Fitted within de Rham complexes for compatibility

Abstract

We investigate discretization of $H(\mathrm{curl})$ and $H(\mathrm{div})$ in two and three space dimensions by partially discontinuous nodal finite elements, i.e., vector-valued Lagrange finite elements with discontinuity in certain directions. These spaces can be implemented as a combination of continuous and discontinuous Lagrange elements and fit in de~Rham complexes. We construct well-conditioned nodal bases.