Distributional Finite Element curl div Complexes and Application to Quad Curl Problems

TL;DR

This work constructs a distributional finite element curl-div complex in 3D by introducing tangential-normal continuity, enabling low-regularity discretizations suitable for the quad-curl problem $-\operatorname{curl}\Delta\operatorname{curl}\boldsymbol{u}=\boldsymbol{f}$ with $\operatorname{div}\boldsymbol{u}=0$ and appropriate boundary conditions. It develops a compatible finite element complex with discrete weak operators, establishes Helmholtz decompositions and discrete stability, and proves optimal-order convergence for a distributional mixed finite element method, along with a post-processing step that yields superconvergent curl-based quantities. The paper also introduces a hybridization strategy that links the method to stabilized weak Galerkin and nonconforming $H(\nabla\,\mathrm{curl})$ elements, and demonstrates equivalences to other discretizations, thereby broadening the practical toolbox for high-order quad-curl problems. Overall, the approach provides a rigorous, flexible framework for discretizing high-order curl-div operators with reduced smoothness while achieving sharp convergence and connections to established nonconforming and weak Galerkin methods, with potential impact on multiphysics simulations requiring quad-curl solvers.

Abstract

The paper addresses the challenge of constructing finite element curl div complexes in three dimensions. Tangential-normal continuity is introduced in order to develop distributional finite element curl div complexes. The spaces constructed are applied to discretize the quad curl problem, demonstrating optimal order of convergence. Furthermore, a hybridization technique is proposed, demonstrating its equivalence to nonconforming finite elements and weak Galerkin methods.

Figures (4)

• Figure 1: The simplest elements $\Sigma_{0,h}^{\rm tn} - \mathring{\mathbb V}_{(1,0), h}^{{\rm curl\,}} - \mathring{\mathbb V}_{1, h}^{{\rm grad\,}}$: the first is a piecewise constant traceless matrix with tangential-normal continuity, the second is the lowest order edge element, and the third is the linear Lagrange element.
• Figure 2: Two intrinsic bases of traceless matrix $\mathbb T$.
• Figure 3: Identities connecting the weak differential operators and interpolation operators.
• Figure 4: An initial perturbed mesh of the uniform mesh with $h=1/2$.

Theorems & Definitions (64)

• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Lemma 3.1
• proof
• Lemma 3.2
• proof