An $H^2$(curl)-conforming finite element in 2D and its applications to the quad-curl problem

TL;DR

This paper introduces new $H^2$(curl)-conforming finite elements for 2D, constructs a finite element space for quad-curl problems, and demonstrates their convergence properties through theoretical analysis and numerical experiments.

Contribution

The paper develops the first $H^2$(curl)-conforming finite elements on rectangles and triangles and applies them to discretize quad-curl problems with proven convergence rates.

Findings

Finite elements possess unique properties proven by rigorous analysis.

Established convergence orders of $O(h^k)$ in $H$(curl) and $O(h^{k-1})$ in $H^2$(curl).

Numerical experiments confirm theoretical convergence results.

Abstract

In this paper, we first construct the $H^2$(curl)-conforming finite elements both on a rectangle and a triangle. They possess some fascinating properties which have been proven by a rigorous theoretical analysis. Then we apply the elements to construct a finite element space for discretizing quad-curl problems. Convergence orders $O(h^k)$ in the $H$(curl) norm and $O(h^{k-1})$ in the $H^2$(curl) norm are established. Numerical experiments are provided to confirm our theoretical results.