An $H^2$(curl)-conforming finite element in 2D and its applications to the quad-curl problem
Provides a new finite element construction for solving quad-curl problems, which are important in electromagnetics and fluid dynamics, but the work is incremental as it extends existing conforming element theory.
This paper constructs H²(curl)-conforming finite elements on rectangles and triangles and applies them to discretize quad-curl problems, achieving convergence orders O(h^k) in the H(curl) norm and O(h^{k-1}) in the H²(curl) norm, confirmed by numerical experiments.
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.