H(curl$^2$)-Conforming Finite Elements in 2 Dimensions and Applications to the Quad-Curl Problem

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. 2000 Mathematics Subject Classification. 65N30 and 35Q60 and 65N15 and 35B45.

Spurious solutions for high order curl problems

“…We consider four finite element methods (FEMs) to solve the problems (2.1) and (2.5): a primal formulation with the H 2 -conforming Argyris element, a mixed formulation with the grad rot-conforming element [18], and the mixed and primal formulations with the H 1 (rot)-conforming element [7]. Let T h be a partition of the domain Ω consisting of shape regular triangles, and let V be the set of vertices.…”

“…According to [2, Theorem 5.4], Scheme 1 is stable if the following discrete Poincaré inequality holds. The discrete Poincaré inequality for V h is due to special structures of the H(grad curl)-conforming elements and the complexes in [12,18], i.e., 1) these elements are subspaces of the Nédélec elements, 2) the 0-forms in the complexes [12] are the Lagrange elements (standard finite element differential forms).…”

A new error analysis of a mixed finite element method for the quad-curl problem

“…A finite element method for the quad-curl problem in two dimensions was studied in [4] based on the Hodge decomposition. Concerning conforming finite element methods, since the curl-curl conforming elements in three dimensions are still unknown (see [31] for curl-curl conforming elements in two dimensions), it would be complicated and far from being obvious (since the conforming elements for the biharmonic problem are quite complicated even in two dimensions, see [9] for example) if the curl-curl conforming elements are considered.…”

A Family of curl-curl Conforming Finite Elements on Tetrahedral Meshes

“…However, the most natural way to solve this problem is the conforming finite element method. In [23], the authors and another collaborator constructed curl-curl conforming or H(curl 2 )-conforming elements in 2 dimensions (2D) to solve the quad-curl problem. In three dimensions (3D), the numerical solution derived by the existing H 2 -conforming (or C 1 -conforming) elements (u ∈ H 1 and ∇u ∈ H 1 ) [25] converges to an H 2 projection of the exact solution.…”