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In this paper, we construct some H(curl 2 )-conforming finite elements on a rectangle (a parallelogram) and a triangle. The proposed elements possess some nice properties which have been proved by a rigorous theoretical analysis. Then we apply our new elements to construct a finite element space for discretizing the quad-curl problem. Convergence orders O(h k ) in the H(curl) norm and O(h k−1 ) in the H(curl 2 ) norm are established. Numerical experiments are provided to confirm the theoretical results.doi:10.1137/18m1199988 fatcat:fgomzfvg6fcpjasd5egtmgvski