In our recent work (Hu et al. in SIAM J Sci Comput 42(6):A3859–A3877, 2020), we observed numerically some superconvergence phenomena of the curlcurl-conforming finite elements on rectangular domains. In this paper, we provide a theoretical justification for our numerical observation and establish a superconvergence theory for the curlcurl-conforming elements on rectangular meshes. For the elements with parameters r (r=k-1\documentclass[12pt]{minimal}
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\begin{document}$$r=k-1$$\end{document}, k, k+1\documentclass[12pt]{minimal}
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\begin{document}$$k+1$$\end{document}) and k (k≥2\documentclass[12pt]{minimal}
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\begin{document}$$k\ge 2$$\end{document}), we show that the first (second) component of the numerical solution uh\documentclass[12pt]{minimal}
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\begin{document}$$\varvec{u}_h$$\end{document} converges with rate r+1\documentclass[12pt]{minimal}
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\begin{document}$$r+1$$\end{document} at r vertical (horizontal) Gaussian lines in each element when r=k-1,k\documentclass[12pt]{minimal}
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\begin{document}$$r=k-1,k$$\end{document} with k≥3\documentclass[12pt]{minimal}
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\begin{document}$$k\ge 3$$\end{document}, ∇×uh\documentclass[12pt]{minimal}
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\begin{document}$$\nabla \times \varvec{u}_h$$\end{document} converges with rate k+1\documentclass[12pt]{minimal}
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\begin{document}$$k^2$$\end{document} Lobatto points in each element when k≥3\documentclass[12pt]{minimal}
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\begin{document}$$k\ge 3$$\end{document}, and the first (second) component of ∇×∇×uh\documentclass[12pt]{minimal}
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\begin{document}$$\nabla \times \nabla \times \varvec{u}_h$$\end{document} converges with rate k at (k-1)\documentclass[12pt]{minimal}
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\begin{document}$$(k-1)$$\end{document} horizontal (vertical) Gaussian lines when k≥2\documentclass[12pt]{minimal}
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\begin{document}$$k\ge 2$$\end{document}. They are all one-order higher than the related optimal rates. More numerical experiments are provided to confirm our theoretical results.
