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In this paper, we propose an $H(\text{curl}^2)$-conforming quadrilateralspectral element method to solve quad-curl problems. Starting with generalizedJacobi polynomials, we first introduce quasi-orthogonal polynomial systems forvector fields over rectangles. $H(\text{curl}^2)$-conforming elements overarbitrary convex quadrilaterals are then constructed explicitly in ahierarchical pattern using the contravariant transform together with thebilinear mapping from the reference square onto each quadrilateral. It isastonishing that both the simplest rectangular and quadrilateral spectralelements possess only 8 degrees of freedom on each physical element. In thesequel, we propose our $H(\text{curl}^2)$-conforming quadrilateral spectralelement approximation based on the mixed weak formulation to solve thequad-curl equation and its eigenvalue problem. Numerical results show theeffectiveness and efficiency of our method.

H($\text{curl}^2$)-conforming quadrilateral spectral element method for quad-curl problems