In this paper, we propose an H(curl2)-conforming quadrilateral spectral element method to solve quad-curl problems. Starting with generalized Jacobi polynomials, we first introduce quasi-orthogonal polynomial systems for vector fields over rectangles. H(curl2)-conforming elements over arbitrary convex quadrilaterals are then constructed explicitly in a hierarchical pattern using the contravariant transform together with the bilinear mapping from the reference square onto each quadrilateral. It is worth noting that both the simplest rectangular and quadrilateral spectral elements possess only 8 degrees of freedom on each physical element. In the sequel, we propose our H(curl2)-conforming quadrilateral spectral element approximation based on the mixed weak formulation to solve the quad-curl equation and its eigenvalue problem. Numerical results show the effectiveness and efficiency of our method.

中文翻译：

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用于四卷曲问题的 H(curl2)-符合四边形谱元法

在本文中，我们提出了一个H(卷曲2)- 符合四边形谱元法来解决四卷曲问题。从广义 Jacobi 多项式开始，我们首先介绍矩形矢量场的准正交多项式系统。H(卷曲2)然后使用逆变变换以及从参考正方形到每个四边形的双线性映射以分层模式显式构造任意凸四边形上的符合元素。值得注意的是，最简单的矩形和四边形光谱元素在每个物理元素上都只有 8 个自由度。在续集中，我们提出了我们的H(卷曲2)-基于混合弱公式的四边形谱元逼近求解四旋曲方程及其特征值问题。数值结果表明了我们方法的有效性和效率。