In this paper, we propose an H ⁢ ( curl 2 ) \boldsymbol{H}(\mathbf{curl}^{2}) -conforming spectral elements to solve the quad-curl problem on cubic meshes in three dimensions. Starting with generalized vectorial Jacobi polynomials, we first construct the basis functions of the H ⁢ ( curl 2 ) \boldsymbol{H}(\mathbf{curl}^{2}) -conforming spectral elements using the contravariant transform together with the affine mapping from the reference cube onto each physical element. Falling into four categories, interior modes, face modes, edge modes, and vertex modes, these H ⁢ ( curl 2 ) \boldsymbol{H}(\mathbf{curl}^{2}) -conforming basis functions are constructed in an arbitrarily high degree with a hierarchical structure. Next, H ⁢ ( curl 2 ) \boldsymbol{H}(\mathbf{curl}^{2}) -conforming spectral element approximation schemes are established to solve the boundary value problem as well as the eigenvalue problem of quad-curl equations. Numerical experiments demonstrate the effectiveness and efficiency of the ℎ-version and the 𝑝-version of our H ⁢ ( curl 2 ) \boldsymbol{H}(\mathbf{curl}^{2}) -conforming spectral element method.

中文翻译：

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𝑯(curl 2)-四卷曲问题的符合谱元方法

在本文中，我们提出了一个符合 H ⁢ ( curl 2 ) \boldsymbol{H}(\mathbf{curl}^{2}) 的谱元素来解决三维立体网格上的四边形卷曲问题。从广义向量雅可比多项式开始，我们首先构造 H ⁢ ( curl 2 ) \boldsymbol{H}(\mathbf{curl}^{2}) - 使用逆变变换和仿射映射的符合谱元素的基函数从参考立方体到每个物理元素。分为四类，内部模式，面模式，边缘模式和顶点模式，这些 H ⁢ ( curl 2 ) \boldsymbol{H}(\mathbf{curl}^{2}) - 符合基函数是任意构造的等级高，层次分明。下一个，H ⁢ ( curl 2 ) \boldsymbol{H}(\mathbf{curl}^{2}) - 符合谱元近似方案被建立来解决边值问题以及quad-curl方程的特征值问题。数值实验证明了我们的 H ⁢ ( curl 2 ) \boldsymbol{H}(\mathbf{curl}^{2}) 符合谱元方法的 ℎ-version 和 𝑝-version 的有效性和效率。