𝑯(curl 2)-Conforming Spectral Element Method for Quad-Curl Problems

Abstract

Abstract 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.