H(curl2)-conforming triangular spectral element method for quad-curl problems

Abstract

In this paper, we consider the $H\left({\text{curl}}^2\right)$-conforming triangular spectral element method to solve the quad-curl problems. We first explicitly construct the $H\left({\text{curl}}^2\right)$-conforming elements on triangles through the contravariant transform and the affine mapping from the reference element to physical elements. These constructed elements possess a hierarchical structure and can be categorized into the kernel space and non-kernel space of the curl operator. We then establish $H\left({\text{curl}}^2\right)$-conforming triangular spectral element spaces and the corresponding mixed formulated spectral element approximation scheme for the quad-curl problems and related eigenvalue problems. Subsequently, we present the best spectral element approximation theory in $H({\text{curl}}^2;\Omega )$-seminorms. Notably, the degrees of polynomials in the kernel space solely impact the convergence rate of the ${\left(L^2\left(\Omega \right)\right)}^2$-norm of $u_h$, without affecting the semi-norm of $H(\text{curl};\Omega )$ and $H({\text{curl}}^2;\Omega )$. This observation enables us to derive eigenvalue approximations from either the upper or lower side by selecting different degrees of polynomials for the kernel space and non-kernel space of the curl operator. Finally, numerical results demonstrate the effectiveness and efficiency of our method.