[formula omitted]-conforming triangular spectral element method for quad-curl problems

Abstract

In this paper, we consider the H(curl2)-conforming triangular spectral element method to solve the quad-curl problems. We first explicitly construct the H(curl2)-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(curl2)-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(curl2;Ω)-seminorms. Notably, the degrees of polynomials in the kernel space solely impact the convergence rate of the (L2(Ω))2-norm of uh, without affecting the semi-norm of H(curl;Ω) and H(curl2;Ω). 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.