Fully H(gradcurl)-nonconforming finite element method for the singularly perturbed quad-curl problem on cubical meshes

In this paper, we develop a fully nonconforming (both H(gradcurl)-nonconforming and H(curl)-nonconforming) finite element on cubical meshes that can fit into the Stokes complex. The newly proposed element has 24 degrees of freedom. Unlike the fully H(gradcurl)-nonconforming tetrahedral finite elements in Huang (2023, Nonconforming finite element stokes complexes in three dimensions. Sci. China Math., 66, 1879–1902), the element in this paper yields a robust finite element method to solve the singularly perturbed quad-curl problem. To confirm this, we prove the optimal convergence of order O(h) for a fixed parameter ε and the uniform convergence of order O(h1/2) for any value of ε in the sense of an H(gradcurl)-norm. When ε tends to 0, we employ the Nitsche’s method to improve the convergence order from O(h1/2) to O(h). Some numerical examples are used to verify the validity of the theoretical analysis.