Fully H(gradcurl)-nonconforming Finite Element Method for The Singularly Perturbed Quad-curl Problem on Cubical Meshes

Abstract

In this paper, we develop two fully nonconforming (both H(grad curl)-nonconforming and H(curl)-nonconforming) finite elements on cubical meshes which can fit into the Stokes complex. The newly proposed elements have 24 and 36 degrees of freedom, respectively. Different from the fully H(grad curl)-nonconforming tetrahedral finite elements in [9], the elements in this paper lead to 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 \(\epsilon\) and the uniform convergence of order \(O(h^{1/2})\) for any value of \(\epsilon\). Some numerical examples are used to verify the correctness of the theoretical analysis.