In this paper, we introduce an \(H^1\) -nonconforming vector-valued finite element whose rot has \(H^1\) -conformity. This element yields a nonconforming interior-penalty finite element method for the primal formulation of the quad-curl Hodge-Laplacian problem. Contrasting with conforming methods based on the primal formulation, our method effectively avoids spurious solutions on non-convex polygonal domains. We establish rigorous error estimates for the method in both the energy norm and the \(L^2\) norm, under graded meshes with various grading parameters. Numerical examples are used to verify our theoretical findings.
