For a \(Q_k\) polynomial, its normal derivatives on the element boundaries are still \(P_k\) ( \(Q_k\) in three dimensions) polynomials. For a Bell \(Q_k\) finite element function, its normal derivatives on the element boundaries are \(P_{k-1}\) ( \(Q_{k-1}\) in three dimensions) polynomials. We construct a Bell \(C^1\) - \(Q_k\) finite element on rectangular meshes in two dimensions and three dimensions for \(k\geqslant 4\) . We show, with a big reduction from the standard Bogner-Fox-Schmit (BFS) \(C^1\) - \(Q_k\) finite element, the \(C^1\) - \(Q_k\) Bell finite element retains the optimal order of convergence. Numerical experiments are performed, comparing the new elements with the original elements.
