A quadrilateral Morley element for biharmonic equations

In this paper, we propose a Morley-type finite element for quadrilateral meshes to solve biharmonic problems. For each quadrilateral $Q$ , the finite element space is defined by the span of $P_2(Q)$ plus two functions in $P_3(Q)$ . Each of the cubic polynomials is the product of a pair of equations of opposite edges and the equation of the bimedian between them. The degrees of freedom consist of the values at vertices and integrals of normal derivatives over edges. Optimal orders of convergence are proved both in discrete $H^2$ and $H^1$ seminorms. Several numerical tests confirm the convergence analysis.