Nonconforming finite element methods on quadrilateral meshes

It is well known that it is comparatively difcult to design nonconforming fnite elements on quadrilateral meshes by using Gauss-Legendre points on each edge of triangulations.One reason lies in that these degrees of freedom associated with these Gauss-Legendre points are not all linearly independent for usual expected polynomial spaces,which explains why only several lower order nonconforming quadrilateral fnite elements can be found in literature.The present paper proposes two families of nonconforming fnite elements of any odd order and one family of nonconforming fnite elements of any even order on quadrilateral meshes.Degrees of freedom are given for these elements,which are proved to be well-defned for their corresponding shape function spaces in a unifying way.These elements generalize three lower order nonconforming fnite elements on quadrilaterals to any order.In addition,these nonconforming fnite element spaces are shown to be full spaces which is somehow not discussed for nonconforming fnite elements in literature before.