Morley–Wang–Xu element
In applied mathematics, the Morlely–Wang–Xu (MWX) element is a canonical construction of a family of piecewise polynomials with the minimal degree elements for any -th order of elliptic and parabolic equations in any spatial-dimension  for . The MWX element provides a consistent approximation of Sobolev space  in . Morley–Wang–Xu element
The Morley–Wang–Xu element  is described as follows. is a simplex and . The set of degrees of freedom will be given next. Given an -simplex  with vertices , for , let  be the set consisting of all -dimensional subsimplexe of . For any , let  denote its measure, and let  be its unit outer normals which are linearly independent. For , any -dimensional
subsimplex  and  with , define 

 

The degrees of freedom are depicted in Table 1. For , we obtain the well-known conforming linear element. For  and , we obtain the well-known nonconforming Crouziex–Raviart element. For   , we recover the well-known Morley element for  and its generalization to . For , we obtain a new cubic element on a simplex that has 20 degrees of freedom. Generalizations
There are two generalizations of Morley–Wang–Xu element (which requires ). : Nonconforming element
As a nontrivial generalization of Morley–Wang–Xu elements, Wu and Xu propose a universal construction for the more difficult case in which . Table 1 depicts the degrees of freedom for the case that . The shape function space is , where  is volume bubble function. This new family of finite element methods provides practical discretization methods for, say, a sixth order elliptic equations in 2D (which only has 12 local degrees of freedom). In addition, Wu and Xu propose an  nonconforming finite element that is robust for the sixth order singularly perturbed problems in 2D. : Interior penalty nonconforming FEMs
An alternative generalization when  is developed by combining the interior penalty and nonconforming methods by Wu and Xu. This family of finite element space consists of piecewise polynomials of degree not greater than . The degrees of freedom are carefully designed to preserve the weak-continuity as much as possible. For the case in which , the corresponding interior penalty terms are applied to obtain the convergence property. As a simple example, the proposed method for the case in which  is to find , such that

 

where the nonconforming element is depicted in Figure 1. . References 

Polynomials