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 $2m$-th order of elliptic and parabolic equations in any spatial-dimension ${ℝ}^n$ for $1\le m\le n$. The MWX element provides a consistent approximation of Sobolev space $H^m$ in ${ℝ}^n$.

Morley–Wang–Xu element

The Morley–Wang–Xu element $(T,P_T,D_T)$ is described as follows. $T$ is a simplex and $P_T=P_m(T)$. The set of degrees of freedom will be given next.

Given an $n$-simplex $T$ with vertices $a_i$, for $1\le k\le n$, let ${ℱ}_{T,k}$ be the set consisting of all $(n-k)$-dimensional subsimplexe of $T$. For any $F\in {ℱ}_{T,k}$, let $|F|$ denote its measure, and let ${\nu }_{F,1},\cdots ,{\nu }_{F,k}$ be its unit outer normals which are linearly independent.

For $1\le k\le m$, any $(n-k)$-dimensional subsimplex $F\in {ℱ}_{T,k}$ and $\beta \in A_k$ with $|\beta |=m-k$, define

$d_{T,F,\beta }(v)=\frac{1}{|F|}\underset{F}{\int }\frac{{\partial }^{|\beta |v}}{\partial {\nu }_{F,1}^{{\beta }_1}\cdots {\nu }_{F,k}^{{\beta }_k}}.$

The degrees of freedom are depicted in Table 1. For $m=n=1$, we obtain the well-known conforming linear element. For $m=1$ and $n\ge 2$, we obtain the well-known nonconforming Crouziex–Raviart element. For $m=2$, we recover the well-known Morley element for $n=2$ and its generalization to $n\ge 2$. For $m=n=3$, we obtain a new cubic element on a simplex that has 20 degrees of freedom.

Table 1: m <= n+1: diagrams of the finite elements

Generalizations

There are two generalizations of Morley–Wang–Xu element (which requires $1\le m\le n$).

$m=n+1$: 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 $m=n+1$. Table 1 depicts the degrees of freedom for the case that $n\le 3,m\le n+1$. The shape function space is ${𝒫}_{n+1}(T)+q_T{𝒫}_1(T)$, where $q_T={\lambda }_1{\lambda }_2\cdots {\lambda }_n+1$ 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 $H^3$ nonconforming finite element that is robust for the sixth order singularly perturbed problems in 2D.

$m,n\ge 1$: Interior penalty nonconforming FEMs

An alternative generalization when $m>n$ 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 $m$. The degrees of freedom are carefully designed to preserve the weak-continuity as much as possible. For the case in which $m>n$, the corresponding interior penalty terms are applied to obtain the convergence property. As a simple example, the proposed method for the case in which $m=3,n=2$ is to find $u_h\in V_h$, such that

$(\underset{h}{\overset{3}{\mathrm{\nabla }}}{u}_h,\underset{h}{\overset{3}{\mathrm{\nabla }}}{v}_h)+\eta \sum _{F\in {ℱ}_h}{h}_F^{-5}\underset{F}{\int }[u_h][v_h]=(f,v_h)\mathrm{\forall }{v}_h\in V_h,$

where the nonconforming element is depicted in Figure 1.

Figure 1: m,n <= 1: The nonconforming element

.