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 $\mathbb {R} ^{n}$ for $1\leq m\leq n$. The MWX element provides a consistent approximation of Sobolev space $H^{m}$ in $\mathbb {R} ^{n}$.

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

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

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

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 $n\geq 2$, 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 $n\geq 2$. 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 $1\leq m\leq n$).

$m=n+1$: 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 $n\leq 3,m\leq n+1$. The shape function space is ${\mathcal {P}}_{n+1}(T)+q_{T}{\mathcal {P}}_{1}(T)$, 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.

$m,n\geq 1$: 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

$(\nabla _{h}^{3}u_{h},\nabla _{h}^{3}v_{h})+\eta \sum _{F\in {\mathcal {F}}_{h}}h_{F}^{-5}\int _{F}[u_{h}][v_{h}]=(f,v_{h})\quad \forall v_{h}\in V_{h},$

where the nonconforming element is depicted in Figure 1.

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

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