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.

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 $$n\leq3, m\leq n+1$$. 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^3_h u_h, \nabla^3_h 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. .