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 [math]\displaystyle{ 2m }[/math]-th order of elliptic and parabolic equations in any spatial-dimension [math]\displaystyle{ \mathbb{R}^n }[/math] for [math]\displaystyle{ 1\leq m \leq n }[/math]. The MWX element provides a consistent approximation of Sobolev space [math]\displaystyle{ H^m }[/math] in [math]\displaystyle{ \mathbb{R}^n }[/math].

Morley–Wang–Xu element

The Morley–Wang–Xu element [math]\displaystyle{ (T,P_T,D_T) }[/math] is described as follows. [math]\displaystyle{ T }[/math] is a simplex and [math]\displaystyle{ P_T = P_m(T) }[/math]. The set of degrees of freedom will be given next.

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

For [math]\displaystyle{ 1\leq k\leq m }[/math], any [math]\displaystyle{ (n-k) }[/math]-dimensional subsimplex [math]\displaystyle{ F\in \mathcal{F}_{T,k} }[/math] and [math]\displaystyle{ \beta\in A_k }[/math] with [math]\displaystyle{ |\beta|=m-k }[/math], define

[math]\displaystyle{ 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}}. }[/math]

The degrees of freedom are depicted in Table 1. For [math]\displaystyle{ m=n=1 }[/math], we obtain the well-known conforming linear element. For [math]\displaystyle{ m=1 }[/math] and [math]\displaystyle{ n\geq 2 }[/math], we obtain the well-known nonconforming Crouziex–Raviart element. For [math]\displaystyle{ m=2 }[/math], we recover the well-known Morley element for [math]\displaystyle{ n=2 }[/math] and its generalization to [math]\displaystyle{ n\geq 2 }[/math]. For [math]\displaystyle{ m=n=3 }[/math], 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 [math]\displaystyle{ 1\leq m \leq n }[/math]).

[math]\displaystyle{ m=n+1 }[/math]: 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 [math]\displaystyle{ m=n+1 }[/math]. Table 1 depicts the degrees of freedom for the case that [math]\displaystyle{ n\leq3, m\leq n+1 }[/math]. The shape function space is [math]\displaystyle{ \mathcal{P}_{n+1}(T)+q_T\mathcal{P}_1(T) }[/math], where [math]\displaystyle{ q_T = \lambda_1\lambda_2\cdots\lambda_n+1 }[/math] 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 [math]\displaystyle{ H^3 }[/math] nonconforming finite element that is robust for the sixth order singularly perturbed problems in 2D.

[math]\displaystyle{ m,n \geq 1 }[/math]: Interior penalty nonconforming FEMs

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

[math]\displaystyle{ (\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, }[/math]

where the nonconforming element is depicted in Figure 1.

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

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