Morley–Wang–Xu element
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In applied mathematics, the Morlely–Wang–Xu (MWX) element[1] 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
[edit]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.
![](http://upload.wikimedia.org/wikipedia/commons/thumb/c/cf/MWX_Table.png/220px-MWX_Table.png)
Generalizations
[edit]There are two generalizations of Morley–Wang–Xu element (which requires ).
: Nonconforming element
[edit]As a nontrivial generalization of Morley–Wang–Xu elements, Wu and Xu propose a universal construction for the more difficult case in which .[2] 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
[edit]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.
![](http://upload.wikimedia.org/wikipedia/commons/thumb/c/cd/MWX_Fig.png/220px-MWX_Fig.png)
.
References
[edit]- ^ Wang, Ming; Xu, Jinchao. "Minimal finite element spaces for 2m-th-order partial differential equations in R^n". Mathematics of Computation. 82 (281): 25–43. doi:10.1090/S0025-5718-2012-02611-1.
- ^ Wu, Shuonan; Xu, Jinchao. "Nonconforming finite element spaces for 2mth order partial differential equations on R^n simplicial grids when m = n + 1". Mathematics of Computation. 88 (316): 531–551. arXiv:1705.10873. doi:10.1090/mcom/3361.