Morley–Wang–Xu element explained
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
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
-dimensionalsubsimplex
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.
Generalizations
There are two generalizations of Morley–Wang–Xu element (which requires
).
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
.
[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
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
uh,
vh)+η\sumF\inh}
\intF[uh][vh]=(f,vh) \forallvh\inVh,
where the nonconforming element is depicted in Figure 1. .
Notes and References
- 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 . 10.1090/S0025-5718-2012-02611-1. free .
- 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 . 10.1090/mcom/3361. 1705.10873 .
