The Morley–Wang–Xu (MWX) element is a nonconforming finite element used in the numerical approximation of fourth‑order partial differential equations, such as the biharmonic equation that models plate bending. It builds upon the classical Morley element, a triangular nonconforming element introduced by M. Morley in the 1960s, by incorporating modifications proposed by J. Wang and J. Xu to improve approximation properties and to simplify implementation.
Key characteristics
| Feature | Description |
|---|---|
| Element type | Nonconforming triangular element |
| Degrees of freedom | Function values at the vertices and normal derivatives at the midpoints of edges (as in the original Morley element), possibly supplemented by additional moments depending on the specific MWX formulation |
| Continuity | The element functions are continuous at the vertices but may be discontinuous across element edges; only certain integral constraints are enforced to achieve weak continuity |
| Target problems | Fourth‑order elliptic problems (e.g., biharmonic equation, Kirchhoff plate models) and, in some extensions, related eigenvalue problems |
| Advantages | Retains the low number of degrees of freedom of the Morley element while providing enhanced convergence rates and stability for certain problem classes; facilitates the use of standard assembly techniques due to its local definition |
Historical development
- Morley element (1968) – Introduced by M. Morley as a simple nonconforming element for the biharmonic equation, employing only vertex values and edge normal derivative averages as degrees of freedom.
- Wang–Xu contributions (2000s–2010s) – J. Wang and J. Xu investigated extensions and refinements of nonconforming elements, focusing on improved error estimates and the construction of stable mixed formulations.
- Morley–Wang‑Xu element (mid‑2010s) – Combining the simplicity of the Morley element with the analytical techniques of Wang and Xu resulted in the MWX element, which has been analyzed in several research articles for its approximation properties and for applications to plate bending and related fourth‑order problems.
Typical formulation
On a triangular element $K$, the MWX shape functions belong to the polynomial space $P_2(K)$ (quadratic polynomials). The degrees of freedom are:
- The function values $v(a_i)$ at the three vertices $a_i$ of $K$.
- The average normal derivative $\frac{1}{|e|}\int_e \frac{\partial v}{\partial n}, ds$ on each edge $e$ of $K$.
These conditions define a unisolvent set for the polynomial space, ensuring that each set of prescribed degrees of freedom corresponds to a unique shape function on the element.
Mathematical properties
- Convergence – Under standard regularity assumptions on the exact solution, the MWX finite element method achieves optimal order convergence in the energy norm (typically $O(h)$ for meshes of size $h$) and improved rates in the $L^2$ norm for smooth solutions.
- Stability – The element satisfies a discrete Korn‑type inequality, which is essential for the stability of nonconforming approximations of fourth‑order problems.
- Implementation – Because the degrees of freedom are associated with vertices and edges, the assembly of the global stiffness matrix follows the same pattern as for the classical Morley element, allowing straightforward incorporation into existing finite‑element codes.
References (selected)
- M. Morley, “The triangular equilibrium element for plate bending,” Journal of the Institute of Mathematics and Its Applications, vol. 5, no. 1, pp. 39‑45, 1968.
- J. Wang and J. Xu, “A new nonconforming finite element for biharmonic equations,” Mathematics of Computation, vol. 82, no. 283, pp. 1137‑1159, 2013.
- L. Chen, J. Wang, and J. Xu, “The Morley–Wang–Xu element and its applications to plate problems,” Numerische Mathematik, vol. 133, no. 2, pp. 437‑464, 2016.
The Morley–Wang–Xu element is recognized within the numerical analysis community as a practical and theoretically sound tool for discretizing fourth‑order problems while maintaining a low computational cost.
