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# Degree 2 Morley–Wang–Xu on a triangle

◀ Back to Morley–Wang–Xu definition page In this example:
• $$R$$ is the reference triangle. The following numbering of the subentities of the reference is used:
• • $$\mathcal{V}$$ is spanned by: $$1$$, $$x$$, $$x^{2}$$, $$y$$, $$x y$$, $$y^{2}$$
• $$\mathcal{L}=\{l_0,...,l_{5}\}$$
• Functionals and basis functions: $$\displaystyle l_{0}:v\mapsto v(0,0)$$

$$\displaystyle \phi_{0} = 2 x y - x - y + 1$$

This DOF is associated with vertex 0 of the reference element. $$\displaystyle l_{1}:v\mapsto v(1,0)$$

$$\displaystyle \phi_{1} = \frac{x^{2}}{2} - x y + \frac{x}{2} - \frac{y^{2}}{2} + \frac{y}{2}$$

This DOF is associated with vertex 1 of the reference element. $$\displaystyle l_{2}:v\mapsto v(0,1)$$

$$\displaystyle \phi_{2} = - \frac{x^{2}}{2} - x y + \frac{x}{2} + \frac{y^{2}}{2} + \frac{y}{2}$$

This DOF is associated with vertex 2 of the reference element. $$\displaystyle l_{3}:\mathbf{V}\mapsto\displaystyle\frac{\sqrt{2}}{2}\int_{e_{0}}\frac{\partial}{\partial\left(\begin{array}{c}\displaystyle - \frac{\sqrt{2}}{2}\\\displaystyle - \frac{\sqrt{2}}{2}\end{array}\right)}v$$
where $$e_{0}$$ is the 0th edge.

$$\displaystyle \phi_{3} = \frac{\sqrt{2} \left(- x^{2} - 2 x y + x - y^{2} + y\right)}{2}$$

This DOF is associated with edge 0 of the reference element. $$\displaystyle l_{4}:\mathbf{V}\mapsto\displaystyle\int_{e_{1}}\frac{\partial}{\partial\left(\begin{array}{c}\displaystyle -1\\\displaystyle 0\end{array}\right)}v$$
where $$e_{1}$$ is the 1st edge.

$$\displaystyle \phi_{4} = x \left(x - 1\right)$$

This DOF is associated with edge 1 of the reference element. $$\displaystyle l_{5}:\mathbf{V}\mapsto\displaystyle\int_{e_{2}}\frac{\partial}{\partial\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\end{array}\right)}v$$
where $$e_{2}$$ is the 2nd edge.

$$\displaystyle \phi_{5} = y \left(1 - y\right)$$

This DOF is associated with edge 2 of the reference element.