Degree 3 Kong–Mulder–Veldhuizen on a triangle

• \(R\) is the reference triangle. The following numbering of the subentities of the reference is used:

• \(\mathcal{V}\) is spanned by: \(1\), \(x\), \(x^{2}\), \(y\), \(x y\), \(y^{2}\), \(x y \left(- x - y + 1\right)\)
• \(\mathcal{L}=\{l_0,...,l_{6}\}\)
• Functionals and basis functions:

\(\displaystyle l_{0}:v\mapsto \frac{1}{40} v(0,0)\)

\(\displaystyle \phi_{0} = - 120 x^{2} y + 80 x^{2} - 120 x y^{2} + 280 x y - 120 x + 80 y^{2} - 120 y + 40\)

This DOF is associated with vertex 0 of the reference element.

\(\displaystyle l_{1}:v\mapsto \frac{1}{40} v(1,0)\)

\(\displaystyle \phi_{1} = 40 x \left(- 3 x y + 2 x - 3 y^{2} + 3 y - 1\right)\)

This DOF is associated with vertex 1 of the reference element.

\(\displaystyle l_{2}:v\mapsto \frac{1}{40} v(0,1)\)

\(\displaystyle \phi_{2} = 40 y \left(- 3 x^{2} - 3 x y + 3 x + 2 y - 1\right)\)

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

\(\displaystyle l_{3}:v\mapsto \frac{1}{15} v(\tfrac{1}{2},\tfrac{1}{2})\)

\(\displaystyle \phi_{3} = 60 x y \left(3 x + 3 y - 2\right)\)

This DOF is associated with edge 0 of the reference element.

\(\displaystyle l_{4}:v\mapsto \frac{1}{15} v(0,\tfrac{1}{2})\)

\(\displaystyle \phi_{4} = 60 y \left(3 x^{2} + 3 x y - 4 x - y + 1\right)\)

This DOF is associated with edge 1 of the reference element.

\(\displaystyle l_{5}:v\mapsto \frac{1}{15} v(\tfrac{1}{2},0)\)

\(\displaystyle \phi_{5} = 60 x \left(3 x y - x + 3 y^{2} - 4 y + 1\right)\)

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

\(\displaystyle l_{6}:v\mapsto \frac{9}{40} v(\tfrac{1}{3},\tfrac{1}{3})\)

\(\displaystyle \phi_{6} = 120 x y \left(- x - y + 1\right)\)

This DOF is associated with face 0 of the reference element.