an encyclopedia of finite element definitions

an encyclopedia of finite element definitions

- \(R\) 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:

\(\displaystyle l_{0}:v\mapsto v(\tfrac{2}{3},\tfrac{1}{3})\)

\(\displaystyle \phi_{0} = \frac{9 x^{2}}{2} - \frac{3 x}{2} - \frac{27 y^{2}}{16} + \frac{27 y}{16} - \frac{3}{8}\)

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

\(\displaystyle \phi_{0} = \frac{9 x^{2}}{2} - \frac{3 x}{2} - \frac{27 y^{2}}{16} + \frac{27 y}{16} - \frac{3}{8}\)

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

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

\(\displaystyle \phi_{1} = 9 x y - 3 x + \frac{27 y^{2}}{4} - \frac{27 y}{4} + \frac{3}{2}\)

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

\(\displaystyle \phi_{1} = 9 x y - 3 x + \frac{27 y^{2}}{4} - \frac{27 y}{4} + \frac{3}{2}\)

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

\(\displaystyle l_{2}:v\mapsto v(0,\tfrac{1}{3})\)

\(\displaystyle \phi_{2} = \frac{9 x^{2}}{2} + 9 x y - \frac{15 x}{2} + \frac{45 y^{2}}{16} - \frac{93 y}{16} + \frac{21}{8}\)

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

\(\displaystyle \phi_{2} = \frac{9 x^{2}}{2} + 9 x y - \frac{15 x}{2} + \frac{45 y^{2}}{16} - \frac{93 y}{16} + \frac{21}{8}\)

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

\(\displaystyle l_{3}:v\mapsto v(0,\tfrac{2}{3})\)

\(\displaystyle \phi_{3} = - 9 x y + 3 x - \frac{9 y^{2}}{4} + \frac{21 y}{4} - \frac{3}{2}\)

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

\(\displaystyle \phi_{3} = - 9 x y + 3 x - \frac{9 y^{2}}{4} + \frac{21 y}{4} - \frac{3}{2}\)

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

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

\(\displaystyle \phi_{4} = \frac{9 y^{2}}{2} - \frac{9 y}{2} + 1\)

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

\(\displaystyle \phi_{4} = \frac{9 y^{2}}{2} - \frac{9 y}{2} + 1\)

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