an encyclopedia of finite element definitions

an encyclopedia of finite element definitions

- \(R\) is the reference interval. The following numbering of the subentities of the reference is used:
- \(\mathcal{V}\) is spanned by: \(1\), \(x\), \(x^{2}\), \(x^{3}\), \(x^{4}\)
- \(\mathcal{L}=\{l_0,...,l_{4}\}\)
- Functionals and basis functions:

\(\displaystyle l_{0}:v\mapsto v(0)\)

\(\displaystyle \phi_{0} = 14 x^{4} - 35 x^{3} + 30 x^{2} - 10 x + 1\)

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

\(\displaystyle \phi_{0} = 14 x^{4} - 35 x^{3} + 30 x^{2} - 10 x + 1\)

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

\(\displaystyle l_{1}:v\mapsto v(1)\)

\(\displaystyle \phi_{1} = x \left(14 x^{3} - 21 x^{2} + 9 x - 1\right)\)

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

\(\displaystyle \phi_{1} = x \left(14 x^{3} - 21 x^{2} + 9 x - 1\right)\)

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

\(\displaystyle l_{2}:v\mapsto v(\tfrac{1}{2} - \tfrac{\sqrt{21}}{14})\)

\(\displaystyle \phi_{2} = \frac{7 x \left(- 28 x^{3} + 2 \sqrt{21} x^{2} + 56 x^{2} - 35 x - 3 \sqrt{21} x + \sqrt{21} + 7\right)}{6}\)

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

\(\displaystyle \phi_{2} = \frac{7 x \left(- 28 x^{3} + 2 \sqrt{21} x^{2} + 56 x^{2} - 35 x - 3 \sqrt{21} x + \sqrt{21} + 7\right)}{6}\)

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

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

\(\displaystyle \phi_{3} = \frac{16 x \left(7 x^{3} - 14 x^{2} + 8 x - 1\right)}{3}\)

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

\(\displaystyle \phi_{3} = \frac{16 x \left(7 x^{3} - 14 x^{2} + 8 x - 1\right)}{3}\)

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