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: \(1\), \(x\), \(x^{2}\), \(x^{3}\)
- \(\mathcal{L}=\{l_0,...,l_{3}\}\)
- Functionals and basis functions:

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

\(\displaystyle \phi_{0} = 1 - x\)

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

\(\displaystyle \phi_{0} = 1 - x\)

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

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

\(\displaystyle \phi_{1} = x\)

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

\(\displaystyle \phi_{1} = x\)

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

\(\displaystyle l_{2}:\mathbf{v}\mapsto\displaystyle\int_{R}(\sqrt{3} \cdot \left(60 s_{0}^{2} - 60 s_{0} + 10\right))v\)

where \(R\) is the reference element;

and \(s_{0}\) is a parametrisation of \(R\).

\(\displaystyle \phi_{2} = \sqrt{3} x \left(x - 1\right)\)

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

where \(R\) is the reference element;

and \(s_{0}\) is a parametrisation of \(R\).

\(\displaystyle \phi_{2} = \sqrt{3} x \left(x - 1\right)\)

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

\(\displaystyle l_{3}:\mathbf{v}\mapsto\displaystyle\int_{R}(\sqrt{5} \cdot \left(280 s_{0}^{3} - 420 s_{0}^{2} + 168 s_{0} - 14\right))v\)

where \(R\) is the reference element;

and \(s_{0}\) is a parametrisation of \(R\).

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

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

where \(R\) is the reference element;

and \(s_{0}\) is a parametrisation of \(R\).

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

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