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: \(\left(\begin{array}{c}\displaystyle 1\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle x\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle y\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle y\end{array}\right)\)
- \(\mathcal{L}=\{l_0,...,l_{5}\}\)
- Functionals and basis functions:

\(\displaystyle l_{0}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{0}}\boldsymbol{v}\cdot(1 - s_{0})\hat{\boldsymbol{n}}_{0}\)

where \(e_{0}\) is the 0th edge;

\(\hat{\boldsymbol{n}}_{0}\) is the normal to facet 0;

and \(s_{0},s_{1}\) is a parametrisation of \(e_{0}\).

\(\displaystyle \boldsymbol{\phi}_{0} = \left(\begin{array}{c}\displaystyle - 4 x\\\displaystyle 2 y\end{array}\right)\)

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

where \(e_{0}\) is the 0th edge;

\(\hat{\boldsymbol{n}}_{0}\) is the normal to facet 0;

and \(s_{0},s_{1}\) is a parametrisation of \(e_{0}\).

\(\displaystyle \boldsymbol{\phi}_{0} = \left(\begin{array}{c}\displaystyle - 4 x\\\displaystyle 2 y\end{array}\right)\)

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

\(\displaystyle l_{1}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{0}}\boldsymbol{v}\cdot(s_{0})\hat{\boldsymbol{n}}_{0}\)

where \(e_{0}\) is the 0th edge;

\(\hat{\boldsymbol{n}}_{0}\) is the normal to facet 0;

and \(s_{0},s_{1}\) is a parametrisation of \(e_{0}\).

\(\displaystyle \boldsymbol{\phi}_{1} = \left(\begin{array}{c}\displaystyle 2 x\\\displaystyle - 4 y\end{array}\right)\)

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

where \(e_{0}\) is the 0th edge;

\(\hat{\boldsymbol{n}}_{0}\) is the normal to facet 0;

and \(s_{0},s_{1}\) is a parametrisation of \(e_{0}\).

\(\displaystyle \boldsymbol{\phi}_{1} = \left(\begin{array}{c}\displaystyle 2 x\\\displaystyle - 4 y\end{array}\right)\)

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

\(\displaystyle l_{2}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{1}}\boldsymbol{v}\cdot(1 - s_{0})\hat{\boldsymbol{n}}_{1}\)

where \(e_{1}\) is the 1st edge;

\(\hat{\boldsymbol{n}}_{1}\) is the normal to facet 1;

and \(s_{0},s_{1}\) is a parametrisation of \(e_{1}\).

\(\displaystyle \boldsymbol{\phi}_{2} = \left(\begin{array}{c}\displaystyle 4 x + 6 y - 4\\\displaystyle - 2 y\end{array}\right)\)

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

where \(e_{1}\) is the 1st edge;

\(\hat{\boldsymbol{n}}_{1}\) is the normal to facet 1;

and \(s_{0},s_{1}\) is a parametrisation of \(e_{1}\).

\(\displaystyle \boldsymbol{\phi}_{2} = \left(\begin{array}{c}\displaystyle 4 x + 6 y - 4\\\displaystyle - 2 y\end{array}\right)\)

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

\(\displaystyle l_{3}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{1}}\boldsymbol{v}\cdot(s_{0})\hat{\boldsymbol{n}}_{1}\)

where \(e_{1}\) is the 1st edge;

\(\hat{\boldsymbol{n}}_{1}\) is the normal to facet 1;

and \(s_{0},s_{1}\) is a parametrisation of \(e_{1}\).

\(\displaystyle \boldsymbol{\phi}_{3} = \left(\begin{array}{c}\displaystyle - 2 x - 6 y + 2\\\displaystyle 4 y\end{array}\right)\)

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

where \(e_{1}\) is the 1st edge;

\(\hat{\boldsymbol{n}}_{1}\) is the normal to facet 1;

and \(s_{0},s_{1}\) is a parametrisation of \(e_{1}\).

\(\displaystyle \boldsymbol{\phi}_{3} = \left(\begin{array}{c}\displaystyle - 2 x - 6 y + 2\\\displaystyle 4 y\end{array}\right)\)

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

\(\displaystyle l_{4}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{2}}\boldsymbol{v}\cdot(1 - s_{0})\hat{\boldsymbol{n}}_{2}\)

where \(e_{2}\) is the 2nd edge;

\(\hat{\boldsymbol{n}}_{2}\) is the normal to facet 2;

and \(s_{0},s_{1}\) is a parametrisation of \(e_{2}\).

\(\displaystyle \boldsymbol{\phi}_{4} = \left(\begin{array}{c}\displaystyle 2 x\\\displaystyle - 6 x - 4 y + 4\end{array}\right)\)

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

where \(e_{2}\) is the 2nd edge;

\(\hat{\boldsymbol{n}}_{2}\) is the normal to facet 2;

and \(s_{0},s_{1}\) is a parametrisation of \(e_{2}\).

\(\displaystyle \boldsymbol{\phi}_{4} = \left(\begin{array}{c}\displaystyle 2 x\\\displaystyle - 6 x - 4 y + 4\end{array}\right)\)

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

\(\displaystyle l_{5}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{2}}\boldsymbol{v}\cdot(s_{0})\hat{\boldsymbol{n}}_{2}\)

where \(e_{2}\) is the 2nd edge;

\(\hat{\boldsymbol{n}}_{2}\) is the normal to facet 2;

and \(s_{0},s_{1}\) is a parametrisation of \(e_{2}\).

\(\displaystyle \boldsymbol{\phi}_{5} = \left(\begin{array}{c}\displaystyle - 4 x\\\displaystyle 6 x + 2 y - 2\end{array}\right)\)

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

where \(e_{2}\) is the 2nd edge;

\(\hat{\boldsymbol{n}}_{2}\) is the normal to facet 2;

and \(s_{0},s_{1}\) is a parametrisation of \(e_{2}\).

\(\displaystyle \boldsymbol{\phi}_{5} = \left(\begin{array}{c}\displaystyle - 4 x\\\displaystyle 6 x + 2 y - 2\end{array}\right)\)

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