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:

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

\(\displaystyle \phi_{0} = 2 x y - x - y + 1\)

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

\(\displaystyle \phi_{0} = 2 x y - x - y + 1\)

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

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

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

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

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

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

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

\(\displaystyle \phi_{2} = - \frac{x^{2}}{2} - x y + \frac{x}{2} + \frac{y^{2}}{2} + \frac{y}{2}\)

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

\(\displaystyle \phi_{2} = - \frac{x^{2}}{2} - x y + \frac{x}{2} + \frac{y^{2}}{2} + \frac{y}{2}\)

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

\(\displaystyle l_{3}:v\mapsto\nabla{v}(\tfrac{1}{2},\tfrac{1}{2})\cdot\hat{\boldsymbol{n}}_{0}\)

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

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

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

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

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

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

\(\displaystyle l_{4}:v\mapsto\nabla{v}(0,\tfrac{1}{2})\cdot\hat{\boldsymbol{n}}_{1}\)

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

\(\displaystyle \phi_{4} = x \left(x - 1\right)\)

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

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

\(\displaystyle \phi_{4} = x \left(x - 1\right)\)

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