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: \(1\), \(x\), \(x^{2}\), \(x^{3}\), \(y\), \(x y\), \(x^{2} y\), \(y^{2}\), \(x y^{2}\), \(y^{3}\)
- \(\mathcal{L}=\{l_0,...,l_{9}\}\)
- Functionals and basis functions:

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

\(\displaystyle \phi_{0} = - x^{3} - 3 x^{2} y + 3 x^{2} - 3 x y^{2} + 6 x y - 3 x - y^{3} + 3 y^{2} - 3 y + 1\)

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

\(\displaystyle \phi_{0} = - x^{3} - 3 x^{2} y + 3 x^{2} - 3 x y^{2} + 6 x y - 3 x - y^{3} + 3 y^{2} - 3 y + 1\)

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

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

\(\displaystyle \phi_{1} = x^{3}\)

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

\(\displaystyle \phi_{1} = x^{3}\)

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

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

\(\displaystyle \phi_{2} = y^{3}\)

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

\(\displaystyle \phi_{2} = y^{3}\)

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

\(\displaystyle l_{3}:v\mapsto c^{e_{0}}_{1}\)

where \(v=\sum_ic^{e_{0}}_iB^{e_{0}}_i\);

\(B^{e_{0}}_1\) to \(B^{e_{0}}_n\) are the degree 3 Bernstein polynomials on \(e_{0}\);

and \(e_{0}\) is the 0th edge.

\(\displaystyle \phi_{3} = 3 x^{2} y\)

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

where \(v=\sum_ic^{e_{0}}_iB^{e_{0}}_i\);

\(B^{e_{0}}_1\) to \(B^{e_{0}}_n\) are the degree 3 Bernstein polynomials on \(e_{0}\);

and \(e_{0}\) is the 0th edge.

\(\displaystyle \phi_{3} = 3 x^{2} y\)

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

\(\displaystyle l_{4}:v\mapsto c^{e_{0}}_{2}\)

where \(v=\sum_ic^{e_{0}}_iB^{e_{0}}_i\);

\(B^{e_{0}}_1\) to \(B^{e_{0}}_n\) are the degree 3 Bernstein polynomials on \(e_{0}\);

and \(e_{0}\) is the 0th edge.

\(\displaystyle \phi_{4} = 3 x y^{2}\)

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

where \(v=\sum_ic^{e_{0}}_iB^{e_{0}}_i\);

\(B^{e_{0}}_1\) to \(B^{e_{0}}_n\) are the degree 3 Bernstein polynomials on \(e_{0}\);

and \(e_{0}\) is the 0th edge.

\(\displaystyle \phi_{4} = 3 x y^{2}\)

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

\(\displaystyle l_{5}:v\mapsto c^{e_{1}}_{1}\)

where \(v=\sum_ic^{e_{1}}_iB^{e_{1}}_i\);

\(B^{e_{1}}_1\) to \(B^{e_{1}}_n\) are the degree 3 Bernstein polynomials on \(e_{1}\);

and \(e_{1}\) is the 1st edge.

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

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

where \(v=\sum_ic^{e_{1}}_iB^{e_{1}}_i\);

\(B^{e_{1}}_1\) to \(B^{e_{1}}_n\) are the degree 3 Bernstein polynomials on \(e_{1}\);

and \(e_{1}\) is the 1st edge.

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

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

\(\displaystyle l_{6}:v\mapsto c^{e_{1}}_{2}\)

where \(v=\sum_ic^{e_{1}}_iB^{e_{1}}_i\);

\(B^{e_{1}}_1\) to \(B^{e_{1}}_n\) are the degree 3 Bernstein polynomials on \(e_{1}\);

and \(e_{1}\) is the 1st edge.

\(\displaystyle \phi_{6} = 3 y^{2} \left(- x - y + 1\right)\)

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

where \(v=\sum_ic^{e_{1}}_iB^{e_{1}}_i\);

\(B^{e_{1}}_1\) to \(B^{e_{1}}_n\) are the degree 3 Bernstein polynomials on \(e_{1}\);

and \(e_{1}\) is the 1st edge.

\(\displaystyle \phi_{6} = 3 y^{2} \left(- x - y + 1\right)\)

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

\(\displaystyle l_{7}:v\mapsto c^{e_{2}}_{1}\)

where \(v=\sum_ic^{e_{2}}_iB^{e_{2}}_i\);

\(B^{e_{2}}_1\) to \(B^{e_{2}}_n\) are the degree 3 Bernstein polynomials on \(e_{2}\);

and \(e_{2}\) is the 2nd edge.

\(\displaystyle \phi_{7} = 3 x \left(x^{2} + 2 x y - 2 x + y^{2} - 2 y + 1\right)\)

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

where \(v=\sum_ic^{e_{2}}_iB^{e_{2}}_i\);

\(B^{e_{2}}_1\) to \(B^{e_{2}}_n\) are the degree 3 Bernstein polynomials on \(e_{2}\);

and \(e_{2}\) is the 2nd edge.

\(\displaystyle \phi_{7} = 3 x \left(x^{2} + 2 x y - 2 x + y^{2} - 2 y + 1\right)\)

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

\(\displaystyle l_{8}:v\mapsto c^{e_{2}}_{2}\)

where \(v=\sum_ic^{e_{2}}_iB^{e_{2}}_i\);

\(B^{e_{2}}_1\) to \(B^{e_{2}}_n\) are the degree 3 Bernstein polynomials on \(e_{2}\);

and \(e_{2}\) is the 2nd edge.

\(\displaystyle \phi_{8} = 3 x^{2} \left(- x - y + 1\right)\)

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

where \(v=\sum_ic^{e_{2}}_iB^{e_{2}}_i\);

\(B^{e_{2}}_1\) to \(B^{e_{2}}_n\) are the degree 3 Bernstein polynomials on \(e_{2}\);

and \(e_{2}\) is the 2nd edge.

\(\displaystyle \phi_{8} = 3 x^{2} \left(- x - y + 1\right)\)

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