an encyclopedia of finite element definitions
Degree 3 Bernstein on a interval
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- \(R\) 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:
\(\displaystyle l_{0}:v\mapsto v(0)\)
\(\displaystyle \phi_{0} = - x^{3} + 3 x^{2} - 3 x + 1\)
This DOF is associated with vertex 0 of the reference element.
\(\displaystyle \phi_{0} = - x^{3} + 3 x^{2} - 3 x + 1\)
This DOF is associated with vertex 0 of the reference element.
\(\displaystyle l_{1}:v\mapsto v(1)\)
\(\displaystyle \phi_{1} = x^{3}\)
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.
\(\displaystyle l_{2}:v\mapsto c_{1}\)
where \(v=\sum_ic_iB_i\);
and \(B_1\) to \(B_n\) are the degree 3 Bernstein polynomials on the cell.
\(\displaystyle \phi_{2} = 3 x \left(x^{2} - 2 x + 1\right)\)
This DOF is associated with edge 0 of the reference element.
where \(v=\sum_ic_iB_i\);
and \(B_1\) to \(B_n\) are the degree 3 Bernstein polynomials on the cell.
\(\displaystyle \phi_{2} = 3 x \left(x^{2} - 2 x + 1\right)\)
This DOF is associated with edge 0 of the reference element.
\(\displaystyle l_{3}:v\mapsto c_{2}\)
where \(v=\sum_ic_iB_i\);
and \(B_1\) to \(B_n\) are the degree 3 Bernstein polynomials on the cell.
\(\displaystyle \phi_{3} = 3 x^{2} \left(1 - x\right)\)
This DOF is associated with edge 0 of the reference element.
where \(v=\sum_ic_iB_i\);
and \(B_1\) to \(B_n\) are the degree 3 Bernstein polynomials on the cell.
\(\displaystyle \phi_{3} = 3 x^{2} \left(1 - x\right)\)
This DOF is associated with edge 0 of the reference element.