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Fortin–Soulie

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Abbreviated namesFS
Orders\(k=2\)
Reference elementstriangle
Polynomial set\(\mathcal{P}_{k}\)
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DOFsOn each facet: point evaluation at two points (but only one point on one of the edges)
On the interior of the reference element: point evaluation at midpoint
Number of DOFstriangle: \(6\)
CategoriesScalar-valued elements

Implementations

Symfem"Fortin-Soulie"
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Examples

triangle
order 2
  • \(R\) 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:
\(\displaystyle l_{0}:v\mapsto\displaystyle\int_{e_{0}}(1 - s_{0})v\)
where \(e_{0}\) is the 0th edge;
and \(s_{0}\) is a parametrisation of \(e_{0}\).

\(\displaystyle \phi_{0} = \sqrt{2} \left(4 x^{2} - 2 x y - x - 2 y^{2} + 2 y - \frac{1}{3}\right)\)

This DOF is associated with edge 0 of the reference element.
\(\displaystyle l_{1}:v\mapsto\displaystyle\int_{e_{0}}(s_{0})v\)
where \(e_{0}\) is the 0th edge;
and \(s_{0}\) is a parametrisation of \(e_{0}\).

\(\displaystyle \phi_{1} = \sqrt{2} \left(12 x y - 3 x + 6 y^{2} - 6 y + 1\right)\)

This DOF is associated with edge 0 of the reference element.
\(\displaystyle l_{2}:v\mapsto\displaystyle\int_{e_{1}}(1 - s_{0})v\)
where \(e_{1}\) is the 1st edge;
and \(s_{0}\) is a parametrisation of \(e_{1}\).

\(\displaystyle \phi_{2} = 12 x^{2} + 24 x y - 18 x - 6 y + 4\)

This DOF is associated with edge 1 of the reference element.
\(\displaystyle l_{3}:v\mapsto\displaystyle\int_{e_{1}}(s_{0})v\)
where \(e_{1}\) is the 1st edge;
and \(s_{0}\) is a parametrisation of \(e_{1}\).

\(\displaystyle \phi_{3} = - 4 x^{2} - 28 x y + 10 x - 4 y^{2} + 10 y - \frac{8}{3}\)

This DOF is associated with edge 1 of the reference element.
\(\displaystyle l_{4}:v\mapsto\displaystyle\int_{e_{2}}(1 - s_{0})v\)
where \(e_{2}\) is the 2nd edge;
and \(s_{0}\) is a parametrisation of \(e_{2}\).

\(\displaystyle \phi_{4} = - 4 x^{2} - 4 x y + 4 x + 8 y^{2} - 8 y + \frac{4}{3}\)

This DOF is associated with edge 2 of the reference element.
\(\displaystyle l_{5}:v\mapsto v(\tfrac{1}{3},\tfrac{1}{3})\)

\(\displaystyle \phi_{5} = - 6 x^{2} - 6 x y + 6 x - 6 y^{2} + 6 y - 1\)

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

References

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Element added06 March 2021
Element last updated09 November 2021