an encyclopedia of finite element definitions

# Fortin–Soulie

 Abbreviated names FS Orders $$k=2$$ Reference elements triangle Polynomial set $$\mathcal{P}_{k}$$↓ Show polynomial set definitions ↓ DOFs On 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 DOFs triangle: $$6$$ Categories Scalar-valued elements

## Implementations

 Symfem "Fortin-Soulie"↓ Show Symfem examples ↓

## 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

• Fortin, M. and Soulie, M. A non‐conforming piecewise quadratic finite element on triangles, International Journal for Numerical Methods in Engineering 19, 505–520, 1983. [DOI: 10.1002/nme.1620190405] [BibTeX]

## DefElement stats

 Element added 06 March 2021 Element last updated 09 November 2021