an encyclopedia of finite element definitions

# Degree 3 serendipity on a quadrilateral

◀ Back to serendipity definition page In this example:
• $$R$$ is the reference quadrilateral. The following numbering of the subentities of the reference is used:
• • $$\mathcal{V}$$ is spanned by: $$1$$, $$x$$, $$x^{2}$$, $$x^{3}$$, $$y$$, $$x y$$, $$x^{2} y$$, $$y^{2}$$, $$x y^{2}$$, $$y^{3}$$, $$x y^{3}$$, $$x^{3} y$$
• $$\mathcal{L}=\{l_0,...,l_{11}\}$$
• Functionals and basis functions: $$\displaystyle l_{0}:v\mapsto v(0,0)$$

$$\displaystyle \phi_{0} = 10 x^{3} y - 10 x^{3} - 18 x^{2} y + 18 x^{2} + 10 x y^{3} - 18 x y^{2} + 17 x y - 9 x - 10 y^{3} + 18 y^{2} - 9 y + 1$$

This DOF is associated with vertex 0 of the reference element. $$\displaystyle l_{1}:v\mapsto v(1,0)$$

$$\displaystyle \phi_{1} = x \left(- 10 x^{2} y + 10 x^{2} + 12 x y - 12 x - 10 y^{3} + 18 y^{2} - 11 y + 3\right)$$

This DOF is associated with vertex 1 of the reference element. $$\displaystyle l_{2}:v\mapsto v(0,1)$$

$$\displaystyle \phi_{2} = y \left(- 10 x^{3} + 18 x^{2} - 10 x y^{2} + 12 x y - 11 x + 10 y^{2} - 12 y + 3\right)$$

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

$$\displaystyle \phi_{3} = x y \left(10 x^{2} - 12 x + 10 y^{2} - 12 y + 5\right)$$

This DOF is associated with vertex 3 of the reference element. $$\displaystyle l_{4}:v\mapsto\displaystyle\int_{e_{0}}(1 - s_{0})v$$
where $$e_{0}$$ is the 0th edge;
and $$s_{0},s_{1}$$ is a parametrisation of $$e_{0}$$.

$$\displaystyle \phi_{4} = 12 x \left(- 5 x^{2} y + 5 x^{2} + 8 x y - 8 x - 3 y + 3\right)$$

This DOF is associated with edge 0 of the reference element. $$\displaystyle l_{5}:v\mapsto\displaystyle\int_{e_{0}}(s_{0})v$$
where $$e_{0}$$ is the 0th edge;
and $$s_{0},s_{1}$$ is a parametrisation of $$e_{0}$$.

$$\displaystyle \phi_{5} = 12 x \left(5 x^{2} y - 5 x^{2} - 7 x y + 7 x + 2 y - 2\right)$$

This DOF is associated with edge 0 of the reference element. $$\displaystyle l_{6}:v\mapsto\displaystyle\int_{e_{1}}(1 - s_{0})v$$
where $$e_{1}$$ is the 1st edge;
and $$s_{0},s_{1}$$ is a parametrisation of $$e_{1}$$.

$$\displaystyle \phi_{6} = 12 y \left(- 5 x y^{2} + 8 x y - 3 x + 5 y^{2} - 8 y + 3\right)$$

This DOF is associated with edge 1 of the reference element. $$\displaystyle l_{7}:v\mapsto\displaystyle\int_{e_{1}}(s_{0})v$$
where $$e_{1}$$ is the 1st edge;
and $$s_{0},s_{1}$$ is a parametrisation of $$e_{1}$$.

$$\displaystyle \phi_{7} = 12 y \left(5 x y^{2} - 7 x y + 2 x - 5 y^{2} + 7 y - 2\right)$$

This DOF is associated with edge 1 of the reference element. $$\displaystyle l_{8}:v\mapsto\displaystyle\int_{e_{2}}(1 - s_{0})v$$
where $$e_{2}$$ is the 2nd edge;
and $$s_{0},s_{1}$$ is a parametrisation of $$e_{2}$$.

$$\displaystyle \phi_{8} = 12 x y \left(5 y^{2} - 8 y + 3\right)$$

This DOF is associated with edge 2 of the reference element. $$\displaystyle l_{9}:v\mapsto\displaystyle\int_{e_{2}}(s_{0})v$$
where $$e_{2}$$ is the 2nd edge;
and $$s_{0},s_{1}$$ is a parametrisation of $$e_{2}$$.

$$\displaystyle \phi_{9} = 12 x y \left(- 5 y^{2} + 7 y - 2\right)$$

This DOF is associated with edge 2 of the reference element. $$\displaystyle l_{10}:v\mapsto\displaystyle\int_{e_{3}}(1 - s_{0})v$$
where $$e_{3}$$ is the 3rd edge;
and $$s_{0},s_{1}$$ is a parametrisation of $$e_{3}$$.

$$\displaystyle \phi_{10} = 12 x y \left(5 x^{2} - 8 x + 3\right)$$

This DOF is associated with edge 3 of the reference element. $$\displaystyle l_{11}:v\mapsto\displaystyle\int_{e_{3}}(s_{0})v$$
where $$e_{3}$$ is the 3rd edge;
and $$s_{0},s_{1}$$ is a parametrisation of $$e_{3}$$.

$$\displaystyle \phi_{11} = 12 x y \left(- 5 x^{2} + 7 x - 2\right)$$

This DOF is associated with edge 3 of the reference element.