an encyclopedia of finite element definitions

Degree 1 Tiniest tensor on a quadrilateral

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In this example:
• $$R$$ is the reference quadrilateral. The following numbering of the subentities of the reference is used:
• $$\mathcal{V}$$ is spanned by: $$1$$, $$y$$, $$x$$, $$x y$$, $$\frac{3 x \left(- x y + x + y - 1\right)}{2}$$, $$\frac{3 x y \left(x - 1\right)}{2}$$, $$\frac{3 y \left(- x y + x + y - 1\right)}{2}$$, $$\frac{3 x y \left(y - 1\right)}{2}$$
• $$\mathcal{L}=\{l_0,...,l_{7}\}$$
• Functionals and basis functions:
$$\displaystyle l_{0}:v\mapsto v(0,0)$$

$$\displaystyle \phi_{0} = - 3 x^{2} y + 3 x^{2} - 3 x y^{2} + 7 x y - 4 x + 3 y^{2} - 4 y + 1$$

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

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

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

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

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

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

This DOF is associated with vertex 3 of the reference element.
$$\displaystyle l_{4}:\mathbf{v}\mapsto\displaystyle\int_{e_{0}}v$$
where $$e_{0}$$ is the 0th edge.

$$\displaystyle \phi_{4} = 6 x \left(x y - x - y + 1\right)$$

This DOF is associated with edge 0 of the reference element.
$$\displaystyle l_{5}:\mathbf{v}\mapsto\displaystyle\int_{e_{1}}v$$
where $$e_{1}$$ is the 1st edge.

$$\displaystyle \phi_{5} = 6 y \left(x y - x - y + 1\right)$$

This DOF is associated with edge 1 of the reference element.
$$\displaystyle l_{6}:\mathbf{v}\mapsto\displaystyle\int_{e_{2}}v$$
where $$e_{2}$$ is the 2nd edge.

$$\displaystyle \phi_{6} = 6 x y \left(1 - y\right)$$

This DOF is associated with edge 2 of the reference element.
$$\displaystyle l_{7}:\mathbf{v}\mapsto\displaystyle\int_{e_{3}}v$$
where $$e_{3}$$ is the 3th edge.

$$\displaystyle \phi_{7} = 6 x y \left(1 - x\right)$$

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