an encyclopedia of finite element definitions

# Degree 1 P1-iso-P2 on a quadrilateral

◀ Back to P1-iso-P2 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: $$\begin{cases} 4 x y - 2 x - 2 y + 1&\text{in }\operatorname{Quadrilateral}(((0, 0), (1/2, 0), (0, 1/2), (1/2, 1/2)))\\0&\text{in }\operatorname{Quadrilateral}(((1/2, 0), (1, 0), (1/2, 1/2), (1, 1/2)))\\0&\text{in }\operatorname{Quadrilateral}(((0, 1/2), (1/2, 1/2), (0, 1), (1/2, 1)))\\0&\text{in }\operatorname{Quadrilateral}(((1/2, 1/2), (1, 1/2), (1/2, 1), (1, 1)))\end{cases}$$, $$\begin{cases} 0&\text{in }\operatorname{Quadrilateral}(((0, 0), (1/2, 0), (0, 1/2), (1/2, 1/2)))\\- 4 x y + 2 x + 2 y - 1&\text{in }\operatorname{Quadrilateral}(((1/2, 0), (1, 0), (1/2, 1/2), (1, 1/2)))\\0&\text{in }\operatorname{Quadrilateral}(((0, 1/2), (1/2, 1/2), (0, 1), (1/2, 1)))\\0&\text{in }\operatorname{Quadrilateral}(((1/2, 1/2), (1, 1/2), (1/2, 1), (1, 1)))\end{cases}$$, $$\begin{cases} 0&\text{in }\operatorname{Quadrilateral}(((0, 0), (1/2, 0), (0, 1/2), (1/2, 1/2)))\\0&\text{in }\operatorname{Quadrilateral}(((1/2, 0), (1, 0), (1/2, 1/2), (1, 1/2)))\\- 4 x y + 2 x + 2 y - 1&\text{in }\operatorname{Quadrilateral}(((0, 1/2), (1/2, 1/2), (0, 1), (1/2, 1)))\\0&\text{in }\operatorname{Quadrilateral}(((1/2, 1/2), (1, 1/2), (1/2, 1), (1, 1)))\end{cases}$$, $$\begin{cases} 0&\text{in }\operatorname{Quadrilateral}(((0, 0), (1/2, 0), (0, 1/2), (1/2, 1/2)))\\0&\text{in }\operatorname{Quadrilateral}(((1/2, 0), (1, 0), (1/2, 1/2), (1, 1/2)))\\0&\text{in }\operatorname{Quadrilateral}(((0, 1/2), (1/2, 1/2), (0, 1), (1/2, 1)))\\4 x y - 2 x - 2 y + 1&\text{in }\operatorname{Quadrilateral}(((1/2, 1/2), (1, 1/2), (1/2, 1), (1, 1)))\end{cases}$$, $$\begin{cases} 2 x \left(1 - 2 y\right)&\text{in }\operatorname{Quadrilateral}(((0, 0), (1/2, 0), (0, 1/2), (1/2, 1/2)))\\4 x y - 2 x - 4 y + 2&\text{in }\operatorname{Quadrilateral}(((1/2, 0), (1, 0), (1/2, 1/2), (1, 1/2)))\\0&\text{in }\operatorname{Quadrilateral}(((0, 1/2), (1/2, 1/2), (0, 1), (1/2, 1)))\\0&\text{in }\operatorname{Quadrilateral}(((1/2, 1/2), (1, 1/2), (1/2, 1), (1, 1)))\end{cases}$$, $$\begin{cases} 2 y \left(1 - 2 x\right)&\text{in }\operatorname{Quadrilateral}(((0, 0), (1/2, 0), (0, 1/2), (1/2, 1/2)))\\0&\text{in }\operatorname{Quadrilateral}(((1/2, 0), (1, 0), (1/2, 1/2), (1, 1/2)))\\4 x y - 4 x - 2 y + 2&\text{in }\operatorname{Quadrilateral}(((0, 1/2), (1/2, 1/2), (0, 1), (1/2, 1)))\\0&\text{in }\operatorname{Quadrilateral}(((1/2, 1/2), (1, 1/2), (1/2, 1), (1, 1)))\end{cases}$$, $$\begin{cases} 0&\text{in }\operatorname{Quadrilateral}(((0, 0), (1/2, 0), (0, 1/2), (1/2, 1/2)))\\2 y \left(2 x - 1\right)&\text{in }\operatorname{Quadrilateral}(((1/2, 0), (1, 0), (1/2, 1/2), (1, 1/2)))\\0&\text{in }\operatorname{Quadrilateral}(((0, 1/2), (1/2, 1/2), (0, 1), (1/2, 1)))\\- 4 x y + 4 x + 2 y - 2&\text{in }\operatorname{Quadrilateral}(((1/2, 1/2), (1, 1/2), (1/2, 1), (1, 1)))\end{cases}$$, $$\begin{cases} 0&\text{in }\operatorname{Quadrilateral}(((0, 0), (1/2, 0), (0, 1/2), (1/2, 1/2)))\\0&\text{in }\operatorname{Quadrilateral}(((1/2, 0), (1, 0), (1/2, 1/2), (1, 1/2)))\\2 x \left(2 y - 1\right)&\text{in }\operatorname{Quadrilateral}(((0, 1/2), (1/2, 1/2), (0, 1), (1/2, 1)))\\- 4 x y + 2 x + 4 y - 2&\text{in }\operatorname{Quadrilateral}(((1/2, 1/2), (1, 1/2), (1/2, 1), (1, 1)))\end{cases}$$, $$\begin{cases} 4 x y&\text{in }\operatorname{Quadrilateral}(((0, 0), (1/2, 0), (0, 1/2), (1/2, 1/2)))\\4 y \left(1 - x\right)&\text{in }\operatorname{Quadrilateral}(((1/2, 0), (1, 0), (1/2, 1/2), (1, 1/2)))\\4 x \left(1 - y\right)&\text{in }\operatorname{Quadrilateral}(((0, 1/2), (1/2, 1/2), (0, 1), (1/2, 1)))\\4 x y - 4 x - 4 y + 4&\text{in }\operatorname{Quadrilateral}(((1/2, 1/2), (1, 1/2), (1/2, 1), (1, 1)))\end{cases}$$
• $$\mathcal{L}=\{l_0,...,l_{8}\}$$
• Functionals and basis functions:
$$\displaystyle l_{0}:v\mapsto v(0,0)$$

$$\displaystyle \phi_{0} = \begin{cases} 4 x y - 2 x - 2 y + 1&\text{in }\operatorname{Quadrilateral}(((0, 0), (1/2, 0), (0, 1/2), (1/2, 1/2)))\\0&\text{in }\operatorname{Quadrilateral}(((1/2, 0), (1, 0), (1/2, 1/2), (1, 1/2)))\\0&\text{in }\operatorname{Quadrilateral}(((0, 1/2), (1/2, 1/2), (0, 1), (1/2, 1)))\\0&\text{in }\operatorname{Quadrilateral}(((1/2, 1/2), (1, 1/2), (1/2, 1), (1, 1)))\end{cases}$$

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

$$\displaystyle \phi_{1} = \begin{cases} 0&\text{in }\operatorname{Quadrilateral}(((0, 0), (1/2, 0), (0, 1/2), (1/2, 1/2)))\\- 4 x y + 2 x + 2 y - 1&\text{in }\operatorname{Quadrilateral}(((1/2, 0), (1, 0), (1/2, 1/2), (1, 1/2)))\\0&\text{in }\operatorname{Quadrilateral}(((0, 1/2), (1/2, 1/2), (0, 1), (1/2, 1)))\\0&\text{in }\operatorname{Quadrilateral}(((1/2, 1/2), (1, 1/2), (1/2, 1), (1, 1)))\end{cases}$$

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

$$\displaystyle \phi_{2} = \begin{cases} 0&\text{in }\operatorname{Quadrilateral}(((0, 0), (1/2, 0), (0, 1/2), (1/2, 1/2)))\\0&\text{in }\operatorname{Quadrilateral}(((1/2, 0), (1, 0), (1/2, 1/2), (1, 1/2)))\\- 4 x y + 2 x + 2 y - 1&\text{in }\operatorname{Quadrilateral}(((0, 1/2), (1/2, 1/2), (0, 1), (1/2, 1)))\\0&\text{in }\operatorname{Quadrilateral}(((1/2, 1/2), (1, 1/2), (1/2, 1), (1, 1)))\end{cases}$$

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

$$\displaystyle \phi_{3} = \begin{cases} 0&\text{in }\operatorname{Quadrilateral}(((0, 0), (1/2, 0), (0, 1/2), (1/2, 1/2)))\\0&\text{in }\operatorname{Quadrilateral}(((1/2, 0), (1, 0), (1/2, 1/2), (1, 1/2)))\\0&\text{in }\operatorname{Quadrilateral}(((0, 1/2), (1/2, 1/2), (0, 1), (1/2, 1)))\\4 x y - 2 x - 2 y + 1&\text{in }\operatorname{Quadrilateral}(((1/2, 1/2), (1, 1/2), (1/2, 1), (1, 1)))\end{cases}$$

This DOF is associated with vertex 3 of the reference element.
$$\displaystyle l_{4}:v\mapsto v(\tfrac{1}{2},0)$$

$$\displaystyle \phi_{4} = \begin{cases} 2 x \left(1 - 2 y\right)&\text{in }\operatorname{Quadrilateral}(((0, 0), (1/2, 0), (0, 1/2), (1/2, 1/2)))\\4 x y - 2 x - 4 y + 2&\text{in }\operatorname{Quadrilateral}(((1/2, 0), (1, 0), (1/2, 1/2), (1, 1/2)))\\0&\text{in }\operatorname{Quadrilateral}(((0, 1/2), (1/2, 1/2), (0, 1), (1/2, 1)))\\0&\text{in }\operatorname{Quadrilateral}(((1/2, 1/2), (1, 1/2), (1/2, 1), (1, 1)))\end{cases}$$

This DOF is associated with edge 0 of the reference element.
$$\displaystyle l_{5}:v\mapsto v(0,\tfrac{1}{2})$$

$$\displaystyle \phi_{5} = \begin{cases} 2 y \left(1 - 2 x\right)&\text{in }\operatorname{Quadrilateral}(((0, 0), (1/2, 0), (0, 1/2), (1/2, 1/2)))\\0&\text{in }\operatorname{Quadrilateral}(((1/2, 0), (1, 0), (1/2, 1/2), (1, 1/2)))\\4 x y - 4 x - 2 y + 2&\text{in }\operatorname{Quadrilateral}(((0, 1/2), (1/2, 1/2), (0, 1), (1/2, 1)))\\0&\text{in }\operatorname{Quadrilateral}(((1/2, 1/2), (1, 1/2), (1/2, 1), (1, 1)))\end{cases}$$

This DOF is associated with edge 1 of the reference element.
$$\displaystyle l_{6}:v\mapsto v(1,\tfrac{1}{2})$$

$$\displaystyle \phi_{6} = \begin{cases} 0&\text{in }\operatorname{Quadrilateral}(((0, 0), (1/2, 0), (0, 1/2), (1/2, 1/2)))\\2 y \left(2 x - 1\right)&\text{in }\operatorname{Quadrilateral}(((1/2, 0), (1, 0), (1/2, 1/2), (1, 1/2)))\\0&\text{in }\operatorname{Quadrilateral}(((0, 1/2), (1/2, 1/2), (0, 1), (1/2, 1)))\\- 4 x y + 4 x + 2 y - 2&\text{in }\operatorname{Quadrilateral}(((1/2, 1/2), (1, 1/2), (1/2, 1), (1, 1)))\end{cases}$$

This DOF is associated with edge 2 of the reference element.
$$\displaystyle l_{7}:v\mapsto v(\tfrac{1}{2},1)$$

$$\displaystyle \phi_{7} = \begin{cases} 0&\text{in }\operatorname{Quadrilateral}(((0, 0), (1/2, 0), (0, 1/2), (1/2, 1/2)))\\0&\text{in }\operatorname{Quadrilateral}(((1/2, 0), (1, 0), (1/2, 1/2), (1, 1/2)))\\2 x \left(2 y - 1\right)&\text{in }\operatorname{Quadrilateral}(((0, 1/2), (1/2, 1/2), (0, 1), (1/2, 1)))\\- 4 x y + 2 x + 4 y - 2&\text{in }\operatorname{Quadrilateral}(((1/2, 1/2), (1, 1/2), (1/2, 1), (1, 1)))\end{cases}$$

This DOF is associated with edge 3 of the reference element.
$$\displaystyle l_{8}:v\mapsto v(\tfrac{1}{2},\tfrac{1}{2})$$

$$\displaystyle \phi_{8} = \begin{cases} 4 x y&\text{in }\operatorname{Quadrilateral}(((0, 0), (1/2, 0), (0, 1/2), (1/2, 1/2)))\\4 y \left(1 - x\right)&\text{in }\operatorname{Quadrilateral}(((1/2, 0), (1, 0), (1/2, 1/2), (1, 1/2)))\\4 x \left(1 - y\right)&\text{in }\operatorname{Quadrilateral}(((0, 1/2), (1/2, 1/2), (0, 1), (1/2, 1)))\\4 x y - 4 x - 4 y + 4&\text{in }\operatorname{Quadrilateral}(((1/2, 1/2), (1, 1/2), (1/2, 1), (1, 1)))\end{cases}$$

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