an encyclopedia of finite element definitions

# Degree 3 Lagrange on a quadrilateral

◀ Back to Lagrange 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: $$1$$, $$y$$, $$y^{2}$$, $$y^{3}$$, $$x$$, $$x y$$, $$x y^{2}$$, $$x y^{3}$$, $$x^{2}$$, $$x^{2} y$$, $$x^{2} y^{2}$$, $$x^{2} y^{3}$$, $$x^{3}$$, $$x^{3} y$$, $$x^{3} y^{2}$$, $$x^{3} y^{3}$$
• $$\mathcal{L}=\{l_0,...,l_{15}\}$$
• Functionals and basis functions: $$\displaystyle l_{0}:v\mapsto v(0,0)$$

$$\displaystyle \phi_{0} = x y - x - y + 1$$

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

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

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

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

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

$$\displaystyle \phi_{3} = x y$$

This DOF is associated with vertex 3 of the reference element. $$\displaystyle l_{4}:\mathbf{v}\mapsto\displaystyle\int_{e_{0}}(\sqrt{3} \cdot \left(60 s_{0}^{2} - 60 s_{0} + 10\right))v$$
where $$e_{0}$$ is the 0th edge;
and $$s_{0},s_{1}$$ is a parametrisation of $$e_{0}$$.

$$\displaystyle \phi_{4} = \sqrt{3} x \left(- x y + x + y - 1\right)$$

This DOF is associated with edge 0 of the reference element. $$\displaystyle l_{5}:\mathbf{v}\mapsto\displaystyle\int_{e_{0}}(\sqrt{5} \cdot \left(280 s_{0}^{3} - 420 s_{0}^{2} + 168 s_{0} - 14\right))v$$
where $$e_{0}$$ is the 0th edge;
and $$s_{0},s_{1}$$ is a parametrisation of $$e_{0}$$.

$$\displaystyle \phi_{5} = \sqrt{5} x \left(- 2 x^{2} y + 2 x^{2} + 3 x y - 3 x - y + 1\right)$$

This DOF is associated with edge 0 of the reference element. $$\displaystyle l_{6}:\mathbf{v}\mapsto\displaystyle\int_{e_{1}}(\sqrt{3} \cdot \left(60 s_{0}^{2} - 60 s_{0} + 10\right))v$$
where $$e_{1}$$ is the 1st edge;
and $$s_{0},s_{1}$$ is a parametrisation of $$e_{1}$$.

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

This DOF is associated with edge 1 of the reference element. $$\displaystyle l_{7}:\mathbf{v}\mapsto\displaystyle\int_{e_{1}}(\sqrt{5} \cdot \left(280 s_{0}^{3} - 420 s_{0}^{2} + 168 s_{0} - 14\right))v$$
where $$e_{1}$$ is the 1st edge;
and $$s_{0},s_{1}$$ is a parametrisation of $$e_{1}$$.

$$\displaystyle \phi_{7} = \sqrt{5} y \left(- 2 x y^{2} + 3 x y - x + 2 y^{2} - 3 y + 1\right)$$

This DOF is associated with edge 1 of the reference element. $$\displaystyle l_{8}:\mathbf{v}\mapsto\displaystyle\int_{e_{2}}(\sqrt{3} \cdot \left(60 s_{0}^{2} - 60 s_{0} + 10\right))v$$
where $$e_{2}$$ is the 2nd edge;
and $$s_{0},s_{1}$$ is a parametrisation of $$e_{2}$$.

$$\displaystyle \phi_{8} = \sqrt{3} x y \left(y - 1\right)$$

This DOF is associated with edge 2 of the reference element. $$\displaystyle l_{9}:\mathbf{v}\mapsto\displaystyle\int_{e_{2}}(\sqrt{5} \cdot \left(280 s_{0}^{3} - 420 s_{0}^{2} + 168 s_{0} - 14\right))v$$
where $$e_{2}$$ is the 2nd edge;
and $$s_{0},s_{1}$$ is a parametrisation of $$e_{2}$$.

$$\displaystyle \phi_{9} = \sqrt{5} x y \left(2 y^{2} - 3 y + 1\right)$$

This DOF is associated with edge 2 of the reference element. $$\displaystyle l_{10}:\mathbf{v}\mapsto\displaystyle\int_{e_{3}}(\sqrt{3} \cdot \left(60 s_{0}^{2} - 60 s_{0} + 10\right))v$$
where $$e_{3}$$ is the 3rd edge;
and $$s_{0},s_{1}$$ is a parametrisation of $$e_{3}$$.

$$\displaystyle \phi_{10} = \sqrt{3} x y \left(x - 1\right)$$

This DOF is associated with edge 3 of the reference element. $$\displaystyle l_{11}:\mathbf{v}\mapsto\displaystyle\int_{e_{3}}(\sqrt{5} \cdot \left(280 s_{0}^{3} - 420 s_{0}^{2} + 168 s_{0} - 14\right))v$$
where $$e_{3}$$ is the 3rd edge;
and $$s_{0},s_{1}$$ is a parametrisation of $$e_{3}$$.

$$\displaystyle \phi_{11} = \sqrt{5} x y \left(2 x^{2} - 3 x + 1\right)$$

This DOF is associated with edge 3 of the reference element. $$\displaystyle l_{12}:\mathbf{v}\mapsto\displaystyle\int_{R}(10800 s_{0}^{2} s_{1}^{2} - 10800 s_{0}^{2} s_{1} + 1800 s_{0}^{2} - 10800 s_{0} s_{1}^{2} + 10800 s_{0} s_{1} - 1800 s_{0} + 1800 s_{1}^{2} - 1800 s_{1} + 300)v$$
where $$R$$ is the reference element;
and $$s_{0},s_{1}$$ is a parametrisation of $$R$$.

$$\displaystyle \phi_{12} = 3 x y \left(x y - x - y + 1\right)$$

This DOF is associated with face 0 of the reference element. $$\displaystyle l_{13}:\mathbf{v}\mapsto\displaystyle\int_{R}(\sqrt{15} \cdot \left(16800 s_{0}^{2} s_{1}^{3} - 25200 s_{0}^{2} s_{1}^{2} + 10080 s_{0}^{2} s_{1} - 840 s_{0}^{2} - 16800 s_{0} s_{1}^{3} + 25200 s_{0} s_{1}^{2} - 10080 s_{0} s_{1} + 840 s_{0} + 2800 s_{1}^{3} - 4200 s_{1}^{2} + 1680 s_{1} - 140\right))v$$
where $$R$$ is the reference element;
and $$s_{0},s_{1}$$ is a parametrisation of $$R$$.

$$\displaystyle \phi_{13} = \sqrt{15} x y \left(2 x y^{2} - 3 x y + x - 2 y^{2} + 3 y - 1\right)$$

This DOF is associated with face 0 of the reference element. $$\displaystyle l_{14}:\mathbf{v}\mapsto\displaystyle\int_{R}(\sqrt{15} \cdot \left(16800 s_{0}^{3} s_{1}^{2} - 16800 s_{0}^{3} s_{1} + 2800 s_{0}^{3} - 25200 s_{0}^{2} s_{1}^{2} + 25200 s_{0}^{2} s_{1} - 4200 s_{0}^{2} + 10080 s_{0} s_{1}^{2} - 10080 s_{0} s_{1} + 1680 s_{0} - 840 s_{1}^{2} + 840 s_{1} - 140\right))v$$
where $$R$$ is the reference element;
and $$s_{0},s_{1}$$ is a parametrisation of $$R$$.

$$\displaystyle \phi_{14} = \sqrt{15} x y \left(2 x^{2} y - 2 x^{2} - 3 x y + 3 x + y - 1\right)$$

This DOF is associated with face 0 of the reference element. $$\displaystyle l_{15}:\mathbf{v}\mapsto\displaystyle\int_{R}(392000 s_{0}^{3} s_{1}^{3} - 588000 s_{0}^{3} s_{1}^{2} + 235200 s_{0}^{3} s_{1} - 19600 s_{0}^{3} - 588000 s_{0}^{2} s_{1}^{3} + 882000 s_{0}^{2} s_{1}^{2} - 352800 s_{0}^{2} s_{1} + 29400 s_{0}^{2} + 235200 s_{0} s_{1}^{3} - 352800 s_{0} s_{1}^{2} + 141120 s_{0} s_{1} - 11760 s_{0} - 19600 s_{1}^{3} + 29400 s_{1}^{2} - 11760 s_{1} + 980)v$$
where $$R$$ is the reference element;
and $$s_{0},s_{1}$$ is a parametrisation of $$R$$.

$$\displaystyle \phi_{15} = 5 x y \left(4 x^{2} y^{2} - 6 x^{2} y + 2 x^{2} - 6 x y^{2} + 9 x y - 3 x + 2 y^{2} - 3 y + 1\right)$$

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