an encyclopedia of finite element definitions

# DPc

 Orders $$0\leqslant k$$ Reference elements interval, quadrilateral, hexahedron Polynomial set $$\mathcal{P}_{k}$$↓ Show polynomial set definitions ↓ DOFs On the interior of the reference element: point evaluations Number of DOFs interval: $$k+1$$ (A000027)quadrilateral: $$(k+1)(k+2)/2$$ (A000217)hexahedron: $$(k+1)(k+2)(k+3)/6$$ (A000292) Categories Scalar-valued elements

## Implementations

 Symfem string "discontinuous Lagrange"↓ Show Symfem examples ↓ Basix string "DPC"↓ Show Basix examples ↓ UFL string "DPC"↓ Show UFL examples ↓

## Examples

interval
order 1
interval
order 2
interval
order 3
order 1
order 2
order 3
• $$R$$ is the reference interval. The following numbering of the subentities of the reference is used:
• $$\mathcal{V}$$ is spanned by: $$1$$, $$x$$
• $$\mathcal{L}=\{l_0,...,l_{1}\}$$
• Functionals and basis functions:
$$\displaystyle l_{0}:v\mapsto v(0)$$

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

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

$$\displaystyle \phi_{1} = x$$

This DOF is associated with edge 0 of the reference element.
• $$R$$ is the reference interval. The following numbering of the subentities of the reference is used:
• $$\mathcal{V}$$ is spanned by: $$1$$, $$x$$, $$x^{2}$$
• $$\mathcal{L}=\{l_0,...,l_{2}\}$$
• Functionals and basis functions:
$$\displaystyle l_{0}:v\mapsto v(0)$$

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

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

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

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

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

This DOF is associated with edge 0 of the reference element.
• $$R$$ is the reference interval. The following numbering of the subentities of the reference is used:
• $$\mathcal{V}$$ is spanned by: $$1$$, $$x$$, $$x^{2}$$, $$x^{3}$$
• $$\mathcal{L}=\{l_0,...,l_{3}\}$$
• Functionals and basis functions:
$$\displaystyle l_{0}:v\mapsto v(0)$$

$$\displaystyle \phi_{0} = - \frac{9 x^{3}}{2} + 9 x^{2} - \frac{11 x}{2} + 1$$

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

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

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

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

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

$$\displaystyle \phi_{3} = \frac{x \left(9 x^{2} - 9 x + 2\right)}{2}$$

This DOF is associated with edge 0 of the reference element.
• $$R$$ is the reference quadrilateral. The following numbering of the subentities of the reference is used:
• $$\mathcal{V}$$ is spanned by: $$1$$, $$x$$, $$y$$
• $$\mathcal{L}=\{l_0,...,l_{2}\}$$
• Functionals and basis functions:
$$\displaystyle l_{0}:v\mapsto v(0,0)$$

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

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

$$\displaystyle \phi_{1} = x$$

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

$$\displaystyle \phi_{2} = y$$

This DOF is associated with face 0 of the reference element.
• $$R$$ is the reference quadrilateral. The following numbering of the subentities of the reference is used:
• $$\mathcal{V}$$ is spanned by: $$1$$, $$x$$, $$x^{2}$$, $$y$$, $$x y$$, $$y^{2}$$
• $$\mathcal{L}=\{l_0,...,l_{5}\}$$
• Functionals and basis functions:
$$\displaystyle l_{0}:v\mapsto v(0,0)$$

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

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

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

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

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

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

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

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

$$\displaystyle \phi_{4} = 4 x y$$

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

$$\displaystyle \phi_{5} = y \left(2 y - 1\right)$$

This DOF is associated with face 0 of the reference element.
• $$R$$ is the reference quadrilateral. The following numbering of the subentities of the reference is used:
• $$\mathcal{V}$$ is spanned by: $$1$$, $$x$$, $$x^{2}$$, $$x^{3}$$, $$y$$, $$x y$$, $$x^{2} y$$, $$y^{2}$$, $$x y^{2}$$, $$y^{3}$$
• $$\mathcal{L}=\{l_0,...,l_{9}\}$$
• Functionals and basis functions:
$$\displaystyle l_{0}:v\mapsto v(0,0)$$

$$\displaystyle \phi_{0} = - \frac{9 x^{3}}{2} - \frac{27 x^{2} y}{2} + 9 x^{2} - \frac{27 x y^{2}}{2} + 18 x y - \frac{11 x}{2} - \frac{9 y^{3}}{2} + 9 y^{2} - \frac{11 y}{2} + 1$$

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

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

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

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

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

$$\displaystyle \phi_{3} = \frac{x \left(9 x^{2} - 9 x + 2\right)}{2}$$

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

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

This DOF is associated with face 0 of the reference element.
$$\displaystyle l_{5}:v\mapsto v(\tfrac{1}{3},\tfrac{1}{3})$$

$$\displaystyle \phi_{5} = 27 x y \left(- x - y + 1\right)$$

This DOF is associated with face 0 of the reference element.
$$\displaystyle l_{6}:v\mapsto v(\tfrac{2}{3},\tfrac{1}{3})$$

$$\displaystyle \phi_{6} = \frac{9 x y \left(3 x - 1\right)}{2}$$

This DOF is associated with face 0 of the reference element.
$$\displaystyle l_{7}:v\mapsto v(0,\tfrac{2}{3})$$

$$\displaystyle \phi_{7} = \frac{9 y \left(- 3 x y + x - 3 y^{2} + 4 y - 1\right)}{2}$$

This DOF is associated with face 0 of the reference element.
$$\displaystyle l_{8}:v\mapsto v(\tfrac{1}{3},\tfrac{2}{3})$$

$$\displaystyle \phi_{8} = \frac{9 x y \left(3 y - 1\right)}{2}$$

This DOF is associated with face 0 of the reference element.
$$\displaystyle l_{9}:v\mapsto v(0,1)$$

$$\displaystyle \phi_{9} = \frac{y \left(9 y^{2} - 9 y + 2\right)}{2}$$

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