an encyclopedia of finite element definitions

DPc

Click here to read what the information on this page means.

Exterior calculus names\(\mathcal{S}_{k}\Lambda^{d}(\square_d)\)
Cockburn–fu names\(\left[S_{1,k}^\square\right]_{d}\)
Orders\(0\leqslant k\)
Reference elementsinterval, quadrilateral, hexahedron
Polynomial set\(\mathcal{P}_{k}\)
↓ Show polynomial set definitions ↓
DOFsOn the interior of the reference element: point evaluations
Number of DOFsinterval: \(k+1\) (A000027)
quadrilateral: \((k+1)(k+2)/2\) (A000217)
hexahedron: \((k+1)(k+2)(k+3)/6\) (A000292)
CategoriesScalar-valued elements

Implementations

Symfem"dPc"
↓ Show Symfem examples ↓
Basixbasix.ElementFamily.DPC
↓ Show Basix examples ↓
UFL"DPC"
↓ Show UFL examples ↓

Examples

interval
order 1
interval
order 2
interval
order 3
quadrilateral
order 1
quadrilateral
order 2
quadrilateral
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.

References

DefElement stats

Element added01 March 2021
Element last updated03 September 2021