an encyclopedia of finite element definitions

Vector dPc

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

Orders\(0\leqslant k\)
Reference elementsinterval, quadrilateral, hexahedron
Polynomial set\(\mathcal{P}_{k}^d\)
↓ Show polynomial set definitions ↓
DOFsOn the interior of the reference element: point evaluations
Number of DOFsquadrilateral: \((k+1)(k+2)\) (A002378)
hexahedron: \((k+1)(k+2)(k+3)/2\) (A027480)
CategoriesScalar-valued elements

Implementations

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

Examples

quadrilateral
order 1
quadrilateral
order 2
quadrilateral
order 3
hexahedron
order 1
hexahedron
order 2
  • \(R\) is the reference quadrilateral. The following numbering of the subentities of the reference is used:
  • \(\mathcal{V}\) is spanned by: \(\left(\begin{array}{c}\displaystyle 1\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle x\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle y\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle y\end{array}\right)\)
  • \(\mathcal{L}=\{l_0,...,l_{5}\}\)
  • Functionals and basis functions:
\(\displaystyle l_{0}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,0)\cdot\left(\begin{array}{c}1\\0\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{0} = \left(\begin{array}{c}\displaystyle - x - y + 1\\\displaystyle 0\end{array}\right)\)

This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{1}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,0)\cdot\left(\begin{array}{c}0\\1\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{1} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle - x - y + 1\end{array}\right)\)

This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{2}:\boldsymbol{v}\mapsto\boldsymbol{v}(1,0)\cdot\left(\begin{array}{c}1\\0\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{2} = \left(\begin{array}{c}\displaystyle x\\\displaystyle 0\end{array}\right)\)

This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{3}:\boldsymbol{v}\mapsto\boldsymbol{v}(1,0)\cdot\left(\begin{array}{c}0\\1\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{3} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle x\end{array}\right)\)

This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{4}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,1)\cdot\left(\begin{array}{c}1\\0\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{4} = \left(\begin{array}{c}\displaystyle y\\\displaystyle 0\end{array}\right)\)

This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{5}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,1)\cdot\left(\begin{array}{c}0\\1\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{5} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle y\end{array}\right)\)

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: \(\left(\begin{array}{c}\displaystyle 1\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle x\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x^{2}\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle x^{2}\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle y\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle y\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x y\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle x y\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle y^{2}\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle y^{2}\end{array}\right)\)
  • \(\mathcal{L}=\{l_0,...,l_{11}\}\)
  • Functionals and basis functions:
\(\displaystyle l_{0}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,0)\cdot\left(\begin{array}{c}1\\0\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{0} = \left(\begin{array}{c}\displaystyle 2 x^{2} + 4 x y - 3 x + 2 y^{2} - 3 y + 1\\\displaystyle 0\end{array}\right)\)

This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{1}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,0)\cdot\left(\begin{array}{c}0\\1\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{1} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 2 x^{2} + 4 x y - 3 x + 2 y^{2} - 3 y + 1\end{array}\right)\)

This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{2}:\boldsymbol{v}\mapsto\boldsymbol{v}(\tfrac{1}{2},0)\cdot\left(\begin{array}{c}1\\0\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{2} = \left(\begin{array}{c}\displaystyle 4 x \left(- x - y + 1\right)\\\displaystyle 0\end{array}\right)\)

This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{3}:\boldsymbol{v}\mapsto\boldsymbol{v}(\tfrac{1}{2},0)\cdot\left(\begin{array}{c}0\\1\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{3} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 4 x \left(- x - y + 1\right)\end{array}\right)\)

This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{4}:\boldsymbol{v}\mapsto\boldsymbol{v}(1,0)\cdot\left(\begin{array}{c}1\\0\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{4} = \left(\begin{array}{c}\displaystyle x \left(2 x - 1\right)\\\displaystyle 0\end{array}\right)\)

This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{5}:\boldsymbol{v}\mapsto\boldsymbol{v}(1,0)\cdot\left(\begin{array}{c}0\\1\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{5} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle x \left(2 x - 1\right)\end{array}\right)\)

This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{6}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,\tfrac{1}{2})\cdot\left(\begin{array}{c}1\\0\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{6} = \left(\begin{array}{c}\displaystyle 4 y \left(- x - y + 1\right)\\\displaystyle 0\end{array}\right)\)

This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{7}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,\tfrac{1}{2})\cdot\left(\begin{array}{c}0\\1\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{7} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 4 y \left(- x - y + 1\right)\end{array}\right)\)

This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{8}:\boldsymbol{v}\mapsto\boldsymbol{v}(\tfrac{1}{2},\tfrac{1}{2})\cdot\left(\begin{array}{c}1\\0\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{8} = \left(\begin{array}{c}\displaystyle 4 x y\\\displaystyle 0\end{array}\right)\)

This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{9}:\boldsymbol{v}\mapsto\boldsymbol{v}(\tfrac{1}{2},\tfrac{1}{2})\cdot\left(\begin{array}{c}0\\1\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{9} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 4 x y\end{array}\right)\)

This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{10}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,1)\cdot\left(\begin{array}{c}1\\0\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{10} = \left(\begin{array}{c}\displaystyle y \left(2 y - 1\right)\\\displaystyle 0\end{array}\right)\)

This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{11}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,1)\cdot\left(\begin{array}{c}0\\1\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{11} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle y \left(2 y - 1\right)\end{array}\right)\)

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: \(\left(\begin{array}{c}\displaystyle 1\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle x\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x^{2}\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle x^{2}\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x^{3}\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle x^{3}\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle y\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle y\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x y\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle x y\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x^{2} y\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle x^{2} y\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle y^{2}\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle y^{2}\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x y^{2}\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle x y^{2}\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle y^{3}\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle y^{3}\end{array}\right)\)
  • \(\mathcal{L}=\{l_0,...,l_{19}\}\)
  • Functionals and basis functions:
\(\displaystyle l_{0}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,0)\cdot\left(\begin{array}{c}1\\0\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{0} = \left(\begin{array}{c}\displaystyle - \frac{9 x^{3}}{2} - \frac{27 x^{2} y}{2} + 9 x^{2} - \frac{27 x y^{2}}{2} + 18 x y - \frac{11 x}{2} - \frac{9 y^{3}}{2} + 9 y^{2} - \frac{11 y}{2} + 1\\\displaystyle 0\end{array}\right)\)

This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{1}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,0)\cdot\left(\begin{array}{c}0\\1\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{1} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle - \frac{9 x^{3}}{2} - \frac{27 x^{2} y}{2} + 9 x^{2} - \frac{27 x y^{2}}{2} + 18 x y - \frac{11 x}{2} - \frac{9 y^{3}}{2} + 9 y^{2} - \frac{11 y}{2} + 1\end{array}\right)\)

This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{2}:\boldsymbol{v}\mapsto\boldsymbol{v}(\tfrac{1}{3},0)\cdot\left(\begin{array}{c}1\\0\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{2} = \left(\begin{array}{c}\displaystyle \frac{9 x \left(3 x^{2} + 6 x y - 5 x + 3 y^{2} - 5 y + 2\right)}{2}\\\displaystyle 0\end{array}\right)\)

This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{3}:\boldsymbol{v}\mapsto\boldsymbol{v}(\tfrac{1}{3},0)\cdot\left(\begin{array}{c}0\\1\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{3} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle \frac{9 x \left(3 x^{2} + 6 x y - 5 x + 3 y^{2} - 5 y + 2\right)}{2}\end{array}\right)\)

This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{4}:\boldsymbol{v}\mapsto\boldsymbol{v}(\tfrac{2}{3},0)\cdot\left(\begin{array}{c}1\\0\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{4} = \left(\begin{array}{c}\displaystyle \frac{9 x \left(- 3 x^{2} - 3 x y + 4 x + y - 1\right)}{2}\\\displaystyle 0\end{array}\right)\)

This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{5}:\boldsymbol{v}\mapsto\boldsymbol{v}(\tfrac{2}{3},0)\cdot\left(\begin{array}{c}0\\1\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{5} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle \frac{9 x \left(- 3 x^{2} - 3 x y + 4 x + y - 1\right)}{2}\end{array}\right)\)

This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{6}:\boldsymbol{v}\mapsto\boldsymbol{v}(1,0)\cdot\left(\begin{array}{c}1\\0\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{6} = \left(\begin{array}{c}\displaystyle \frac{x \left(9 x^{2} - 9 x + 2\right)}{2}\\\displaystyle 0\end{array}\right)\)

This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{7}:\boldsymbol{v}\mapsto\boldsymbol{v}(1,0)\cdot\left(\begin{array}{c}0\\1\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{7} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle \frac{x \left(9 x^{2} - 9 x + 2\right)}{2}\end{array}\right)\)

This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{8}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,\tfrac{1}{3})\cdot\left(\begin{array}{c}1\\0\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{8} = \left(\begin{array}{c}\displaystyle \frac{9 y \left(3 x^{2} + 6 x y - 5 x + 3 y^{2} - 5 y + 2\right)}{2}\\\displaystyle 0\end{array}\right)\)

This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{9}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,\tfrac{1}{3})\cdot\left(\begin{array}{c}0\\1\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{9} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle \frac{9 y \left(3 x^{2} + 6 x y - 5 x + 3 y^{2} - 5 y + 2\right)}{2}\end{array}\right)\)

This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{10}:\boldsymbol{v}\mapsto\boldsymbol{v}(\tfrac{1}{3},\tfrac{1}{3})\cdot\left(\begin{array}{c}1\\0\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{10} = \left(\begin{array}{c}\displaystyle 27 x y \left(- x - y + 1\right)\\\displaystyle 0\end{array}\right)\)

This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{11}:\boldsymbol{v}\mapsto\boldsymbol{v}(\tfrac{1}{3},\tfrac{1}{3})\cdot\left(\begin{array}{c}0\\1\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{11} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 27 x y \left(- x - y + 1\right)\end{array}\right)\)

This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{12}:\boldsymbol{v}\mapsto\boldsymbol{v}(\tfrac{2}{3},\tfrac{1}{3})\cdot\left(\begin{array}{c}1\\0\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{12} = \left(\begin{array}{c}\displaystyle \frac{9 x y \left(3 x - 1\right)}{2}\\\displaystyle 0\end{array}\right)\)

This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{13}:\boldsymbol{v}\mapsto\boldsymbol{v}(\tfrac{2}{3},\tfrac{1}{3})\cdot\left(\begin{array}{c}0\\1\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{13} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle \frac{9 x y \left(3 x - 1\right)}{2}\end{array}\right)\)

This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{14}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,\tfrac{2}{3})\cdot\left(\begin{array}{c}1\\0\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{14} = \left(\begin{array}{c}\displaystyle \frac{9 y \left(- 3 x y + x - 3 y^{2} + 4 y - 1\right)}{2}\\\displaystyle 0\end{array}\right)\)

This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{15}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,\tfrac{2}{3})\cdot\left(\begin{array}{c}0\\1\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{15} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle \frac{9 y \left(- 3 x y + x - 3 y^{2} + 4 y - 1\right)}{2}\end{array}\right)\)

This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{16}:\boldsymbol{v}\mapsto\boldsymbol{v}(\tfrac{1}{3},\tfrac{2}{3})\cdot\left(\begin{array}{c}1\\0\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{16} = \left(\begin{array}{c}\displaystyle \frac{9 x y \left(3 y - 1\right)}{2}\\\displaystyle 0\end{array}\right)\)

This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{17}:\boldsymbol{v}\mapsto\boldsymbol{v}(\tfrac{1}{3},\tfrac{2}{3})\cdot\left(\begin{array}{c}0\\1\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{17} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle \frac{9 x y \left(3 y - 1\right)}{2}\end{array}\right)\)

This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{18}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,1)\cdot\left(\begin{array}{c}1\\0\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{18} = \left(\begin{array}{c}\displaystyle \frac{y \left(9 y^{2} - 9 y + 2\right)}{2}\\\displaystyle 0\end{array}\right)\)

This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{19}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,1)\cdot\left(\begin{array}{c}0\\1\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{19} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle \frac{y \left(9 y^{2} - 9 y + 2\right)}{2}\end{array}\right)\)

This DOF is associated with face 0 of the reference element.
  • \(R\) is the reference hexahedron. The following numbering of the subentities of the reference is used:
  • \(\mathcal{V}\) is spanned by: \(\left(\begin{array}{c}\displaystyle 1\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 1\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle x\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle y\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle y\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle y\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle z\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle z\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle z\end{array}\right)\)
  • \(\mathcal{L}=\{l_0,...,l_{11}\}\)
  • Functionals and basis functions:
\(\displaystyle l_{0}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,0,0)\cdot\left(\begin{array}{c}1\\0\\0\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{0} = \left(\begin{array}{c}\displaystyle - x - y - z + 1\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)

This DOF is associated with volume 0 of the reference element.
\(\displaystyle l_{1}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,0,0)\cdot\left(\begin{array}{c}0\\1\\0\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{1} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle - x - y - z + 1\\\displaystyle 0\end{array}\right)\)

This DOF is associated with volume 0 of the reference element.
\(\displaystyle l_{2}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,0,0)\cdot\left(\begin{array}{c}0\\0\\1\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{2} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle - x - y - z + 1\end{array}\right)\)

This DOF is associated with volume 0 of the reference element.
\(\displaystyle l_{3}:\boldsymbol{v}\mapsto\boldsymbol{v}(1,0,0)\cdot\left(\begin{array}{c}1\\0\\0\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{3} = \left(\begin{array}{c}\displaystyle x\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)

This DOF is associated with volume 0 of the reference element.
\(\displaystyle l_{4}:\boldsymbol{v}\mapsto\boldsymbol{v}(1,0,0)\cdot\left(\begin{array}{c}0\\1\\0\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{4} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle x\\\displaystyle 0\end{array}\right)\)

This DOF is associated with volume 0 of the reference element.
\(\displaystyle l_{5}:\boldsymbol{v}\mapsto\boldsymbol{v}(1,0,0)\cdot\left(\begin{array}{c}0\\0\\1\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{5} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x\end{array}\right)\)

This DOF is associated with volume 0 of the reference element.
\(\displaystyle l_{6}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,1,0)\cdot\left(\begin{array}{c}1\\0\\0\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{6} = \left(\begin{array}{c}\displaystyle y\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)

This DOF is associated with volume 0 of the reference element.
\(\displaystyle l_{7}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,1,0)\cdot\left(\begin{array}{c}0\\1\\0\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{7} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle y\\\displaystyle 0\end{array}\right)\)

This DOF is associated with volume 0 of the reference element.
\(\displaystyle l_{8}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,1,0)\cdot\left(\begin{array}{c}0\\0\\1\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{8} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle y\end{array}\right)\)

This DOF is associated with volume 0 of the reference element.
\(\displaystyle l_{9}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,0,1)\cdot\left(\begin{array}{c}1\\0\\0\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{9} = \left(\begin{array}{c}\displaystyle z\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)

This DOF is associated with volume 0 of the reference element.
\(\displaystyle l_{10}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,0,1)\cdot\left(\begin{array}{c}0\\1\\0\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{10} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle z\\\displaystyle 0\end{array}\right)\)

This DOF is associated with volume 0 of the reference element.
\(\displaystyle l_{11}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,0,1)\cdot\left(\begin{array}{c}0\\0\\1\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{11} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle z\end{array}\right)\)

This DOF is associated with volume 0 of the reference element.
  • \(R\) is the reference hexahedron. The following numbering of the subentities of the reference is used:
  • \(\mathcal{V}\) is spanned by: \(\left(\begin{array}{c}\displaystyle 1\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 1\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle x\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x^{2}\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle x^{2}\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x^{2}\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle y\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle y\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle y\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x y\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle x y\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x y\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle y^{2}\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle y^{2}\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle y^{2}\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle z\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle z\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle z\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x z\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle x z\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x z\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle y z\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle y z\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle y z\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle z^{2}\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle z^{2}\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle z^{2}\end{array}\right)\)
  • \(\mathcal{L}=\{l_0,...,l_{29}\}\)
  • Functionals and basis functions:
\(\displaystyle l_{0}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,0,0)\cdot\left(\begin{array}{c}1\\0\\0\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{0} = \left(\begin{array}{c}\displaystyle 2 x^{2} + 4 x y + 4 x z - 3 x + 2 y^{2} + 4 y z - 3 y + 2 z^{2} - 3 z + 1\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)

This DOF is associated with volume 0 of the reference element.
\(\displaystyle l_{1}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,0,0)\cdot\left(\begin{array}{c}0\\1\\0\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{1} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 2 x^{2} + 4 x y + 4 x z - 3 x + 2 y^{2} + 4 y z - 3 y + 2 z^{2} - 3 z + 1\\\displaystyle 0\end{array}\right)\)

This DOF is associated with volume 0 of the reference element.
\(\displaystyle l_{2}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,0,0)\cdot\left(\begin{array}{c}0\\0\\1\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{2} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 2 x^{2} + 4 x y + 4 x z - 3 x + 2 y^{2} + 4 y z - 3 y + 2 z^{2} - 3 z + 1\end{array}\right)\)

This DOF is associated with volume 0 of the reference element.
\(\displaystyle l_{3}:\boldsymbol{v}\mapsto\boldsymbol{v}(\tfrac{1}{2},0,0)\cdot\left(\begin{array}{c}1\\0\\0\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{3} = \left(\begin{array}{c}\displaystyle 4 x \left(- x - y - z + 1\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)

This DOF is associated with volume 0 of the reference element.
\(\displaystyle l_{4}:\boldsymbol{v}\mapsto\boldsymbol{v}(\tfrac{1}{2},0,0)\cdot\left(\begin{array}{c}0\\1\\0\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{4} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 4 x \left(- x - y - z + 1\right)\\\displaystyle 0\end{array}\right)\)

This DOF is associated with volume 0 of the reference element.
\(\displaystyle l_{5}:\boldsymbol{v}\mapsto\boldsymbol{v}(\tfrac{1}{2},0,0)\cdot\left(\begin{array}{c}0\\0\\1\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{5} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 4 x \left(- x - y - z + 1\right)\end{array}\right)\)

This DOF is associated with volume 0 of the reference element.
\(\displaystyle l_{6}:\boldsymbol{v}\mapsto\boldsymbol{v}(1,0,0)\cdot\left(\begin{array}{c}1\\0\\0\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{6} = \left(\begin{array}{c}\displaystyle x \left(2 x - 1\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)

This DOF is associated with volume 0 of the reference element.
\(\displaystyle l_{7}:\boldsymbol{v}\mapsto\boldsymbol{v}(1,0,0)\cdot\left(\begin{array}{c}0\\1\\0\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{7} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle x \left(2 x - 1\right)\\\displaystyle 0\end{array}\right)\)

This DOF is associated with volume 0 of the reference element.
\(\displaystyle l_{8}:\boldsymbol{v}\mapsto\boldsymbol{v}(1,0,0)\cdot\left(\begin{array}{c}0\\0\\1\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{8} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x \left(2 x - 1\right)\end{array}\right)\)

This DOF is associated with volume 0 of the reference element.
\(\displaystyle l_{9}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,\tfrac{1}{2},0)\cdot\left(\begin{array}{c}1\\0\\0\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{9} = \left(\begin{array}{c}\displaystyle 4 y \left(- x - y - z + 1\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)

This DOF is associated with volume 0 of the reference element.
\(\displaystyle l_{10}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,\tfrac{1}{2},0)\cdot\left(\begin{array}{c}0\\1\\0\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{10} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 4 y \left(- x - y - z + 1\right)\\\displaystyle 0\end{array}\right)\)

This DOF is associated with volume 0 of the reference element.
\(\displaystyle l_{11}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,\tfrac{1}{2},0)\cdot\left(\begin{array}{c}0\\0\\1\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{11} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 4 y \left(- x - y - z + 1\right)\end{array}\right)\)

This DOF is associated with volume 0 of the reference element.
\(\displaystyle l_{12}:\boldsymbol{v}\mapsto\boldsymbol{v}(\tfrac{1}{2},\tfrac{1}{2},0)\cdot\left(\begin{array}{c}1\\0\\0\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{12} = \left(\begin{array}{c}\displaystyle 4 x y\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)

This DOF is associated with volume 0 of the reference element.
\(\displaystyle l_{13}:\boldsymbol{v}\mapsto\boldsymbol{v}(\tfrac{1}{2},\tfrac{1}{2},0)\cdot\left(\begin{array}{c}0\\1\\0\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{13} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 4 x y\\\displaystyle 0\end{array}\right)\)

This DOF is associated with volume 0 of the reference element.
\(\displaystyle l_{14}:\boldsymbol{v}\mapsto\boldsymbol{v}(\tfrac{1}{2},\tfrac{1}{2},0)\cdot\left(\begin{array}{c}0\\0\\1\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{14} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 4 x y\end{array}\right)\)

This DOF is associated with volume 0 of the reference element.
\(\displaystyle l_{15}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,1,0)\cdot\left(\begin{array}{c}1\\0\\0\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{15} = \left(\begin{array}{c}\displaystyle y \left(2 y - 1\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)

This DOF is associated with volume 0 of the reference element.
\(\displaystyle l_{16}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,1,0)\cdot\left(\begin{array}{c}0\\1\\0\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{16} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle y \left(2 y - 1\right)\\\displaystyle 0\end{array}\right)\)

This DOF is associated with volume 0 of the reference element.
\(\displaystyle l_{17}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,1,0)\cdot\left(\begin{array}{c}0\\0\\1\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{17} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle y \left(2 y - 1\right)\end{array}\right)\)

This DOF is associated with volume 0 of the reference element.
\(\displaystyle l_{18}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,0,\tfrac{1}{2})\cdot\left(\begin{array}{c}1\\0\\0\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{18} = \left(\begin{array}{c}\displaystyle 4 z \left(- x - y - z + 1\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)

This DOF is associated with volume 0 of the reference element.
\(\displaystyle l_{19}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,0,\tfrac{1}{2})\cdot\left(\begin{array}{c}0\\1\\0\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{19} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 4 z \left(- x - y - z + 1\right)\\\displaystyle 0\end{array}\right)\)

This DOF is associated with volume 0 of the reference element.
\(\displaystyle l_{20}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,0,\tfrac{1}{2})\cdot\left(\begin{array}{c}0\\0\\1\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{20} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 4 z \left(- x - y - z + 1\right)\end{array}\right)\)

This DOF is associated with volume 0 of the reference element.
\(\displaystyle l_{21}:\boldsymbol{v}\mapsto\boldsymbol{v}(\tfrac{1}{2},0,\tfrac{1}{2})\cdot\left(\begin{array}{c}1\\0\\0\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{21} = \left(\begin{array}{c}\displaystyle 4 x z\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)

This DOF is associated with volume 0 of the reference element.
\(\displaystyle l_{22}:\boldsymbol{v}\mapsto\boldsymbol{v}(\tfrac{1}{2},0,\tfrac{1}{2})\cdot\left(\begin{array}{c}0\\1\\0\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{22} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 4 x z\\\displaystyle 0\end{array}\right)\)

This DOF is associated with volume 0 of the reference element.
\(\displaystyle l_{23}:\boldsymbol{v}\mapsto\boldsymbol{v}(\tfrac{1}{2},0,\tfrac{1}{2})\cdot\left(\begin{array}{c}0\\0\\1\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{23} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 4 x z\end{array}\right)\)

This DOF is associated with volume 0 of the reference element.
\(\displaystyle l_{24}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,\tfrac{1}{2},\tfrac{1}{2})\cdot\left(\begin{array}{c}1\\0\\0\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{24} = \left(\begin{array}{c}\displaystyle 4 y z\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)

This DOF is associated with volume 0 of the reference element.
\(\displaystyle l_{25}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,\tfrac{1}{2},\tfrac{1}{2})\cdot\left(\begin{array}{c}0\\1\\0\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{25} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 4 y z\\\displaystyle 0\end{array}\right)\)

This DOF is associated with volume 0 of the reference element.
\(\displaystyle l_{26}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,\tfrac{1}{2},\tfrac{1}{2})\cdot\left(\begin{array}{c}0\\0\\1\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{26} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 4 y z\end{array}\right)\)

This DOF is associated with volume 0 of the reference element.
\(\displaystyle l_{27}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,0,1)\cdot\left(\begin{array}{c}1\\0\\0\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{27} = \left(\begin{array}{c}\displaystyle z \left(2 z - 1\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)

This DOF is associated with volume 0 of the reference element.
\(\displaystyle l_{28}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,0,1)\cdot\left(\begin{array}{c}0\\1\\0\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{28} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle z \left(2 z - 1\right)\\\displaystyle 0\end{array}\right)\)

This DOF is associated with volume 0 of the reference element.
\(\displaystyle l_{29}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,0,1)\cdot\left(\begin{array}{c}0\\0\\1\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{29} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle z \left(2 z - 1\right)\end{array}\right)\)

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