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Raviart–Thomas

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Alternative namesRao–Wilton–Glisson, \(\mathcal{P}^-_k\Lambda^{d-1}(\Delta_d)\)
Abbreviated namesRT, RWG
Orders\(1\leqslant k\)
Reference elementstriangle, tetrahedron
Polynomial set\(\mathcal{P}_{k-1}^d \oplus \mathcal{Z}^{(15)}_{k}\)
↓ Show polynomial set definitions ↓
DOFsOn each facet: normal integral moments with an order \(k-1\) Lagrange space
On the interior of the reference element: integral moments with an order \(k-2\) vector Lagrange space
Number of DOFstriangle: \(k(k+2)\) (A005563)
tetrahedron: \(k(k+1)(k+3)/2\) (A077414)
CategoriesVector-valued elements, H(div) conforming elements

Implementations

Symfem string"N1div"
↓ Show Symfem examples ↓
Basix string"Raviart-Thomas"
↓ Show Basix examples ↓
UFL string"RT"
↓ Show UFL examples ↓
Bempp string"RWG" (triangle)
↓ Show Bempp examples ↓

Examples

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

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

This DOF is associated with edge 0 of the reference element.
\(\displaystyle l_{1}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{1}}\boldsymbol{v}\cdot(1)\hat{\boldsymbol{n}}_{1}\)

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

This DOF is associated with edge 1 of the reference element.
\(\displaystyle l_{2}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{2}}\boldsymbol{v}\cdot(1)\hat{\boldsymbol{n}}_{2}\)

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

This DOF is associated with edge 2 of the reference element.
  • \(R\) is the reference triangle. The following numbering of the subentities of the reference is used:
  • \(\mathcal{V}\) is spanned by: \(\left(\begin{array}{c}\displaystyle 1\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle x\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle y\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle y\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x^{2}\\\displaystyle x y\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x y\\\displaystyle y^{2}\end{array}\right)\)
  • \(\mathcal{L}=\{l_0,...,l_{7}\}\)
  • Functionals and basis functions:
\(\displaystyle l_{0}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{0}}\boldsymbol{v}\cdot(1 - t_{0})\hat{\boldsymbol{n}}_{0}\)

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

This DOF is associated with edge 0 of the reference element.
\(\displaystyle l_{1}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{0}}\boldsymbol{v}\cdot(t_{0})\hat{\boldsymbol{n}}_{0}\)

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

This DOF is associated with edge 0 of the reference element.
\(\displaystyle l_{2}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{1}}\boldsymbol{v}\cdot(1 - t_{0})\hat{\boldsymbol{n}}_{1}\)

\(\displaystyle \boldsymbol{\phi}_{2} = \left(\begin{array}{c}\displaystyle - 8 x^{2} - 8 x y + 12 x + 6 y - 4\\\displaystyle 2 y \left(- 4 x - 4 y + 3\right)\end{array}\right)\)

This DOF is associated with edge 1 of the reference element.
\(\displaystyle l_{3}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{1}}\boldsymbol{v}\cdot(t_{0})\hat{\boldsymbol{n}}_{1}\)

\(\displaystyle \boldsymbol{\phi}_{3} = \left(\begin{array}{c}\displaystyle 8 x y - 2 x - 6 y + 2\\\displaystyle 4 y \left(2 y - 1\right)\end{array}\right)\)

This DOF is associated with edge 1 of the reference element.
\(\displaystyle l_{4}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{2}}\boldsymbol{v}\cdot(1 - t_{0})\hat{\boldsymbol{n}}_{2}\)

\(\displaystyle \boldsymbol{\phi}_{4} = \left(\begin{array}{c}\displaystyle 2 x \left(4 x + 4 y - 3\right)\\\displaystyle 8 x y - 6 x + 8 y^{2} - 12 y + 4\end{array}\right)\)

This DOF is associated with edge 2 of the reference element.
\(\displaystyle l_{5}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{2}}\boldsymbol{v}\cdot(t_{0})\hat{\boldsymbol{n}}_{2}\)

\(\displaystyle \boldsymbol{\phi}_{5} = \left(\begin{array}{c}\displaystyle 4 x \left(1 - 2 x\right)\\\displaystyle - 8 x y + 6 x + 2 y - 2\end{array}\right)\)

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

\(\displaystyle \boldsymbol{\phi}_{6} = \left(\begin{array}{c}\displaystyle 8 x \left(- 2 x - y + 2\right)\\\displaystyle 8 y \left(- 2 x - y + 1\right)\end{array}\right)\)

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

\(\displaystyle \boldsymbol{\phi}_{7} = \left(\begin{array}{c}\displaystyle 8 x \left(- x - 2 y + 1\right)\\\displaystyle 8 y \left(- x - 2 y + 2\right)\end{array}\right)\)

This DOF is associated with face 0 of the reference element.
  • \(R\) is the reference tetrahedron. The following numbering of the subentities of the reference is used:
  • \(\mathcal{V}\) is spanned by: \(\left(\begin{array}{c}\displaystyle 1\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 1\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x\\\displaystyle y\\\displaystyle z\end{array}\right)\)
  • \(\mathcal{L}=\{l_0,...,l_{3}\}\)
  • Functionals and basis functions:
\(\displaystyle l_{0}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{0}}\boldsymbol{v}\cdot(1)\hat{\boldsymbol{n}}_{0}\)

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

This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{1}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{1}}\boldsymbol{v}\cdot(1)\hat{\boldsymbol{n}}_{1}\)

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

This DOF is associated with face 1 of the reference element.
\(\displaystyle l_{2}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{2}}\boldsymbol{v}\cdot(1)\hat{\boldsymbol{n}}_{2}\)

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

This DOF is associated with face 2 of the reference element.
\(\displaystyle l_{3}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{3}}\boldsymbol{v}\cdot(1)\hat{\boldsymbol{n}}_{3}\)

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

This DOF is associated with face 3 of the reference element.
  • \(R\) is the reference tetrahedron. The following numbering of the subentities of the reference is used:
  • \(\mathcal{V}\) is spanned by: \(\left(\begin{array}{c}\displaystyle 1\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 1\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle x\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle y\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle y\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle y\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle z\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle z\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle z\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x^{2}\\\displaystyle x y\\\displaystyle x z\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x z\\\displaystyle y z\\\displaystyle z^{2}\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x y\\\displaystyle y^{2}\\\displaystyle y z\end{array}\right)\)
  • \(\mathcal{L}=\{l_0,...,l_{14}\}\)
  • Functionals and basis functions:
\(\displaystyle l_{0}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{0}}\boldsymbol{v}\cdot(- t_{0} - t_{1} + 1)\hat{\boldsymbol{n}}_{0}\)

\(\displaystyle \boldsymbol{\phi}_{0} = \left(\begin{array}{c}\displaystyle 6 x \left(5 x - 2\right)\\\displaystyle 6 y \left(5 x - 1\right)\\\displaystyle 6 z \left(5 x - 1\right)\end{array}\right)\)

This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{1}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{0}}\boldsymbol{v}\cdot(t_{0})\hat{\boldsymbol{n}}_{0}\)

\(\displaystyle \boldsymbol{\phi}_{1} = \left(\begin{array}{c}\displaystyle 6 x \left(5 y - 1\right)\\\displaystyle 6 y \left(5 y - 2\right)\\\displaystyle 6 z \left(5 y - 1\right)\end{array}\right)\)

This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{2}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{0}}\boldsymbol{v}\cdot(t_{1})\hat{\boldsymbol{n}}_{0}\)

\(\displaystyle \boldsymbol{\phi}_{2} = \left(\begin{array}{c}\displaystyle 6 x \left(5 z - 1\right)\\\displaystyle 6 y \left(5 z - 1\right)\\\displaystyle 6 z \left(5 z - 2\right)\end{array}\right)\)

This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{3}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{1}}\boldsymbol{v}\cdot(- t_{0} - t_{1} + 1)\hat{\boldsymbol{n}}_{1}\)

\(\displaystyle \boldsymbol{\phi}_{3} = \left(\begin{array}{c}\displaystyle 30 x^{2} + 30 x y + 30 x z - 48 x - 24 y - 24 z + 18\\\displaystyle 6 y \left(5 x + 5 y + 5 z - 4\right)\\\displaystyle 6 z \left(5 x + 5 y + 5 z - 4\right)\end{array}\right)\)

This DOF is associated with face 1 of the reference element.
\(\displaystyle l_{4}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{1}}\boldsymbol{v}\cdot(t_{0})\hat{\boldsymbol{n}}_{1}\)

\(\displaystyle \boldsymbol{\phi}_{4} = \left(\begin{array}{c}\displaystyle - 30 x y + 6 x + 24 y - 6\\\displaystyle 6 y \left(2 - 5 y\right)\\\displaystyle 6 z \left(1 - 5 y\right)\end{array}\right)\)

This DOF is associated with face 1 of the reference element.
\(\displaystyle l_{5}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{1}}\boldsymbol{v}\cdot(t_{1})\hat{\boldsymbol{n}}_{1}\)

\(\displaystyle \boldsymbol{\phi}_{5} = \left(\begin{array}{c}\displaystyle - 30 x z + 6 x + 24 z - 6\\\displaystyle 6 y \left(1 - 5 z\right)\\\displaystyle 6 z \left(2 - 5 z\right)\end{array}\right)\)

This DOF is associated with face 1 of the reference element.
\(\displaystyle l_{6}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{2}}\boldsymbol{v}\cdot(- t_{0} - t_{1} + 1)\hat{\boldsymbol{n}}_{2}\)

\(\displaystyle \boldsymbol{\phi}_{6} = \left(\begin{array}{c}\displaystyle 6 x \left(- 5 x - 5 y - 5 z + 4\right)\\\displaystyle - 30 x y + 24 x - 30 y^{2} - 30 y z + 48 y + 24 z - 18\\\displaystyle 6 z \left(- 5 x - 5 y - 5 z + 4\right)\end{array}\right)\)

This DOF is associated with face 2 of the reference element.
\(\displaystyle l_{7}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{2}}\boldsymbol{v}\cdot(t_{0})\hat{\boldsymbol{n}}_{2}\)

\(\displaystyle \boldsymbol{\phi}_{7} = \left(\begin{array}{c}\displaystyle 6 x \left(5 x - 2\right)\\\displaystyle 30 x y - 24 x - 6 y + 6\\\displaystyle 6 z \left(5 x - 1\right)\end{array}\right)\)

This DOF is associated with face 2 of the reference element.
\(\displaystyle l_{8}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{2}}\boldsymbol{v}\cdot(t_{1})\hat{\boldsymbol{n}}_{2}\)

\(\displaystyle \boldsymbol{\phi}_{8} = \left(\begin{array}{c}\displaystyle 6 x \left(5 z - 1\right)\\\displaystyle 30 y z - 6 y - 24 z + 6\\\displaystyle 6 z \left(5 z - 2\right)\end{array}\right)\)

This DOF is associated with face 2 of the reference element.
\(\displaystyle l_{9}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{3}}\boldsymbol{v}\cdot(- t_{0} - t_{1} + 1)\hat{\boldsymbol{n}}_{3}\)

\(\displaystyle \boldsymbol{\phi}_{9} = \left(\begin{array}{c}\displaystyle 6 x \left(5 x + 5 y + 5 z - 4\right)\\\displaystyle 6 y \left(5 x + 5 y + 5 z - 4\right)\\\displaystyle 30 x z - 24 x + 30 y z - 24 y + 30 z^{2} - 48 z + 18\end{array}\right)\)

This DOF is associated with face 3 of the reference element.
\(\displaystyle l_{10}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{3}}\boldsymbol{v}\cdot(t_{0})\hat{\boldsymbol{n}}_{3}\)

\(\displaystyle \boldsymbol{\phi}_{10} = \left(\begin{array}{c}\displaystyle 6 x \left(2 - 5 x\right)\\\displaystyle 6 y \left(1 - 5 x\right)\\\displaystyle - 30 x z + 24 x + 6 z - 6\end{array}\right)\)

This DOF is associated with face 3 of the reference element.
\(\displaystyle l_{11}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{3}}\boldsymbol{v}\cdot(t_{1})\hat{\boldsymbol{n}}_{3}\)

\(\displaystyle \boldsymbol{\phi}_{11} = \left(\begin{array}{c}\displaystyle 6 x \left(1 - 5 y\right)\\\displaystyle 6 y \left(2 - 5 y\right)\\\displaystyle - 30 y z + 24 y + 6 z - 6\end{array}\right)\)

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

\(\displaystyle \boldsymbol{\phi}_{12} = \left(\begin{array}{c}\displaystyle 30 x \left(- 2 x - y - z + 2\right)\\\displaystyle 30 y \left(- 2 x - y - z + 1\right)\\\displaystyle 30 z \left(- 2 x - y - z + 1\right)\end{array}\right)\)

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

\(\displaystyle \boldsymbol{\phi}_{13} = \left(\begin{array}{c}\displaystyle 30 x \left(- x - 2 y - z + 1\right)\\\displaystyle 30 y \left(- x - 2 y - z + 2\right)\\\displaystyle 30 z \left(- x - 2 y - z + 1\right)\end{array}\right)\)

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

\(\displaystyle \boldsymbol{\phi}_{14} = \left(\begin{array}{c}\displaystyle 30 x \left(- x - y - 2 z + 1\right)\\\displaystyle 30 y \left(- x - y - 2 z + 1\right)\\\displaystyle 30 z \left(- x - y - 2 z + 2\right)\end{array}\right)\)

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

References