Regge

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Orders	\(1\leqslant k\)
Reference elements	triangle, tetrahedron
Polynomial set	\(\mathcal{Z}^{(3)}_{k}\) ↓ Show polynomial set definitions ↓↑ Hide polynomial set definitions ↑ \(\mathcal{Z}^{(3)}_k=\left\{\mathbf{M}\in\mathcal{P}_{k}^{d\times d}\middle\|\mathbf{M}^t=\mathbf{M}\right\}\) \(\mathcal{P}_k=\operatorname{span}\left\{\prod_{i=1}^dx_i^{p_i}\middle\|\sum_{i=1}^dp_i\leqslant k\right\}\)
DOFs	On each edge: point evaluations of inner products with direction of edge On each face: point evaluations of inner products with direction of edges On each volume: point evaluations of inner products with direction of edges
Number of DOFs	triangle: \(3(k+1)(k+2)/2\) (A045943) tetrahedron: \((k+1)(k+2)(k+3)\) (A007531)
Categories	Matrix-valued elements

Implementations

Symfem string	`"Regge"` ↓ Show Symfem examples ↓↑ Hide Symfem examples ↑ Before trying this example, you must install Symfem: pip3 install symfem This element can then be created with the following lines of Python: import symfem # Create Regge order 1 on a triangle element = symfem.create_element("triangle", "Regge", 1) # Create Regge order 2 on a triangle element = symfem.create_element("triangle", "Regge", 2)
Basix string	`"Regge"` ↓ Show Basix examples ↓↑ Hide Basix examples ↑ Before trying this example, you must install Basix. This element can then be created with the following lines of Python: import basix # Create Regge order 1 on a triangle element = basix.create_element("Regge", "triangle", 1) # Create Regge order 2 on a triangle element = basix.create_element("Regge", "triangle", 2)
UFL string	`"Regge"` ↓ Show UFL examples ↓↑ Hide UFL examples ↑ Before trying this example, you must install UFL: pip3 install UFL This element can then be created with the following lines of Python: import ufl # Create Regge order 1 on a triangle element = ufl.FiniteElement("Regge", "triangle", 1) # Create Regge order 2 on a triangle element = ufl.FiniteElement("Regge", "triangle", 2)

Examples

triangle
order 1 triangle
order 2

\(R\) is the reference triangle. The following numbering of the subentities of the reference is used:

\(\mathcal{V}\) is spanned by: \(\left(\begin{matrix}1 & 0\\0 & 0\end{matrix}\right)\), \(\left(\begin{matrix}0 & 1\\1 & 0\end{matrix}\right)\), \(\left(\begin{matrix}0 & 0\\0 & 1\end{matrix}\right)\), \(\left(\begin{matrix}x & 0\\0 & 0\end{matrix}\right)\), \(\left(\begin{matrix}0 & x\\x & 0\end{matrix}\right)\), \(\left(\begin{matrix}0 & 0\\0 & x\end{matrix}\right)\), \(\left(\begin{matrix}y & 0\\0 & 0\end{matrix}\right)\), \(\left(\begin{matrix}0 & y\\y & 0\end{matrix}\right)\), \(\left(\begin{matrix}0 & 0\\0 & y\end{matrix}\right)\)
\(\mathcal{L}=\{l_0,...,l_{8}\}\)
Functionals and basis functions:

\(\displaystyle l_{0}:\mathbf{V}\mapsto\left(\begin{array}{c}-1\\1\end{array}\right)^{\text{t}}\mathbf{V}(\tfrac{2}{3},\tfrac{1}{3})\left(\begin{array}{c}-1\\1\end{array}\right)\)

\(\displaystyle \mathbf{\Phi}_{0} = \left(\begin{matrix}0 & \frac{1}{2} - \frac{3 x}{2}\\\frac{1}{2} - \frac{3 x}{2} & 0\end{matrix}\right)\)

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

\(\displaystyle l_{1}:\mathbf{V}\mapsto\left(\begin{array}{c}-1\\1\end{array}\right)^{\text{t}}\mathbf{V}(\tfrac{1}{3},\tfrac{2}{3})\left(\begin{array}{c}-1\\1\end{array}\right)\)

\(\displaystyle \mathbf{\Phi}_{1} = \left(\begin{matrix}0 & \frac{1}{2} - \frac{3 y}{2}\\\frac{1}{2} - \frac{3 y}{2} & 0\end{matrix}\right)\)

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

\(\displaystyle l_{2}:\mathbf{V}\mapsto\left(\begin{array}{c}0\\1\end{array}\right)^{\text{t}}\mathbf{V}(0,\tfrac{1}{3})\left(\begin{array}{c}0\\1\end{array}\right)\)

\(\displaystyle \mathbf{\Phi}_{2} = \left(\begin{matrix}0 & - \frac{3 x}{2} - \frac{3 y}{2} + 1\\- \frac{3 x}{2} - \frac{3 y}{2} + 1 & - 3 x - 3 y + 2\end{matrix}\right)\)

This DOF is associated with edge 1 of the reference element.

\(\displaystyle l_{3}:\mathbf{V}\mapsto\left(\begin{array}{c}0\\1\end{array}\right)^{\text{t}}\mathbf{V}(0,\tfrac{2}{3})\left(\begin{array}{c}0\\1\end{array}\right)\)

\(\displaystyle \mathbf{\Phi}_{3} = \left(\begin{matrix}0 & \frac{3 y}{2} - \frac{1}{2}\\\frac{3 y}{2} - \frac{1}{2} & 3 y - 1\end{matrix}\right)\)

This DOF is associated with edge 1 of the reference element.

\(\displaystyle l_{4}:\mathbf{V}\mapsto\left(\begin{array}{c}1\\0\end{array}\right)^{\text{t}}\mathbf{V}(\tfrac{1}{3},0)\left(\begin{array}{c}1\\0\end{array}\right)\)

\(\displaystyle \mathbf{\Phi}_{4} = \left(\begin{matrix}- 3 x - 3 y + 2 & - \frac{3 x}{2} - \frac{3 y}{2} + 1\\- \frac{3 x}{2} - \frac{3 y}{2} + 1 & 0\end{matrix}\right)\)

This DOF is associated with edge 2 of the reference element.

\(\displaystyle l_{5}:\mathbf{V}\mapsto\left(\begin{array}{c}1\\0\end{array}\right)^{\text{t}}\mathbf{V}(\tfrac{2}{3},0)\left(\begin{array}{c}1\\0\end{array}\right)\)

\(\displaystyle \mathbf{\Phi}_{5} = \left(\begin{matrix}3 x - 1 & \frac{3 x}{2} - \frac{1}{2}\\\frac{3 x}{2} - \frac{1}{2} & 0\end{matrix}\right)\)

This DOF is associated with edge 2 of the reference element.

\(\displaystyle l_{6}:\mathbf{V}\mapsto\left(\begin{array}{c}1\\0\end{array}\right)^{\text{t}}\mathbf{V}(\tfrac{1}{3},\tfrac{1}{3})\left(\begin{array}{c}1\\0\end{array}\right)\)

\(\displaystyle \mathbf{\Phi}_{6} = \left(\begin{matrix}3 y & \frac{3 y}{2}\\\frac{3 y}{2} & 0\end{matrix}\right)\)

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

\(\displaystyle l_{7}:\mathbf{V}\mapsto\left(\begin{array}{c}0\\1\end{array}\right)^{\text{t}}\mathbf{V}(\tfrac{1}{3},\tfrac{1}{3})\left(\begin{array}{c}0\\1\end{array}\right)\)

\(\displaystyle \mathbf{\Phi}_{7} = \left(\begin{matrix}0 & \frac{3 x}{2}\\\frac{3 x}{2} & 3 x\end{matrix}\right)\)

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

\(\displaystyle l_{8}:\mathbf{V}\mapsto\left(\begin{array}{c}-1\\1\end{array}\right)^{\text{t}}\mathbf{V}(\tfrac{1}{3},\tfrac{1}{3})\left(\begin{array}{c}-1\\1\end{array}\right)\)

\(\displaystyle \mathbf{\Phi}_{8} = \left(\begin{matrix}0 & \frac{3 x}{2} + \frac{3 y}{2} - \frac{3}{2}\\\frac{3 x}{2} + \frac{3 y}{2} - \frac{3}{2} & 0\end{matrix}\right)\)

This DOF is associated with face 0 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: \(\left(\begin{matrix}1 & 0\\0 & 0\end{matrix}\right)\), \(\left(\begin{matrix}0 & 1\\1 & 0\end{matrix}\right)\), \(\left(\begin{matrix}0 & 0\\0 & 1\end{matrix}\right)\), \(\left(\begin{matrix}x & 0\\0 & 0\end{matrix}\right)\), \(\left(\begin{matrix}0 & x\\x & 0\end{matrix}\right)\), \(\left(\begin{matrix}0 & 0\\0 & x\end{matrix}\right)\), \(\left(\begin{matrix}x^{2} & 0\\0 & 0\end{matrix}\right)\), \(\left(\begin{matrix}0 & x^{2}\\x^{2} & 0\end{matrix}\right)\), \(\left(\begin{matrix}0 & 0\\0 & x^{2}\end{matrix}\right)\), \(\left(\begin{matrix}y & 0\\0 & 0\end{matrix}\right)\), \(\left(\begin{matrix}0 & y\\y & 0\end{matrix}\right)\), \(\left(\begin{matrix}0 & 0\\0 & y\end{matrix}\right)\), \(\left(\begin{matrix}x y & 0\\0 & 0\end{matrix}\right)\), \(\left(\begin{matrix}0 & x y\\x y & 0\end{matrix}\right)\), \(\left(\begin{matrix}0 & 0\\0 & x y\end{matrix}\right)\), \(\left(\begin{matrix}y^{2} & 0\\0 & 0\end{matrix}\right)\), \(\left(\begin{matrix}0 & y^{2}\\y^{2} & 0\end{matrix}\right)\), \(\left(\begin{matrix}0 & 0\\0 & y^{2}\end{matrix}\right)\)
\(\mathcal{L}=\{l_0,...,l_{17}\}\)
Functionals and basis functions:

\(\displaystyle l_{0}:\mathbf{V}\mapsto\left(\begin{array}{c}-1\\1\end{array}\right)^{\text{t}}\mathbf{V}(\tfrac{3}{4},\tfrac{1}{4})\left(\begin{array}{c}-1\\1\end{array}\right)\)

\(\displaystyle \mathbf{\Phi}_{0} = \left(\begin{matrix}0 & - 4 x^{2} + 3 x - \frac{1}{2}\\- 4 x^{2} + 3 x - \frac{1}{2} & 0\end{matrix}\right)\)

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

\(\displaystyle l_{1}:\mathbf{V}\mapsto\left(\begin{array}{c}-1\\1\end{array}\right)^{\text{t}}\mathbf{V}(\tfrac{1}{2},\tfrac{1}{2})\left(\begin{array}{c}-1\\1\end{array}\right)\)

\(\displaystyle \mathbf{\Phi}_{1} = \left(\begin{matrix}0 & - 8 x y + 2 x + 2 y - \frac{1}{2}\\- 8 x y + 2 x + 2 y - \frac{1}{2} & 0\end{matrix}\right)\)

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

\(\displaystyle l_{2}:\mathbf{V}\mapsto\left(\begin{array}{c}-1\\1\end{array}\right)^{\text{t}}\mathbf{V}(\tfrac{1}{4},\tfrac{3}{4})\left(\begin{array}{c}-1\\1\end{array}\right)\)

\(\displaystyle \mathbf{\Phi}_{2} = \left(\begin{matrix}0 & - 4 y^{2} + 3 y - \frac{1}{2}\\- 4 y^{2} + 3 y - \frac{1}{2} & 0\end{matrix}\right)\)

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

\(\displaystyle l_{3}:\mathbf{V}\mapsto\left(\begin{array}{c}0\\1\end{array}\right)^{\text{t}}\mathbf{V}(0,\tfrac{1}{4})\left(\begin{array}{c}0\\1\end{array}\right)\)

\(\displaystyle \mathbf{\Phi}_{3} = \left(\begin{matrix}0 & 4 x^{2} + 8 x y - 5 x + 4 y^{2} - 5 y + \frac{3}{2}\\4 x^{2} + 8 x y - 5 x + 4 y^{2} - 5 y + \frac{3}{2} & 8 x^{2} + 16 x y - 10 x + 8 y^{2} - 10 y + 3\end{matrix}\right)\)

This DOF is associated with edge 1 of the reference element.

\(\displaystyle l_{4}:\mathbf{V}\mapsto\left(\begin{array}{c}0\\1\end{array}\right)^{\text{t}}\mathbf{V}(0,\tfrac{1}{2})\left(\begin{array}{c}0\\1\end{array}\right)\)

\(\displaystyle \mathbf{\Phi}_{4} = \left(\begin{matrix}0 & - 8 x y + 2 x - 8 y^{2} + 8 y - \frac{3}{2}\\- 8 x y + 2 x - 8 y^{2} + 8 y - \frac{3}{2} & - 16 x y + 4 x - 16 y^{2} + 16 y - 3\end{matrix}\right)\)

This DOF is associated with edge 1 of the reference element.

\(\displaystyle l_{5}:\mathbf{V}\mapsto\left(\begin{array}{c}0\\1\end{array}\right)^{\text{t}}\mathbf{V}(0,\tfrac{3}{4})\left(\begin{array}{c}0\\1\end{array}\right)\)

\(\displaystyle \mathbf{\Phi}_{5} = \left(\begin{matrix}0 & 4 y^{2} - 3 y + \frac{1}{2}\\4 y^{2} - 3 y + \frac{1}{2} & 8 y^{2} - 6 y + 1\end{matrix}\right)\)

This DOF is associated with edge 1 of the reference element.

\(\displaystyle l_{6}:\mathbf{V}\mapsto\left(\begin{array}{c}1\\0\end{array}\right)^{\text{t}}\mathbf{V}(\tfrac{1}{4},0)\left(\begin{array}{c}1\\0\end{array}\right)\)

\(\displaystyle \mathbf{\Phi}_{6} = \left(\begin{matrix}8 x^{2} + 16 x y - 10 x + 8 y^{2} - 10 y + 3 & 4 x^{2} + 8 x y - 5 x + 4 y^{2} - 5 y + \frac{3}{2}\\4 x^{2} + 8 x y - 5 x + 4 y^{2} - 5 y + \frac{3}{2} & 0\end{matrix}\right)\)

This DOF is associated with edge 2 of the reference element.

\(\displaystyle l_{7}:\mathbf{V}\mapsto\left(\begin{array}{c}1\\0\end{array}\right)^{\text{t}}\mathbf{V}(\tfrac{1}{2},0)\left(\begin{array}{c}1\\0\end{array}\right)\)

\(\displaystyle \mathbf{\Phi}_{7} = \left(\begin{matrix}- 16 x^{2} - 16 x y + 16 x + 4 y - 3 & - 8 x^{2} - 8 x y + 8 x + 2 y - \frac{3}{2}\\- 8 x^{2} - 8 x y + 8 x + 2 y - \frac{3}{2} & 0\end{matrix}\right)\)

This DOF is associated with edge 2 of the reference element.

\(\displaystyle l_{8}:\mathbf{V}\mapsto\left(\begin{array}{c}1\\0\end{array}\right)^{\text{t}}\mathbf{V}(\tfrac{3}{4},0)\left(\begin{array}{c}1\\0\end{array}\right)\)

\(\displaystyle \mathbf{\Phi}_{8} = \left(\begin{matrix}8 x^{2} - 6 x + 1 & 4 x^{2} - 3 x + \frac{1}{2}\\4 x^{2} - 3 x + \frac{1}{2} & 0\end{matrix}\right)\)

This DOF is associated with edge 2 of the reference element.

\(\displaystyle l_{9}:\mathbf{V}\mapsto\left(\begin{array}{c}1\\0\end{array}\right)^{\text{t}}\mathbf{V}(\tfrac{1}{4},\tfrac{1}{4})\left(\begin{array}{c}1\\0\end{array}\right)\)

\(\displaystyle \mathbf{\Phi}_{9} = \left(\begin{matrix}4 y \left(- 4 x - 4 y + 3\right) & 2 y \left(- 4 x - 4 y + 3\right)\\2 y \left(- 4 x - 4 y + 3\right) & 0\end{matrix}\right)\)

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

\(\displaystyle l_{10}:\mathbf{V}\mapsto\left(\begin{array}{c}0\\1\end{array}\right)^{\text{t}}\mathbf{V}(\tfrac{1}{4},\tfrac{1}{4})\left(\begin{array}{c}0\\1\end{array}\right)\)

\(\displaystyle \mathbf{\Phi}_{10} = \left(\begin{matrix}0 & 2 x \left(- 4 x - 4 y + 3\right)\\2 x \left(- 4 x - 4 y + 3\right) & 4 x \left(- 4 x - 4 y + 3\right)\end{matrix}\right)\)

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

\(\displaystyle l_{11}:\mathbf{V}\mapsto\left(\begin{array}{c}-1\\1\end{array}\right)^{\text{t}}\mathbf{V}(\tfrac{1}{4},\tfrac{1}{4})\left(\begin{array}{c}-1\\1\end{array}\right)\)

\(\displaystyle \mathbf{\Phi}_{11} = \left(\begin{matrix}0 & - 4 x^{2} - 8 x y + 7 x - 4 y^{2} + 7 y - 3\\- 4 x^{2} - 8 x y + 7 x - 4 y^{2} + 7 y - 3 & 0\end{matrix}\right)\)

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

\(\displaystyle l_{12}:\mathbf{V}\mapsto\left(\begin{array}{c}1\\0\end{array}\right)^{\text{t}}\mathbf{V}(\tfrac{1}{2},\tfrac{1}{4})\left(\begin{array}{c}1\\0\end{array}\right)\)

\(\displaystyle \mathbf{\Phi}_{12} = \left(\begin{matrix}4 y \left(4 x - 1\right) & 2 y \left(4 x - 1\right)\\2 y \left(4 x - 1\right) & 0\end{matrix}\right)\)

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

\(\displaystyle l_{13}:\mathbf{V}\mapsto\left(\begin{array}{c}0\\1\end{array}\right)^{\text{t}}\mathbf{V}(\tfrac{1}{2},\tfrac{1}{4})\left(\begin{array}{c}0\\1\end{array}\right)\)

\(\displaystyle \mathbf{\Phi}_{13} = \left(\begin{matrix}0 & x \left(4 x - 1\right)\\x \left(4 x - 1\right) & 2 x \left(4 x - 1\right)\end{matrix}\right)\)

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

\(\displaystyle l_{14}:\mathbf{V}\mapsto\left(\begin{array}{c}-1\\1\end{array}\right)^{\text{t}}\mathbf{V}(\tfrac{1}{2},\tfrac{1}{4})\left(\begin{array}{c}-1\\1\end{array}\right)\)

\(\displaystyle \mathbf{\Phi}_{14} = \left(\begin{matrix}0 & 8 x^{2} + 8 x y - 10 x - 2 y + 2\\8 x^{2} + 8 x y - 10 x - 2 y + 2 & 0\end{matrix}\right)\)

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

\(\displaystyle l_{15}:\mathbf{V}\mapsto\left(\begin{array}{c}1\\0\end{array}\right)^{\text{t}}\mathbf{V}(\tfrac{1}{4},\tfrac{1}{2})\left(\begin{array}{c}1\\0\end{array}\right)\)

\(\displaystyle \mathbf{\Phi}_{15} = \left(\begin{matrix}2 y \left(4 y - 1\right) & y \left(4 y - 1\right)\\y \left(4 y - 1\right) & 0\end{matrix}\right)\)

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

\(\displaystyle l_{16}:\mathbf{V}\mapsto\left(\begin{array}{c}0\\1\end{array}\right)^{\text{t}}\mathbf{V}(\tfrac{1}{4},\tfrac{1}{2})\left(\begin{array}{c}0\\1\end{array}\right)\)

\(\displaystyle \mathbf{\Phi}_{16} = \left(\begin{matrix}0 & 2 x \left(4 y - 1\right)\\2 x \left(4 y - 1\right) & 4 x \left(4 y - 1\right)\end{matrix}\right)\)

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

\(\displaystyle l_{17}:\mathbf{V}\mapsto\left(\begin{array}{c}-1\\1\end{array}\right)^{\text{t}}\mathbf{V}(\tfrac{1}{4},\tfrac{1}{2})\left(\begin{array}{c}-1\\1\end{array}\right)\)

\(\displaystyle \mathbf{\Phi}_{17} = \left(\begin{matrix}0 & 8 x y - 2 x + 8 y^{2} - 10 y + 2\\8 x y - 2 x + 8 y^{2} - 10 y + 2 & 0\end{matrix}\right)\)

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

References

Regge, T. General relativity without coordinates, Il Nuovo Cimento 19(3), 558–571, 1961. [DOI: 10.1007/BF02733251] [BibTeX]
Christiansen, S. H. On the linearization of Regge calculus, Numerische Mathematik 119(4), 613–640, 2011. [DOI: 10.1007/s00211-011-0394-z] [BibTeX]