an encyclopedia of finite element definitions

Regge

 Orders $$1\leqslant k$$ Reference elements triangle, tetrahedron Polynomial set $$\mathcal{Z}^{(3)}_{k}$$↓ Show polynomial set definitions ↓ 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 ↓ Basix string "Regge"↓ Show Basix examples ↓ UFL string "Regge"↓ Show UFL examples ↓

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]