an encyclopedia of finite element definitions

# Bernstein

 Alternative names Bernstein–Bézier Orders $$0\leqslant k$$ Reference elements interval, triangle, tetrahedron Polynomial set $$\mathcal{P}_{k}$$↓ Show polynomial set definitions ↓ DOFs On the interior of the reference element: integral moments with order $$k$$ Bernstein polynomials Number of DOFs interval: $$k+1$$ (A000027)triangle: $$(k+1)(k+2)/2$$ (A000217)tetrahedron: $$(k+1)(k+2)(k+3)/6$$ (A000292) Categories Scalar-valued elements

## Implementations

 Symfem "Bernstein"↓ Show Symfem examples ↓ UFL "Bernstein"↓ Show UFL examples ↓

## Examples

interval
order 1
interval
order 2
interval
order 3
triangle
order 1
triangle
order 2
triangle
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:
$$\displaystyle l_{0}:\mathbf{V}\mapsto\displaystyle\int_{R}(s_{0})v$$
where $$R$$ is the reference element;
and $$s_{0}$$ is a parametrisation of $$R(1)$$.

$$\displaystyle \phi_{0} = 6 x - 2$$

This DOF is associated with edge 0 of the reference element.
$$\displaystyle l_{1}:\mathbf{V}\mapsto\displaystyle\int_{R}(1 - s_{0})v$$
where $$R$$ is the reference element;
and $$s_{0}$$ is a parametrisation of $$R(1)$$.

$$\displaystyle \phi_{1} = 4 - 6 x$$

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:
$$\displaystyle l_{0}:\mathbf{V}\mapsto\displaystyle\int_{R}(s_{0}^{2})v$$
where $$R$$ is the reference element;
and $$s_{0}$$ is a parametrisation of $$R(1)$$.

$$\displaystyle \phi_{0} = 30 x^{2} - 24 x + 3$$

This DOF is associated with edge 0 of the reference element.
$$\displaystyle l_{1}:\mathbf{V}\mapsto\displaystyle\int_{R}(2 s_{0} \left(1 - s_{0}\right))v$$
where $$R$$ is the reference element;
and $$s_{0}$$ is a parametrisation of $$R(1)$$.

$$\displaystyle \phi_{1} = - 60 x^{2} + 60 x - 9$$

This DOF is associated with edge 0 of the reference element.
$$\displaystyle l_{2}:\mathbf{V}\mapsto\displaystyle\int_{R}(s_{0}^{2} - 2 s_{0} + 1)v$$
where $$R$$ is the reference element;
and $$s_{0}$$ is a parametrisation of $$R(1)$$.

$$\displaystyle \phi_{2} = 30 x^{2} - 36 x + 9$$

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:
$$\displaystyle l_{0}:\mathbf{V}\mapsto\displaystyle\int_{R}(s_{0}^{3})v$$
where $$R$$ is the reference element;
and $$s_{0}$$ is a parametrisation of $$R(1)$$.

$$\displaystyle \phi_{0} = 140 x^{3} - 180 x^{2} + 60 x - 4$$

This DOF is associated with edge 0 of the reference element.
$$\displaystyle l_{1}:\mathbf{V}\mapsto\displaystyle\int_{R}(3 s_{0}^{2} \left(1 - s_{0}\right))v$$
where $$R$$ is the reference element;
and $$s_{0}$$ is a parametrisation of $$R(1)$$.

$$\displaystyle \phi_{1} = - 420 x^{3} + 600 x^{2} - 220 x + 16$$

This DOF is associated with edge 0 of the reference element.
$$\displaystyle l_{2}:\mathbf{V}\mapsto\displaystyle\int_{R}(3 s_{0} \left(s_{0}^{2} - 2 s_{0} + 1\right))v$$
where $$R$$ is the reference element;
and $$s_{0}$$ is a parametrisation of $$R(1)$$.

$$\displaystyle \phi_{2} = 420 x^{3} - 660 x^{2} + 280 x - 24$$

This DOF is associated with edge 0 of the reference element.
$$\displaystyle l_{3}:\mathbf{V}\mapsto\displaystyle\int_{R}(- s_{0}^{3} + 3 s_{0}^{2} - 3 s_{0} + 1)v$$
where $$R$$ is the reference element;
and $$s_{0}$$ is a parametrisation of $$R(1)$$.

$$\displaystyle \phi_{3} = - 140 x^{3} + 240 x^{2} - 120 x + 16$$

This DOF is associated with edge 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: $$1$$, $$x$$, $$y$$
• $$\mathcal{L}=\{l_0,...,l_{2}\}$$
• Functionals and basis functions:
$$\displaystyle l_{0}:\mathbf{V}\mapsto\displaystyle\int_{R}(s_{1})v$$
where $$R$$ is the reference element;
and $$(s_{0},s_{1})$$ is a parametrisation of $$R(2)$$.

$$\displaystyle \phi_{0} = 24 y - 6$$

This DOF is associated with face 0 of the reference element.
$$\displaystyle l_{1}:\mathbf{V}\mapsto\displaystyle\int_{R}(s_{0})v$$
where $$R$$ is the reference element;
and $$(s_{0},s_{1})$$ is a parametrisation of $$R(2)$$.

$$\displaystyle \phi_{1} = 24 x - 6$$

This DOF is associated with face 0 of the reference element.
$$\displaystyle l_{2}:\mathbf{V}\mapsto\displaystyle\int_{R}(- s_{0} - s_{1} + 1)v$$
where $$R$$ is the reference element;
and $$(s_{0},s_{1})$$ is a parametrisation of $$R(2)$$.

$$\displaystyle \phi_{2} = - 24 x - 24 y + 18$$

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: $$1$$, $$x$$, $$x^{2}$$, $$y$$, $$x y$$, $$y^{2}$$
• $$\mathcal{L}=\{l_0,...,l_{5}\}$$
• Functionals and basis functions:
$$\displaystyle l_{0}:\mathbf{V}\mapsto\displaystyle\int_{R}(s_{1}^{2})v$$
where $$R$$ is the reference element;
and $$(s_{0},s_{1})$$ is a parametrisation of $$R(2)$$.

$$\displaystyle \phi_{0} = 180 y^{2} - 120 y + 12$$

This DOF is associated with face 0 of the reference element.
$$\displaystyle l_{1}:\mathbf{V}\mapsto\displaystyle\int_{R}(2 s_{0} s_{1})v$$
where $$R$$ is the reference element;
and $$(s_{0},s_{1})$$ is a parametrisation of $$R(2)$$.

$$\displaystyle \phi_{1} = 360 x y - 60 x - 60 y + 12$$

This DOF is associated with face 0 of the reference element.
$$\displaystyle l_{2}:\mathbf{V}\mapsto\displaystyle\int_{R}(s_{0}^{2})v$$
where $$R$$ is the reference element;
and $$(s_{0},s_{1})$$ is a parametrisation of $$R(2)$$.

$$\displaystyle \phi_{2} = 180 x^{2} - 120 x + 12$$

This DOF is associated with face 0 of the reference element.
$$\displaystyle l_{3}:\mathbf{V}\mapsto\displaystyle\int_{R}(2 s_{1} \left(- s_{0} - s_{1} + 1\right))v$$
where $$R$$ is the reference element;
and $$(s_{0},s_{1})$$ is a parametrisation of $$R(2)$$.

$$\displaystyle \phi_{3} = - 360 x y + 60 x - 360 y^{2} + 360 y - 48$$

This DOF is associated with face 0 of the reference element.
$$\displaystyle l_{4}:\mathbf{V}\mapsto\displaystyle\int_{R}(2 s_{0} \left(- s_{0} - s_{1} + 1\right))v$$
where $$R$$ is the reference element;
and $$(s_{0},s_{1})$$ is a parametrisation of $$R(2)$$.

$$\displaystyle \phi_{4} = - 360 x^{2} - 360 x y + 360 x + 60 y - 48$$

This DOF is associated with face 0 of the reference element.
$$\displaystyle l_{5}:\mathbf{V}\mapsto\displaystyle\int_{R}(s_{0}^{2} + 2 s_{0} s_{1} - 2 s_{0} + s_{1}^{2} - 2 s_{1} + 1)v$$
where $$R$$ is the reference element;
and $$(s_{0},s_{1})$$ is a parametrisation of $$R(2)$$.

$$\displaystyle \phi_{5} = 180 x^{2} + 360 x y - 240 x + 180 y^{2} - 240 y + 72$$

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: $$1$$, $$x$$, $$x^{2}$$, $$x^{3}$$, $$y$$, $$x y$$, $$x^{2} y$$, $$y^{2}$$, $$x y^{2}$$, $$y^{3}$$
• $$\mathcal{L}=\{l_0,...,l_{9}\}$$
• Functionals and basis functions:
$$\displaystyle l_{0}:\mathbf{V}\mapsto\displaystyle\int_{R}(s_{1}^{3})v$$
where $$R$$ is the reference element;
and $$(s_{0},s_{1})$$ is a parametrisation of $$R(2)$$.

$$\displaystyle \phi_{0} = 1120 y^{3} - 1260 y^{2} + 360 y - 20$$

This DOF is associated with face 0 of the reference element.
$$\displaystyle l_{1}:\mathbf{V}\mapsto\displaystyle\int_{R}(3 s_{0} s_{1}^{2})v$$
where $$R$$ is the reference element;
and $$(s_{0},s_{1})$$ is a parametrisation of $$R(2)$$.

$$\displaystyle \phi_{1} = 3360 x y^{2} - 1680 x y + 120 x - 420 y^{2} + 240 y - 20$$

This DOF is associated with face 0 of the reference element.
$$\displaystyle l_{2}:\mathbf{V}\mapsto\displaystyle\int_{R}(3 s_{0}^{2} s_{1})v$$
where $$R$$ is the reference element;
and $$(s_{0},s_{1})$$ is a parametrisation of $$R(2)$$.

$$\displaystyle \phi_{2} = 3360 x^{2} y - 420 x^{2} - 1680 x y + 240 x + 120 y - 20$$

This DOF is associated with face 0 of the reference element.
$$\displaystyle l_{3}:\mathbf{V}\mapsto\displaystyle\int_{R}(s_{0}^{3})v$$
where $$R$$ is the reference element;
and $$(s_{0},s_{1})$$ is a parametrisation of $$R(2)$$.

$$\displaystyle \phi_{3} = 1120 x^{3} - 1260 x^{2} + 360 x - 20$$

This DOF is associated with face 0 of the reference element.
$$\displaystyle l_{4}:\mathbf{V}\mapsto\displaystyle\int_{R}(3 s_{1}^{2} \left(- s_{0} - s_{1} + 1\right))v$$
where $$R$$ is the reference element;
and $$(s_{0},s_{1})$$ is a parametrisation of $$R(2)$$.

$$\displaystyle \phi_{4} = - 3360 x y^{2} + 1680 x y - 120 x - 3360 y^{3} + 4620 y^{2} - 1560 y + 100$$

This DOF is associated with face 0 of the reference element.
$$\displaystyle l_{5}:\mathbf{V}\mapsto\displaystyle\int_{R}(6 s_{0} s_{1} \left(- s_{0} - s_{1} + 1\right))v$$
where $$R$$ is the reference element;
and $$(s_{0},s_{1})$$ is a parametrisation of $$R(2)$$.

$$\displaystyle \phi_{5} = - 6720 x^{2} y + 840 x^{2} - 6720 x y^{2} + 7560 x y - 840 x + 840 y^{2} - 840 y + 100$$

This DOF is associated with face 0 of the reference element.
$$\displaystyle l_{6}:\mathbf{V}\mapsto\displaystyle\int_{R}(3 s_{0}^{2} \left(- s_{0} - s_{1} + 1\right))v$$
where $$R$$ is the reference element;
and $$(s_{0},s_{1})$$ is a parametrisation of $$R(2)$$.

$$\displaystyle \phi_{6} = - 3360 x^{3} - 3360 x^{2} y + 4620 x^{2} + 1680 x y - 1560 x - 120 y + 100$$

This DOF is associated with face 0 of the reference element.
$$\displaystyle l_{7}:\mathbf{V}\mapsto\displaystyle\int_{R}(3 s_{1} \left(s_{0}^{2} + 2 s_{0} s_{1} - 2 s_{0} + s_{1}^{2} - 2 s_{1} + 1\right))v$$
where $$R$$ is the reference element;
and $$(s_{0},s_{1})$$ is a parametrisation of $$R(2)$$.

$$\displaystyle \phi_{7} = 3360 x^{2} y - 420 x^{2} + 6720 x y^{2} - 5880 x y + 600 x + 3360 y^{3} - 5460 y^{2} + 2400 y - 200$$

This DOF is associated with face 0 of the reference element.
$$\displaystyle l_{8}:\mathbf{V}\mapsto\displaystyle\int_{R}(3 s_{0} \left(s_{0}^{2} + 2 s_{0} s_{1} - 2 s_{0} + s_{1}^{2} - 2 s_{1} + 1\right))v$$
where $$R$$ is the reference element;
and $$(s_{0},s_{1})$$ is a parametrisation of $$R(2)$$.

$$\displaystyle \phi_{8} = 3360 x^{3} + 6720 x^{2} y - 5460 x^{2} + 3360 x y^{2} - 5880 x y + 2400 x - 420 y^{2} + 600 y - 200$$

This DOF is associated with face 0 of the reference element.
$$\displaystyle l_{9}:\mathbf{V}\mapsto\displaystyle\int_{R}(- s_{0}^{3} - 3 s_{0}^{2} s_{1} + 3 s_{0}^{2} - 3 s_{0} s_{1}^{2} + 6 s_{0} s_{1} - 3 s_{0} - s_{1}^{3} + 3 s_{1}^{2} - 3 s_{1} + 1)v$$
where $$R$$ is the reference element;
and $$(s_{0},s_{1})$$ is a parametrisation of $$R(2)$$.

$$\displaystyle \phi_{9} = - 1120 x^{3} - 3360 x^{2} y + 2100 x^{2} - 3360 x y^{2} + 4200 x y - 1200 x - 1120 y^{3} + 2100 y^{2} - 1200 y + 200$$

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

## DefElement stats

 Element added 20 February 2021 Element last updated 05 June 2021