Raviart–Thomas

\(\mathcal{P}_k=\operatorname{span}\left\{\prod_{i=1}^dx_i^{p_i}\middle|\sum_{i=1}^dp_i\leqslant k\right\}\)

\(\mathcal{Z}^{(26)}_k=\left\{\left(\begin{array}{c}px_1\\\vdots\\px_d\end{array}\right)\middle|p\in\hat{\mathcal{P}}_{k-1}\right\}\)

\(\hat{\mathcal{P}}_k=\operatorname{span}\left\{\prod_{i=1}^dx_i^{p_i}\middle|\sum_{i=1}^dp_i=k\right\}=\mathcal{P}_k\setminus\mathcal{P}_{k-1}\)

Before trying this example, you must install Basix:

pip3 install git+https://github.com/FEniCS/basix.git

This element can then be created with the following lines of Python:

import basix

# Create Raviart-Thomas (Lagrange variant) order 1 on a triangle
element = basix.create_element(basix.ElementFamily.RT, basix.CellType.triangle, 1)

# Create Raviart-Thomas (Lagrange variant) order 2 on a triangle
element = basix.create_element(basix.ElementFamily.RT, basix.CellType.triangle, 2)

# Create Raviart-Thomas (Lagrange variant) order 1 on a tetrahedron
element = basix.create_element(basix.ElementFamily.RT, basix.CellType.tetrahedron, 1)

# Create Raviart-Thomas (Lagrange variant) order 2 on a tetrahedron
element = basix.create_element(basix.ElementFamily.RT, basix.CellType.tetrahedron, 2)

# Create Raviart-Thomas (Legendre variant) order 1 on a triangle
element = basix.create_element(basix.ElementFamily.RT, basix.CellType.triangle, 1, lagrange_variant=basix.LagrangeVariant.legendre)

# Create Raviart-Thomas (Legendre variant) order 2 on a triangle
element = basix.create_element(basix.ElementFamily.RT, basix.CellType.triangle, 2, lagrange_variant=basix.LagrangeVariant.legendre)

# Create Raviart-Thomas (Legendre variant) order 1 on a tetrahedron
element = basix.create_element(basix.ElementFamily.RT, basix.CellType.tetrahedron, 1, lagrange_variant=basix.LagrangeVariant.legendre)

# Create Raviart-Thomas (Legendre variant) order 2 on a tetrahedron
element = basix.create_element(basix.ElementFamily.RT, basix.CellType.tetrahedron, 2, lagrange_variant=basix.LagrangeVariant.legendre)

Before trying this example, you must install Basix.UFL:

pip3 install git+https://github.com/FEniCS/basix.git git+https://github.com/FEniCS/ufl.git

This element can then be created with the following lines of Python:

import basix
import basix.ufl

# Create Raviart-Thomas (Lagrange variant) order 1 on a triangle
element = basix.ufl.element(basix.ElementFamily.RT, basix.CellType.triangle, 1, lagrange_variant=basix.LagrangeVariant.equispaced)

# Create Raviart-Thomas (Lagrange variant) order 2 on a triangle
element = basix.ufl.element(basix.ElementFamily.RT, basix.CellType.triangle, 2, lagrange_variant=basix.LagrangeVariant.equispaced)

# Create Raviart-Thomas (Lagrange variant) order 1 on a tetrahedron
element = basix.ufl.element(basix.ElementFamily.RT, basix.CellType.tetrahedron, 1, lagrange_variant=basix.LagrangeVariant.equispaced)

# Create Raviart-Thomas (Lagrange variant) order 2 on a tetrahedron
element = basix.ufl.element(basix.ElementFamily.RT, basix.CellType.tetrahedron, 2, lagrange_variant=basix.LagrangeVariant.equispaced)

# Create Raviart-Thomas (Legendre variant) order 1 on a triangle
element = basix.ufl.element(basix.ElementFamily.RT, basix.CellType.triangle, 1)

# Create Raviart-Thomas (Legendre variant) order 2 on a triangle
element = basix.ufl.element(basix.ElementFamily.RT, basix.CellType.triangle, 2)

# Create Raviart-Thomas (Legendre variant) order 1 on a tetrahedron
element = basix.ufl.element(basix.ElementFamily.RT, basix.CellType.tetrahedron, 1)

# Create Raviart-Thomas (Legendre variant) order 2 on a tetrahedron
element = basix.ufl.element(basix.ElementFamily.RT, basix.CellType.tetrahedron, 2)

Before trying this example, you must install Bempp:

pip3 install bempp-cl

This element can then be created with the following lines of Python:

import bempp.api
grid = bempp.api.shapes.regular_sphere(1)

Before trying this example, you must install FIAT:

pip3 install git+https://github.com/firedrakeproject/fiat.git

This element can then be created with the following lines of Python:

import FIAT

# Create Raviart-Thomas (Legendre variant) order 1
element = FIAT.RaviartThomas(FIAT.ufc_cell("triangle"), 1, variant="integral")

# Create Raviart-Thomas (Legendre variant) order 2
element = FIAT.RaviartThomas(FIAT.ufc_cell("triangle"), 2, variant="integral")

# Create Raviart-Thomas (Legendre variant) order 1
element = FIAT.RaviartThomas(FIAT.ufc_cell("tetrahedron"), 1, variant="integral")

# Create Raviart-Thomas (Legendre variant) order 2
element = FIAT.RaviartThomas(FIAT.ufc_cell("tetrahedron"), 2, variant="integral")

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 Raviart-Thomas (Lagrange variant) order 1 on a triangle
element = symfem.create_element("triangle", "N1div", 1)

# Create Raviart-Thomas (Lagrange variant) order 2 on a triangle
element = symfem.create_element("triangle", "N1div", 2)

# Create Raviart-Thomas (Lagrange variant) order 1 on a tetrahedron
element = symfem.create_element("tetrahedron", "N1div", 1)

# Create Raviart-Thomas (Lagrange variant) order 2 on a tetrahedron
element = symfem.create_element("tetrahedron", "N1div", 2)

# Create Raviart-Thomas (Legendre variant) order 1 on a triangle
element = symfem.create_element("triangle", "N1div", 1, variant="legendre")

# Create Raviart-Thomas (Legendre variant) order 2 on a triangle
element = symfem.create_element("triangle", "N1div", 2, variant="legendre")

# Create Raviart-Thomas (Legendre variant) order 1 on a tetrahedron
element = symfem.create_element("tetrahedron", "N1div", 1, variant="legendre")

# Create Raviart-Thomas (Legendre variant) order 2 on a tetrahedron
element = symfem.create_element("tetrahedron", "N1div", 2, variant="legendre")

Before trying this example, you must install (legacy) UFL:

pip3 install fenics-ufl-legacy

This element can then be created with the following lines of Python:

import ufl_legacy

# Create Raviart-Thomas (Lagrange variant) order 1 on a triangle
element = ufl_legacy.FiniteElement("RT", "triangle", 1)

# Create Raviart-Thomas (Lagrange variant) order 2 on a triangle
element = ufl_legacy.FiniteElement("RT", "triangle", 2)

# Create Raviart-Thomas (Lagrange variant) order 1 on a tetrahedron
element = ufl_legacy.FiniteElement("RT", "tetrahedron", 1)

# Create Raviart-Thomas (Lagrange variant) order 2 on a tetrahedron
element = ufl_legacy.FiniteElement("RT", "tetrahedron", 2)