Serendipity

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

\(\mathcal{X}_k=\operatorname{span}\left\{\prod_{i=1}^dx_i^{p_i}\middle|k<\sum_{i=1}^da_i\leqslant k+\#\left\{i\in\{1,\dots,d\}\middle| a_i=1\right\}\right\}\)

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 serendipity order 1 on a interval
element = basix.create_element(basix.ElementFamily.serendipity, basix.CellType.interval, 1, lagrange_variant=basix.LagrangeVariant.equispaced, dpc_variant=basix.DPCVariant.simplex_equispaced)

# Create serendipity order 2 on a interval
element = basix.create_element(basix.ElementFamily.serendipity, basix.CellType.interval, 2, lagrange_variant=basix.LagrangeVariant.equispaced, dpc_variant=basix.DPCVariant.simplex_equispaced)

# Create serendipity order 3 on a interval
element = basix.create_element(basix.ElementFamily.serendipity, basix.CellType.interval, 3, lagrange_variant=basix.LagrangeVariant.equispaced, dpc_variant=basix.DPCVariant.simplex_equispaced)

# Create serendipity order 1 on a quadrilateral
element = basix.create_element(basix.ElementFamily.serendipity, basix.CellType.quadrilateral, 1, lagrange_variant=basix.LagrangeVariant.equispaced, dpc_variant=basix.DPCVariant.simplex_equispaced)

# Create serendipity order 2 on a quadrilateral
element = basix.create_element(basix.ElementFamily.serendipity, basix.CellType.quadrilateral, 2, lagrange_variant=basix.LagrangeVariant.equispaced, dpc_variant=basix.DPCVariant.simplex_equispaced)

# Create serendipity order 3 on a quadrilateral
element = basix.create_element(basix.ElementFamily.serendipity, basix.CellType.quadrilateral, 3, lagrange_variant=basix.LagrangeVariant.equispaced, dpc_variant=basix.DPCVariant.simplex_equispaced)

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 serendipity order 1 on a interval
element = symfem.create_element("interval", "serendipity", 1)

# Create serendipity order 2 on a interval
element = symfem.create_element("interval", "serendipity", 2)

# Create serendipity order 3 on a interval
element = symfem.create_element("interval", "serendipity", 3)

# Create serendipity order 1 on a quadrilateral
element = symfem.create_element("quadrilateral", "serendipity", 1)

# Create serendipity order 2 on a quadrilateral
element = symfem.create_element("quadrilateral", "serendipity", 2)

# Create serendipity order 3 on a quadrilateral
element = symfem.create_element("quadrilateral", "serendipity", 3)

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 serendipity order 1 on a interval
element = ufl.FiniteElement("S", "interval", 1)

# Create serendipity order 2 on a interval
element = ufl.FiniteElement("S", "interval", 2)

# Create serendipity order 3 on a interval
element = ufl.FiniteElement("S", "interval", 3)

# Create serendipity order 1 on a quadrilateral
element = ufl.FiniteElement("S", "quadrilateral", 1)

# Create serendipity order 2 on a quadrilateral
element = ufl.FiniteElement("S", "quadrilateral", 2)

# Create serendipity order 3 on a quadrilateral
element = ufl.FiniteElement("S", "quadrilateral", 3)