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There are a bunch of more sophisticated splits that deliver spline-like elements of higher continuity. For an example on quads, Lai and Shumaker (2002) https://epubs.siam.org/doi/epdf/10.1137/S0036141000377134 covers construction of $C^r, r \ge 1$ spaces on triangularly subdivided quads (the four-way splitting you have as a diagram). That split is sometimes called a "Malkus split" or "Malkus quad".
Macroelements are defined by splitting the reference cell into smaller cells, then defining the polyset using piecewise polynomials on these subcells.
The hard assumption that basis functions are polynomial on the cell in Basix prevent these from being used currently.
There are two types of macro element that I am aware of:
1. Macro elements obtained by dividing each cell by splitting edges in half
For example: P1-iso-P2
The diagram for the tetrahedron is taken from Snoeijer, J., Numerical Reconstruction of Fundamental Solutions of the Stokes System with Finite Elements (with labels added)
2. Macro elements obtained by dividing cell by splitting angles in half
For example, Guzman-Neilan, HCT
I am yet to find a source that uses this type of macro element for a quad. Two possible dissections could be:
![image](https://private-user-images.githubusercontent.com/9850599/261271947-13f904ab-eebd-42be-9c47-91c5943a6406.png?jwt=eyJhbGciOiJIUzI1NiIsInR5cCI6IkpXVCJ9.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.vfxfyCUHryNNBCCnLjVzyQG4EnwKk-zsNB4LosUzRqE)
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