Approximation by piecwise linear functions, Accuracy Table. Triangle

General
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In the book Mark S. Gockenbach (Understanding the Finite Element Method). In Chapter 5, starting mostly 5.1.3 there is a heading Approximation by piecwise linear functions.

First Mark S. talks about the interpolant and how one can be used to estimate the smallest error. So in my attempt to understand this I looked in dolfinx’s book and I do see an area that does talk about an error analysis ( a priori is one).

So in terms of a triangle what I would like to have come out of dolfinx one thing is the interpolant of a triangle. The other thing ultimately that I would like to look for a triangle mesh is:

accuracy_table1
accuracy_table1704×310 34 KB

It does seem as though there should be a way to get these conclusions. It’s a bit of a long read through the whole thing so I thought maybe to try and get some help on this.

Is there some direct way with dolfinx to get the interpolant and the error table out directly or with some code way in python?

Where might it be stated for this type of calculation how that dolfinx does this?

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see for example Error control: Computing convergence rates — FEniCSx tutorial

Except instead of solving the FE problem, just compute the error of the interpolant in whichever norm you like on whichever mesh you like.

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To add to this comment there is also: Variants of Lagrange elements — DOLFINx 0.6.0 documentation
which looks at interpolation error between different variants of Lagrange elements.