from fenics import *
mesh = UnitSquareMesh(8,8)
V = FunctionSpace(mesh, 'P', 1)
print('done')

So then a function space V is opened… The mesh is part of the formulation… So far I take it as a LaGrange element of the first order of complexity… So then if one would like to check (mostly by hand.) how one gets formulated for a 2D and or a 3D shape what might I look to first? I looked a bit at source but maybe that isn’t the best place to first start.

In terms of the links provided… I noticed there is a polynomial set which at this time I would assume it represents a polynomial of some length and number of terms. So as a direct question it would be…

Is the polynomial used to build a function space V as in the code snippet, if so can you tell me a little bit more about how that gets done maybe with some LaTex?

(So far I also have a reference to Chapter 2. The weak form of a BVP pg. 36 Gochenbach, Google Play Books. There is a dot product of as I take it vector x and y.)

\vec{x} \cdot \vec{y} = \Sigma_{i=1}^n x_i y_i

He states the above which I have seen a similar type be used before used as a building a polynomial mechanism. So far that is a far as I got.

Your questions are not particularly clear and give me the impression that perhaps you should consider the fundamentals before diving into FEniCS. The DOLFINx tutorial should point you in the right direction. Tom Hughes’s book is a good point to start. I also like Claes Johnson’s book. The FEniCS book is very outdated but useful as a guide for what’s going on under the hood.