Can you explain why your code does produce a conformal map? I don't understand arguments from the link. TIA.

Actually this is numerical model, therefore it could be conformal in exact arithmetic only. In the real case as above we have some numerical approximation to conformal map. In this example, you can see that only one corner of the triangle is reproduced quite accurately. To reproduce the other 2 angles, you need to increase the number of points in certain places of reg2.

@AlexTrounnev: Thank you. I still don't understand why "we have some numerical approximation to conformal map". Any picture is not a serious argument.

@azerbajdzan Do you mean to map mesh to see how it preserves angles?

@AlexTrounnev: Sorry, why is f1 an approximation of a conformal map from reg1 to the unit disk?

@user64494 We use Riemann mapping theorem to compute f1,f2`. Your question probably concerns the corners of the triangle, near which the solution to Laplace's equation should be singular. Is it correct?

@AlexTrounnev: How do we use Riemann mapping theorem to compute f1? Please give us details. No, my request does not concern the corners of the triangle.

@azerbajdzan Please pay attention that in this example we use mesh of 4830 triangle elements. But even this number is not enough to reproduce the border of the triangle reg1 using f2 and inverse f1. If we take a small number of triangular elements then the display will not even look like the desired triangle reg1. Apparently, we can only theoretically assume that angles are preserved with such a map.

@user64494 Do you want me to retell the post that Heinrich Schumacher wrote? :)

@AlexTrounnev: As I wrote I don't understand the argument from the link. As I understand it, you, following the linked answer by Henrik Schumacher, solve the diffusion equation in reg1. Why is it a conformal map from reg1 to the open unit disk?

@user64494 This is standard proof to the Riemann theorem considered in any complex analysis book. See for example Sketch proof via Dirichlet problem on en.wikipedia.org/wiki/Riemann_mapping_theorem

There is no diffusion equation in my answer. We solve Dirichlet problem for the Laplace equation $\nabla^2 u=0$. But using Mathematica FEM we call standard utility InitializePDECoefficients with parameters "DiffusionCoefficients" and "MassCoefficients" that can be used to solve diffusion equation as well.