I am looking to derive a conformal map for the problem illustrated in [this](https://i.stack.imgur.com/T4Ep5.jpg) image. I've read a bit about how to map a square onto a circle, but I'm struggling to extend the concepts for the domain at hand. I don't have a rigorous mathematical background (mech. engineer in computational fluid dynamics), so I would appreciate if someone here could advise me on the route that I should take in order to derive such a conformal mapping.


The end application is to generate a smooth computational mesh that looks like [this](https://i.stack.imgur.com/VFtiC.jpg). I have generated a mesh like this using other means, but the smoothness of the mesh vertices is not sufficient for extremely fine meshes. This results in spurious oscillations in the numerical problem I am trying to solve.



Conformal map onto a circle, from an identification space composed of five squares
Edited. Using symmetry lines, break the original $D$ (5 squares) into 8 trapezoids with angles $\pi/4,\pi/2,\pi/2,3\pi/4$. This trapezoid must be mapped conformally onto a sector which makes $1/8$-th of the disk. Under this map all angles at the corners are preserved, except $3\pi/4$ which becomes $\pi$. Then Christoffel-Schwarz leads to the following integral
$$C\int\_0^z \zeta^{-3/4}(\zeta-a)^{-1/2}(\zeta-1)^{-1/2}d\zeta,$$
where $a$ is the accessory parameter, and it is determined from the equation
$$\int\_0^a=-3i\int\_a^1.$$
This equation can be easily solved numerically since the absolute values of
right and left sides are monotone in the opposite directions (one increasing another decreasiing as functions of $a$). The definite integral involved in
the equation for the accessory parameter can be expressed in terms of hypergeometric function, Gradshtein, Ryzhik, p. 317, item 3.197, but I am not sure how much this helps. I would just find $a$ numerically.


