I want to implement a numerical method for conformal mapping of a mesh on the disk to some simply connected bounded Jordan region whose boundary is parametrized by a Fourier series. I was looking at different methods, see an overview here.
One of the conditions placed upon my yet-to-be-chosen method is that it should allow for efficient point evaluations of both the conformal map $\psi$ and its derivative, i.e. we want to be able to efficiently compute the value of $\psi(z)$ and $D\psi(z)$ for any z on the mesh.
Now the conformal map constructed by the Zipper method is essentially a composition of elementary maps, which individually should be possible to compute and derive efficiently. What I am unsure of is if that is still true for the composition, i.e. if the map output by the Zipper algorithm is actually a viable choice for my project.
Note that the number of compositions depends on the mesh size (i.e. number of grid points on the boundary of the disk) and I would like to have an efficient algorithm independently of my mesh size (as I might want to refine the mesh in applications). This is why I am thinking that the Zipper method might not be suited, but since I'm not too sure I would like some qualified opinions.
