I want to map two regions conformally in order to solve Poisson's equation on arbitrary, doubly-connected regions. My case of particular interest is to map the annulus to a square with a circular hole on its center.
For the mapping, I propose to find first a holomorphic function that maps the boundaries, and then solve Laplace's equation given those boundary conditions in order to find the mapping of the interior region. All this numerically.
Since the square onto which the outer circumference is going to be mapped can be described by 4 straight lines, I was planning to construct a piecewise map with 4 Möbius transformations that turned each quarter of the circumference into each of the square's sides, and leave the inner circumference as it is.
Now my question, is this method going to produce a conformal mapping almost everywhere? Are there similar or better methods for doing so? I know that the mapping will fail to be conformal at the corners, but that is not a big deal for my purposes.