I would like to get some insight into the practicalities of applying conformal mapping techniques for the numerical solution of PDEs. Up until now I had the impression that conformal mapping techniques basically enable the solution of specific PDEs on complex domains by transforming the $\mathbf{solution}$ on a simpler one via the conformal map. Is this true?
Assuming I have Laplace's equation in 2D I can solve it based on a cartesian grid using finite differences. In case the computational domain is more complicated, lets say a half annulus the solution to the equation should be obtained by "warping" the obtained solution with the conformal map. However if we consider the same problem and use a solution method based on a curvilinear coordinate system fitted to the half annulus and the finite difference method then the finite difference formulas that have to be applied change considerably yielding a different solution.
What I was wondering about is if the initial assumption is false? I have heard about the Joukowski transform being applied for the computation of streamlines around airfoils using the transformed analytical solution around a cylinder, that's the reason I had this assumption.