Harmonic functions and conformal mappings

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.

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I would put in a slightly different way: "Conformal mapping enables the solution of specific boundary value problems by transforming the BVP to a simpler domain via a conformal map, and then transporting the solution back."

Using the chain rule, you can check that $\Delta (u\circ f) = (\Delta(u)\circ f) |f'|^2$ whenever $f$ is conformal and $u$ is sufficiently smooth. In particular, $\Delta u\equiv 0$ if and only if $\Delta (u\circ f)\equiv 0$. Thus, instead of solving Laplace's equation on the original (complicated) domain, we may decide to solve it on a disk or half-plane instead. Of course, we must keep track of what happens to the boundary conditions. And we must be able to compute a conformal map, e.g., by the Schwarz-Christoffel method.

To get the general idea of what is going on, I recommend this illustrated explanation by Tobin Driscoll.

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