From searching online, I have only found methods for solving the Laplace equation with boundary conditions for rectangular, or circular domains. Is there a method for solving it for more complicated domains?
I would like to solve it for a L shaped domain described by the set $$\Omega=\{(x,y)\,|\,0\le x\le 2L,\,0\le y\le L\}\cup\{(x,y)\,|\,0\le x\le L,\,0\le y\le 2L\}$$ with the boundary conditions $u(x,L)=C(2L-x)$ for $L\le x\le 2L$ and $u(L,y)=C(2L-y)$ for $L\le y\le 2L$ and $0$ for all other boundaries.
My task is to solve this problem numerically, which is pretty straight forward, but I was curious whether I can find an analytical solution for it. Here is an image that visualizes the boundary conditions:
Further research showed that I could define a (holomorphic?) function to transform the domain into a square, for example, solve it on the square, and then transform it back. I tried some things, but I do not think that I defined my transformation well.