I have the solution $u(x,y) = \sin(k\pi x) \sinh(k\pi (1-y))$ of the poisson problem
$ \nabla^2u(x,y) = 0 \\ u(x, 1) = u(1, y) = u(0, y) = 0 \\ u(x, 0) = \sin(k\pi x) $
Now I thought when my domain is not square, the solution of the problem
$ \nabla^2u(x,y) = 0 \\ u(x, y_{max}) = u(1, y) = u(0, y) = 0 \\ u(x, 0) = \sin(k\pi x) $
would be scaled by the y-range: $u(x,y) = \sin(k\pi x) \sinh(k\pi (y_{max}-y)$
but this does not match my numeric solution.
I wonder if the solution is incorrect, if my numeric solution is wrong or if the solution is no longer unique and the numeric solution is another function than the analytic one.