Consider Poisson's equation $\nabla^2 u = 1$ on a square of side-length 1 centered at the origin. Cut out a circle of radius 1/3 at the center of this square.
Impose a von Neumann boundary condition $\frac{\partial u}{\partial n} = 0$ at the outer (squared) boundary, and a Dirichlet boundary condition $u = 0$ at the inner (circular) boundary.
I would like to find either of:
- An analytic solution (of course!).
- Asymptotics at any of the corners.
Thank you.