Numerical solution of the 2-D Poisson equation on an irregular domain with Robin boundary conditions

We describe a 2-D finite difference algorithm for inverting the Poisson equation on an irregularly shaped domain, with mixed boundary conditions, with the domain embedded in a rectangular Cartesian grid. We give both linear and quadratic boundary treatments and derive 1D error expressions for both cases. The linear approach uses a 5-point formulation and is first-order accurate while the quadratic treatment uses a 9-point stencil and is second-order accurate. The key aspect of the quadratic treatment is the use of a suitably chosen directional derivative to find the second order accurate approximation to the normal derivative at the boundary.