A second order Cartesian finite volume method for elliptic interface and embedded Dirichlet problems

We present a finite volume method to solve elliptic equations with immersed interface conditions. This method allows discontinuities on the solution and its normal derivatives on an interface inside the domain on a Cartesian grid. The main idea is to use a piecewise polynomial representation of the solution on a dual grid that avoid distinctions between the different interface configurations. The method achieves second order accuracy with a compact nine-point stencil. Moreover, we show that this method applies to solve embedded Dirichlet and Neumann problems.