The embedded finite difference method for the Poisson equation in a domain with an irregular boundary and Dirichlet boundary conditions

The Poisson equation subject to Dirichlet boundary conditions on an irregular domain can be treated by embedding the region in a rectangular domain and solving using finite differences over the rectangle. The crucial issue is the discretization of the boundaries of the irregular domain. In the past, both linear and quadratic boundary treatments have been used and error bounds have been derived in both cases, showing that the linear case gives uniform second-order accuracy, whereas the quadratic case gives third-order accuracy at the boundaries and second-order accuracy internally. Thus, it has been recommended that the linear boundary treatment be used, as it is simpler, gives rise to a symmetric matrix formulation and has uniform accuracy. The present work shows that this argument is inadequate, because the coefficients of the error terms also play an important role. We demonstrate this in the 1-D case by determining explicit expressions for the error for both the linear and quadratic boundary treatments. It is shown that for the linear case the coefficient of error is in general large enough to dominate the calculation and that therefore it is necessary to use a quadratic boundary treatment in order to obtain errors comparable with those obtained for a regular domain. We go on to show that the 1-D expressions for error can be used to approximate the boundary error for 2-D problems, and that for the linear treatment, the boundary error again dominates.