A geometric multigrid approach to solving the 2D inhomogeneous Laplace equation with internal Dirichlet boundary conditions

The inhomogeneous Laplace (Poisson) equation with internal Dirichlet boundary conditions has recently appeared in several applications to image processing and analysis. Although these approaches have demonstrated quality results, the computational burden of solution demands an efficient solver. Design of an efficient multigrid solver is difficult for these problems due to unpredictable inhomogeneity in the equation coefficients and internal Dirichlet conditions with arbitrary location and value. We present a geometric multigrid approach to solving these systems designed around weighted prolongation/restriction operators and an appropriate system coarsening. This approach is compared against a modified incomplete Cholesky conjugate gradient solver for a range of image sizes. We note that this approach applies equally well to the anisotropic diffusion problem and offers an alternative method to the classic multigrid approach of Acton (1998).