A Cartesian grid embedded boundary method for the heat equation and Poisson’s equation in three dimensions

We present an algorithm for solving Poisson’s equation and the heat equation on irregular domains in three dimensions. Our work uses the Cartesian grid embedded boundary algorithm for 2D problems of Johansen and Colella [A Cartesian grid embedded boundary method for Poisson’s equation on irregular domains, J. Comput. Phys. 147(2) (1998) 60–85] and extends work of McCorquodale, Colella and Johansen [A Cartesian grid embedded boundary method for the heat equation on irregular domains, J. Comput. Phys. 173 (2001) 620–635]. Our method is based on a finite-volume discretization of the operator, on the control volumes formed by intersecting the Cartesian grid cells with the domain, combined with a second-order accurate discretization of the fluxes. The resulting method provides uniformly second-order accurate solutions and gradients and is amenable to geometric multigrid solvers.