A Neumann–Neumann preconditioned iterative substructuring approach for computing solutions to Poisson’s equation with prescribed jumps on an embedded boundary

In multifluid problems with surface tension the fluid pressure and its derivative are discontinuous at fluid interfaces. We present a Cartesian grid embedded boundary method for numerically resolving these discontinuities in which we use Neumann–Neumann preconditioned iterative substructuring to solve the governing equations. We validate this method by computing several well-known Poisson problems with discontinuous coefficients, and we compare its performance to an approach based on simple iteration. By analogy with the conjugate gradient method, we hypothesize that the scaling of the Neumann–Neumann preconditioned iterative substructuring is O ( h - D ln h - 1 ) where h is the cell size and D = 2 , 3 is the dimensionality of the problem. In contrast, we show that the simple iterative procedure scales like O ( h - ( D + 1 ) ) and is slower by a factor of 4000 for a small (i.e., 64 × 64 cell) model calculation with physical parameters corresponding to a 1.5 mm air bubble in water. We present an analytical model to explain the scaling of this iterative procedure.