We present a numerical method for the variable coefficient Poisson equation in three dimensional irregular domains and with interfacial discontinuities. The discretization embeds the domain and interface into a uniform Cartesian grid augmented with virtual degrees of freedom to provide accurate treatment of jump and boundary conditions. The matrix associated with the discretization is symmetric positive definite and equal to the standard 7-point finite difference stencil away from embedded interfaces and boundaries. Numerical evidence suggests second order accuracy in the L1-norm. Our approach improves the treatment of Dirichlet and jump constraints in the recent work of Bedrossian et al. and provides novel aspects necessary for problems in three dimensions. Specifically, we construct new constraint-based Lagrange multiplier spaces that significantly improve the conditioning of the associated linear system of equations; we provide a method for sub-cell polyhedral approximation to the zero isocontour surface of a level set needed for three dimensional embedding; and we show that the new Lagrange multiplier spaces naturally lead to a class of easy-to-implement multigrid methods that achieve near-optimal efficiency, as shown by numerical examples. For the specific case of a continuous Poisson coefficient in interface problems, we provide an expansive treatment of the construction of a particular solution that satisfies the value jump and flux jump constraints. As in Bedrossian et al 2010, this is used in a discontinuity removal technique that yields the standard 7-point Poisson stencil across the interface and only requires a modification to the right-hand side of the linear system.

@Article{HWST12,
  author       = "Hellrung, Jeffrey and Wang, Luming and Sifakis, Eftychios and Teran, Joseph",
  title        = "A Second-Order Virtual Node Method for Elliptic Problems with Interfaces and Irregular Domains in Three Dimensions",
  journal      = "Journal of Computational Physics",
  number       = "4",
  volume       = "231",
  pages        = "2015--2048",
  year         = "2012",
  url          = "http://graphics.cs.wisc.edu/Papers/2012/HWST12"
}