  Sakurai et al. (J Comput Phys, 2019) presented a flux-based volume
penalization (VP) approach for imposing inhomogeneous Neumann boundary
conditions on embedded interfaces. The flux-based VP method modifies the
diffusion coefficient of the original elliptic (Poisson) equation and uses a
flux-forcing function as a source term in the equation to impose the desired
Neumann boundary conditions. As such, the flux-based VP method can be easily
incorporated into existing fictitious domain codes. Sakurai et al. relied on
analytical construction of flux-forcing functions, which limits the
practicality of the approach. Because of the analytical approach taken in the
prior work, only (spatially) constant flux values along simple interfaces were
considered. In this paper, we present a numerical approach to construct
flux-forcing functions for arbitrarily complex boundaries. The imposed flux
values are also allowed to vary spatially in our approach. Furthermore, the
flux-based VP method is extended to include (spatially varying) Robin boundary
conditions, which makes the flux-based method even more general. We consider
several two- and three-dimensional test examples to access the spatial accuracy
of the numerical solutions. We formally derive flux-based volume penalized
Poisson equation satisfying Neumann/Robin boundary condition in strong form;
such a derivation was not presented in Sakurai et al., where the equation first
appeared for the Neumann problem. The derivation reveals that the flux-based VP
approach relies on a surface delta function to impose inhomogeneous
Neumann/Robin boundary conditions. However, an explicit construction of the
delta function is not necessary for the flux-based VP method, which makes it
different from other diffuse domain equations presented in the literature.
