Mathematical Modeling and Analysis New Discretization Methodology for Diffusion Problems on Polyhedral

Tetrahedral and structured hexahedral meshes have been used for decades in a majority of en-gineering simulations; they are relatively easy to generate and there exists an enormous repos-itory of numerical methods designed for these meshes. Nowadays, a growing number of com-plex simulations shows advantage of using poly-hedral meshes. For example, in the simulation of ow through a water jacket of an engine [1], the results obtained on a polyhedral mesh are more accurate than the results obtained on a tetrahe-dral mesh with a comparable number of cells. In oil reservoir simulations, the polyhedral mesh topology offers unlimited possibilities: cells can be automatically joined, split, or modied by in-troducing additional points, edges and faces to model complex geological features. Unfortu-nately, most of the existing numerical methods cannot be extended to polyhedral meshes, espe-cially to meshes with cells having strongly curved (non-planar) faces. In [2], we considered a diffusion problem, which appears in computational uid dynamics, heat conduction, radiation transport, etc., and de-veloped a new discretization methodology that has no analogs in literature. The methodology follows the general principle of the mimetic -nite difference (MFD) method to mimic the es-sential underlying properties of the original con-tinuum differential operators such as the conser-vation laws, solution symmetries, and the funda-mental identities and theorems of vector and ten-sor calculus