A pyramid scheme for three-dimensional diffusion equations on polyhedral meshes

In this paper, a new cell-centered finite volume scheme is proposed for three-dimensional diffusion equations on polyhedral meshes, which is called as pyramid scheme (P-scheme). The scheme is designed for polyhedral cells with nonplanar cell-faces. The normal flux on a nonplanar cell-face is discretized on a planar face, which is determined by a simple optimization procedure. The resulted discrete form of the normal flux involves only cell-centered and cell-vertex unknowns, and is free from face-centered unknowns. In the case of hexahedral meshes with skewed nonplanar cell-faces, a quite simple expression is obtained for the discrete normal flux. Compared with the second order accurate O-scheme [31], the P-scheme is more robust and the discretization cost is reduced remarkably. Numerical results are presented to show the performance of the P-scheme on various kinds of distorted meshes. In particular, the P-scheme is shown to be second order accurate.