A Decoupled and Positivity-Preserving DDFVS Scheme for Diffusion Problems on Polyhedral Meshes

Abstract

We propose a decoupled and positivity-preserving discrete duality finite volume (DDFV) scheme for anisotropic diffusion problems on polyhedral meshes with star-shaped cells and planar faces. Under the generalized DDFV framework, two sets of finite volume (FV) equations are respectively constructed on the dual and primary meshes, where the ones on the dual mesh are derived from the ingenious combination of a geometric relationship with the construction of the cell matrix. The resulting system on the dual mesh is symmetric and positive definite, while the one on the primary mesh possesses an M-matrix structure. To guarantee the positivity of the two categories of unknowns, a cutoff technique is introduced. As for the local conservation, it is conditionally maintained on the dual mesh while strictly preserved on the primary mesh. More interesting is that the FV equations on the dual mesh can be solved independently, so that the two sets of FV equations are decoupled. As a result, no nonlinear iteration is required for linear problems and a general nonlinear solver could be used for nonlinear problems. In addition, we analyze the well-posedness of numerical solutions for linear problems. The properties of the presented scheme are examined by numerical experiments. The efficiency of the Newton method is also demonstrated by comparison with those of the fixed-point iteration method and its Anderson acceleration.