A Decoupled and Positivity-Preserving DDFVS Scheme for Diffusion Problems on Polyhedral Meshes (Communications in Computational Physics, 2020)
We propose a decoupled and positivity-preserving discrete duality finitevolume (DDFV) scheme for anisotropic diffusion problems on polyhedral meshes withstar-shaped cells and planar faces. Under the generalized DDFV framework, two setsof finite volume (FV) equations are respectively constructed on the dual and primarymeshes, where the ones on the dual mesh are derived from the ingenious combinationof 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 primarymesh 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, itis conditionally maintained on the dual mesh while strictly preserved on the primarymesh. 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 nonlineariteration is required for linear problems and a general nonlinear solver could be usedfor nonlinear problems. In addition, we analyze the well-posedness of numerical solutions for linear problems. The properties of the presented scheme are examined bynumerical experiments. The efficiency of the Newton method is also demonstrated bycomparison with those of the fixed-point iteration method and its Anderson acceleration.