Relationship between the vertex-centered linearity-preserving scheme and the lowest-order virtual element method for diffusion problems on star-shaped polygons

A virtual element method‐based positivity‐preserving conservative scheme for convection–diffusion problems on polygonal meshes

“…This procedure will not result in the loss of numerical accuracy. In fact, as explained in [39], if the analytic solution u(x) to ( 1) is nonnegative, we have…”

“…This procedure will not result in the loss of numerical accuracy. In fact, as explained in [39], if the analytic solution u ( x ) to (1) is nonnegative, we have $$$$ \mid u\left({\mathbf{x}}_P\right)-{\hat{u}}_P\mid \le \mid u\left({\mathbf{x}}_P\right)-{u}_P\mid +\varepsilon . $$$$ Step Post processing to preserve the positivity of numerical solution and the local conservation .…”

A novel vertex-centered finite volume method for solving Richards' equation and its adaptation to local mesh refinement

“…Our approach is based on a Vertex-Centered Finite Volume Method (hereinafter referred to as VCFVM) and sets all unknowns on grid vertices. It should be noted that the vertex-centered finite volume method has been widely studied for solving anisotropic diffusion/parabolic equations on general polygonal/polyhedral meshes in the past decade, see [41,40,45,36,5,29]. The numerical scheme proposed in this paper is partially inherited from [29], but substantial modifications in the spatial and temporal discretization have been made to consider the properties of Richards' equation, including the advection flux involved and the high nonlinearity between the soil hydraulic parameters and the soil water potential.…”