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

Abstract

As a generalization of the finite element method, the virtual element method (VEM) has made remarkable achievements in the algorithm and analysis of simulating various problems recently. Vertex-centered linearity-preserving scheme (VLPS, Wu et al., 2016) is a linear finite volume scheme which is currently only applied to the numerical simulation of diffusion problems. In this paper, we study the relationship between VLPS and the lowest-order VEM for diffusion problems on star-shaped polygonal meshes from an algebraic point of view. The global stiffness matrix of VLPS with a special stabilization term coincides with that of the lowest-order VEM while the load terms are generally different. Specifically, we find that the global stiffness matrices of the two methods can be split as the consistency parts and the stability parts, and the consistency parts are the same while the stability parts coincide under some assumptions. As a by-product, a new stability term is obtained for VLPS. Besides, a post-processing procedure is suggested for the lowest-order VEM to preserve the positivity of the cell-centered unknowns and maintain at the same time the local conservation on the primary meshes. The positivity, existence and uniqueness of the cell-centered unknowns are studied. Numerical experiments confirm the theoretical findings and demonstrate the efficiency of the post-processing procedure on various polygonal meshes.