Analysis of linear and quadratic simplicial finite volume methods for elliptic equations

This paper is devoted to analysis of some convergent properties of both linear and quadratic simplicial finite volume methods (FVMs) for elliptic equations. For linear FVM on domains in any dimensions, the inf-sup condition is established in a simple fashion. It is also proved that the solution of a linear FVM is super-close to that of a relevant finite element method (FEM). As a result, some a posterior error estimates and also algebraic solvers for FEM are extended to FVM. For quadratic FVM on domains in two dimensions, the inf-sup condition is established under some weak condition on the grid.