Three dimensional high order finite volume element schemes for elliptic equations

Abstract

AbstractIn this work, we construct and analyze three dimensional high order finite volume element schemes for elliptic equations. In these schemes, the trial function space is chosen as the standard rth order Lagrange finite element space, where is an arbitrary positive integer; the test function space is chosen as the piecewise constant space with respect to the dual mesh of which the control volumes are constructed using Gauss points in each element of the primal mesh. We investigate the inf–sup property of these schemes and based on it, we prove that , and , where is the exact solution, is the rth order finite volume element solution and is the piecewise rth order Lagrange interpolation of . Several numerical examples are presented to verify the theoretical findings.