Piecewise Linear Petrov–Galerkin Error Estimates for the Box Method

Abstract

The perpendicular bisector box-method (PBM) discretization of the Laplacian on a simplicial element is equivalent to a piecewise linear Petrov–Galerkin method with piecewise constant test functions. This PBM element matrix is expressed in terms of Galerkin’s piecewise linear (PLG) element matrix and the average values assumed by test functions over element faces only. These results are shown to be applicable to the accuracy analysis of PBM discretizations of extremum stable elliptic gradient equations, subject to regularity conditions on variable coefficients in these equations. These gradient equations are shown to satisfy sufficient conditions to validate a discrete extremum principle on N-dimensional Delaunay triangulations. This conclusion need not hold for the PLG equations. The dependence of the discretized Laplacian on only the element-face averages of test functions enables the substitution of piecewise polynomial test functions for the piecewise constant functions. The latter are used to establish piecewise linear accuracy for the PBM by an adaptation of an error estimate for Petrov–Galerkin methods by Strang.