Inverse-average-type finite element discretizations of selfadjoint second-order elliptic problems

Abstract

This paper is concerned with the analysis of a class of "special purpose" piecewise linear finite element discretizations of selfadjoint second-order elliptic boundary value problems. The discretization differs from standard finite element methods by inverse-average-type approximations (along element sides) of the coefficient functiona(x)a(x)in the operator−div⁡(a(x)gradu)- \operatorname {div}(a(x)\,{\operatorname {grad}}\,u). The derivation of the discretization is based on approximating the flux densityJ=agraduJ = a\,{\operatorname {grad}}{\mkern 1mu} uby constants on each element. In many cases the flux density is well behaved (moderately varying) even ifa(x)a(x)andu(x)u(x)are fast varying. Discretization methods of this type have been used successfully in semiconductor device simulation for many years; however, except in the one-dimensional case, the mathematical understanding of these methods was rather limited. We analyze the stiffness matrix and prove that—under a rather mild restriction on the mesh—it is a diagonally dominant Stieltjes matrix. Most importantly, we derive an estimate which asserts that the piecewise linear interpolant of the solutionuis approximated to order 1 by the finite element solution in theH1{H^1}-norm. The estimate depends only on the mesh width and on derivatives of the flux density and of a possibly occurring inhomogeneity.