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 function $a(x)$ in the operator $- \operatorname {div}(a(x) {\operatorname {grad}} u)$. The derivation of the discretization is based on approximating the flux density $J = a {\operatorname {grad}}{\mkern 1mu} u$ by constants on each element. In many cases the flux density is well behaved (moderately varying) even if $a(x)$ and $u(x)$ 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 solution u is approximated to order 1 by the finite element solution in the ${H^1}$-norm. The estimate depends only on the mesh width and on derivatives of the flux density and of a possibly occurring inhomogeneity.