On the Eigenvalues of Second Order Elliptic Difference Operators

Abstract

While there is now an extensive literature on the approximation of the eigenvalue problem for compact operators, these theories do not include the classical finite-difference approximation. In the self-adjoint case the problem is resolved through the variational characterizations of the eigenvalue and eigenfunctions. The problem is unresolved in the non-self -adjoint case. In this work we consider a model problem: $\Delta u + au_x + bu_y = \lambda Nu$ the unit square. The operator N is a first order operator. The left-hand side is approximated by straightforward central differences. The operator N is approximated “weakly”.