Generalized local maximum principles for finite-difference operators

Abstract

The generalized local maximum principle for a difference operator L h {L_h} asserts that if L h u ( x ) > 0 {L_h}u(x) > 0 then Γ u \Gamma u cannot attain its positive maximum at the net-point x. Here Γ \Gamma is a local net-operator such that Γ u = u + O ( h ) \Gamma u = u + O(h) for any smooth function u. This principle, with simple forms of Γ \Gamma , is proved for some quite general classes of second-order elliptic operators L h {L_h} , whose associated global matrices are not necessarily monotone. It is shown that these generalized principles can be used for easy derivation of global a priori estimates to the solutions of elliptic difference equations and to their difference-quotients. Some examples of parabolic difference equations are also treated.