Numerical spectral analysis of a difference operator with non-local boundary conditions

The spectrum of a finite difference operator, subject to non-local Robin type boundary conditions, is dealt with. We analyse the spectral properties that relate to the stability of finite difference schemes for parabolic equations. The impact of functions and parameters, defining non-local conditions, on a spectral structure is examined and theoretical study is supported by numerical experiments. Also, for a difference scheme, applied to a parabolic equation with non-local conditions, a sufficient stability criterion, based on spectral properties of the difference operator, is discussed. Numerical evidence suggests that such a criterion is not only sufficient for stability, but necessary, too.