Stability of difference schemes for two-dimensional parabolic equations with non-local boundary conditions

The stability of difference schemes for one-dimensional and two-dimensional parabolic equations, subject to non-local (Bitsadze–Samarskii type) boundary conditions is dealt with. To analyze the stability of difference schemes, the structure of the spectrum of the matrix that defines the linear system of difference equations for a respective stationary problem is studied. Depending on the values of parameters in non-local conditions, this matrix can have one zero, one negative or complex eigenvalues. The stepwise stability is proved and the domain of stability of difference schemes is found.