On convergence of methods with different time steps in subdomains (Q1375012)

For the parabolic equation \[ \frac{\partial u}{\partial t}+Au= f(x,t), \qquad x\in\Omega, \quad t\in[0,T] \tag{1} \] in a multidimensional domain \(\Omega\) with boundary and initial conditions we introduce two nets with different time steps. Then we construct an implicit balanced monotonic difference scheme. The obtained system of difference equations has no constructive algorithm for its realization. Therefore the solution of the difference problem is obtained by interpolation. Convergence to the solution of the differential problem (1) is proved.