Convergence of finite difference method for the parabolic problem with concentrated capacity and variable operator

We investigate the convergence of difference schemes for the one-dimensional heat equation with variable (time-dependent) operator and the coefficient at the time derivative (heat capacity) containing Dirac delta distribution. An abstract operator method is developed for analyzing this equation. Estimate for the rate of the convergence in special discrete energetic Sobolev norm, compatible with the smoothness of the solution is obtained.