Advantages and difficulties of this method are presented. In particular, it is shown that the proposed approach is significantly better in many cases than previous adaptive schemes not taking into account the quantity of interest. Cases involving nonlinear functionals and in which the approach fails, are presented along with theoretical explanations and numerical evidence of the reasons for this.
The proposed methods are applied to a simple model from the physics of the solar atmosphere and the propagation of linear acoustic waves is computed. The fraction of the wave energy that passes the chromosphere-corona transition is computed to good accuracy.