> So, in "valueOUT" variable, I have the outflow (m³/s), don't I?

I suppose that you ask this question because this doesn't work. Or does it? 
Why don't you describe what is happening?

BTW, The way I would write this would be as follows:

  FEFaceValues fe_values_total;
  FEValuesViews::Extractors velocity(0);
  std::vector<Tensor<1,dim> > velocity_values (n_q_points);
  for (cell=...)
    for (face=0...)
     {
         fe_values_total[velocity].get_function_values (total_solution,
                              velocity_values);
        for (q_point=0...)
           outflow += velocity_values[q_point] *
                            fe_values_total.normal_vector(q_point);
    }
         

W.
-- 
-------------------------------------------------------------------------
Wolfgang Bangerth                email:            [email protected]
                                 www: http://www.math.tamu.edu/~bangerth/

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