Thank for the reply! 
I just fixed my code according to your suggestion. 
Now, I am evaluating my error with following code lines. 

template <int dim>

void msurface<dim>::evaluate_error()

{

    Vector<float> difference_per_cell (triangulation.n_active_cells());

    

    *const MappingQ<dim> mapping (degree);*

    

    VectorTools::integrate_difference (*mapping*, 
dof_handler,solution,Solution<dim>(),

                                       
difference_per_cell,QGauss<dim>(degree+2),VectorTools::L2_norm);

    

    const double L2_error = difference_per_cell.l2_norm();


    std::cout << "   L2_error : " << L2_error << std::endl;

   

    error=L2_error;

}

But still. I have following results......

<https://lh3.googleusercontent.com/-zRDyAouVeVM/WBl0QqXkAVI/AAAAAAAAA8M/YNTlDe7KlXo9KVUuCFM3ts1s2_cx_ZYlACLcB/s1600/Screen%2BShot%2B2016-11-01%2Bat%2B11.51.24%2BPM.png>


do you have any other idea? 

Thanks again for your time

Regards, 
Jaekwang Kim 

2016년 11월 1일 화요일 오후 5시 19분 35초 UTC-5, Daniel Arndt 님의 말:
>
> Jaekwang, 
>
> [...]
>>
>>     VectorTools::integrate_difference 
>> (dof_handler,solution,Solution<dim>(),
>>
>>                                        difference_per_cell
>> *,QGauss<dim>(degree+**2**)*,VectorTools::L2_norm);
>>
> somegthing that immediately comes to mind is that you don't use a 
> non-default Mapping in here although your boundary is curved.
> Make sure that you use an appropriate Mapping everywhere!
>
> Best,
> Daniel
>

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