Jaekwang,

It seems that I don't need to add_periodicity to the triangulation unless 
> it is parallel distributed triangulation? 
>
I would be heavily surprised if you get correct results without calling 
Triangulation::add_periodicity() if you are using discontinuous elements 
and a serial Triangulation.
Otherwise, you are not providing any kind of information to these objects 
that you want to treat specific boundaries as periodic ones. 
 

> By the way, when I 'add_periodicity', I have compiler error on the red 
>> line of the code. 
>>
>> Can you give me a clue to resolve this error? 
>>
>>   //original argument
>>
>>         std::vector<dealii::GridTools::
>>
>>         PeriodicFacePair<typename DoFHandler<dim>::cell_iterator>> pairs;
>>
>>     
>>
>>         GridTools::collect_periodic_faces(dof_handler,
>>
>>                                           /*b_id1*/ 6,
>>
>>                                           /*b_id2*/ 5,
>>
>>                                           /*direction*/ 0,
>>
>>                                           pairs);
>>
>>         
>>
>>         triangulation.add_periodicity(pairs);
>>
> You need to call GridTools::collect_periodic_faces with a parameter of 
type 

std::vector<dealii::GridTools::PeriodicFacePair<typename 
Triangulation<dim>::cell_iterator>> pairs;

if you want to use the resulting object for 
Triangulation::add_periodicity().
 
Best,
Daniel

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