Chucui,

Maybe a question more : how to make sure your period boundary coditions are 
> added truly?
>
How would you do it for Dirichlet boundary conditions?

There are mutiple possibilities. 
- You can just print the constraints (ConstraintMatrx.print(std::cout)),
print the locations of degrees of freedom via 
DoFTools::write_gnuplot_dof_support_point_info 
(https://www.dealii.org/9.0.0/doxygen/deal.II/namespaceDoFTools.html#a69d19d6d574269cc6e69fa5c5b2d89e2)
and compare. 
- You can visually confirm that your solution looks periodic.
- You can calculate the solution for all the support points at the periodic 
faces and compare the values.

Best,
Daniel

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