Dear Daniel and Wolfgang,

Thank you very much! In fact, I only want the gradient of periodic, not the 
Dirichlet periodic boundary. I thought they are same in periodic boundary 
conditions before, i.e. all informations of the solution on the face pairs 
are same. Now, I get that I have made mistake.

Why I need the gradient of periodic only? Because in my case, I have some 
boundary integrals in my cost functional, I need to make all of them equal 
to zero, so I can choose the gradient of periodic only.

But in the analysis of weak solution of original PDEs in my case, the space 
of trial and test funcution are (H^1) * (H^1) * (L^2) * (H^1) (sobolev 
space), as there are the gradient of 1st, 2nd, 4th component in the weak 
form. And I choose C0 * C0 * C0 * C0 FEM space

So I cannot make sure the continuity of the gradient in boundary or 
internal faces. But I don' know how to implement what I need? Maybe some 
penalty part like in Discontinuous Gradient Method to make up for 
continuity of the gradient?

Thank you very much!

Best,
Chucui

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