Hamed, Let consider a situation when we don't want to apply Neumann BC at all, > namely, we do not want to eliminate the Neumann term, (v, n.grad(u)), > considering zero value for that or we do not have a known g function to > consider (v, n.grad(u))=g. In this case, how we can include the Neumann > term, (v, n.grad(u)), in the week formulation and assembly object, no > matter we have periodic condition or not. > The question in the end is just how the weak formulation you are interested in looks like and what the appropriate function spaces are. If you are using periodic boundary conditions, you are saying that there is no boundary but that these faces should be treated the same as internal faces. Hence, it sounds weird to want to do something more there regarding boundary values. What else do you want to achieve with respect to your formulation?
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