But if "periodic boundary condition" means that all information of solution on periodic faces pair are equal, not only the value of solution (like Dirichlet boundary condition above), but also gradient value of solution (like Neuman boundary condition, as normal vector is opposite, so ifgrad_u_on_left = grad_u_on_right, for example, so n_left *grad_u_on_left + n_right * grad_u_on_right, top and bottom as well) . In my case, periodic boundary condition should include

46.JPG <about:invalid#zClosurez>

Do you mean that you want *both* Dirichlet and Neumann conditions to be satisfied? Or that you don't want Dirichlet periodic boundary conditions and only want the gradient to be periodic?

Daniel already answered the question from the *discrete* point of view. But from the perspective of the PDE, if you have an elliptic PDE with smooth solutions in the interior of the domain, then the solution of the PDE with periodic boundary conditions is also smooth across the boundary. That means that by requiring that the *value* of the solution is periodic across the boundary, you also get that the *gradient* of the solution is periodic across the boundary. This does not have to explicitly prescribed: It just happens, in the same way as it is the case in the interior of the domain.

Of course, since we only approximate these solutions with piecewise polynomials, continuity of value does not imply continuity of the gradient across periodic boundaries. But this is the same as between faces of the mesh: the solution is continuous across a face, but the gradient of the finite element approximation is not.

Best
 W.


--
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Wolfgang Bangerth          email:                 bange...@colostate.edu
                           www: http://www.math.colostate.edu/~bangerth/

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