But if "periodic boundary condition" means that all information of solution onperiodic faces pair are equal, not only the value of solution (like Dirichletboundary condition above), but also gradient value of solution (like Neumanboundary 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 boundarycondition should include46.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. -- ------------------------------------------------------------------------ Wolfgang Bangerth email: bange...@colostate.edu www: http://www.math.colostate.edu/~bangerth/ -- The deal.II project is located at http://www.dealii.org/ For mailing list/forum options, see https://groups.google.com/d/forum/dealii?hl=en

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