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: [email protected]
www: http://www.math.colostate.edu/~bangerth/
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