Hi, everyone,
I’m trying to solve a 2D mixed variational problem with periodic boundaries with discontinuous finite elements. In the variational formulation, there are variational forms on the inner boundaries. In theory, I would like to treat the periodic boundaries just as inner boundaries and define the variational forms on these boundaries.
For example, there are variational forms defined on inner boundaries such as:
ufl.jump(E) * ufl.avg(D) * ufl.dS
How can I define such a form on periodic boundaries?
In priori, I have generated periodic mesh with Gmsh so the pairs of periodic boundaries are discretised in the same manner.
A natural idea is to create new cell connectivity for the cells sharing the same facet on periodic boundaries. But this seems very tricky to me.
Another idea is to define the variational forms on the periodic boundaries just like defining forms on outer boundaries. But a form like this will require to call the values of the functions from the opposite side of the boundary, which are on the opposite side of the domain geometrically. I also don’t know how it is possible to do that.
I’ll appreciate any help or ideas.