I am working on integrating a set of PDEs on the surface of a sphere, but the question could be applied to any close manifold. In this case, the boundary conditions are, by the nature of the geometry, periodic. Is it then necessary to specify somehow the boundary conditions or are they then by default working properly within the algorithm? How would that behave mathematically in the integral over the boundary arising from the integration by parts?
If the connectivity of your mesh is also such that the manifold is closed, then also from a computational perspective there is no boundary. No boundary simply means no boundary condition; nothing to be done. No boundary integrals arise in the integration by parts.
I appreciate your view that you could consider that a ‘periodic boundary condition’, but I’d say that is giving it too much credit 
There truly simply is no boundary.
If the mesh does not have the appropriate connectivity, and there is a ‘ficticous’ boundary purely due to ‘faulty’ meshing, then yes, you’d need the dolfinx_mpc module to ‘tie’ those ‘loose ends’.
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Okay that does make a lot of sense. Usually when we are considering the spatial derivatives we apply integration by parts and as a result one of the remaining terms is dependent on a single gradient -if we are considering second order differential equations such as the laplacian- and that term is simplified by using the divergence theorem so we can apply the boundary conditions. Since the mesh has a connectivity such that the manifold is closed we would just no apply the divergence theorem in the term and we remain with two integrals.
Thanks a lot!
Oh no, that’s not entirely what I mean. You would do integration by parts, but that would simply yield zero. In fact, that is telling you that on a closed manifold, the integral on which you want to perform the divergence theorem is identically zero.