Periodic boundary conditions - a need for studying documents

General
1

Good evening, Sirs.

I’m able to apply Periodic boundary conditions (PBCs) using FEniCS for 2-D and 3-D.
Thanks to the instructions in the following
https://fenicsproject.org/olddocs/dolfin/1.4.0/python/demo/documented/periodic/python/documentation.html
https://github.com/jorgensd/dolfinx_mpc
https://github.com/jorgensd/dolfinx_mpc/blob/main/python/demos/demo_periodic_geometrical.py

However, I have to admit that I do not clearly understand how the PBCs work behind the user-defined codes.

I’m making a minimal working example which implementing the PBCs to a PDE from scratch. Only for understanding and studying purposes. It is well known for the PBCs as follows (2-D)
u(0,x2) = u(L,x2)
u(x1,0) = u(x1,L)
where L is the length of a unit-cell.
I am stuck with how to solve the Au=f system that has the PBCs constraint.

I attempted to find literature on implementing PBCs and solving the Au = f system with the PBC constraints. However, most papers directly present the above equations, assuming that PBCs are implicitly included in the system.

I need essential documents for studying. Please suggest any documents, articles, tech reports, or books that include

  • Direct implementation of PBCs.
  • Solving the Au=f system that has the PBCs constraint.

I would like to thank you in advance for your time and support.

2

See for instance

which goes through the implementation details of DOLFINx MPC

Legacy DOLFIN uses a different approach where it reduces the size of the dofmap, which only works for u_i=u_j constraints on matching grids.

1 Like
3

Thank you very much. This is helpful information.