Two scalar PDEs coupled via boundary


I have a set of two scalar PDEs that are coupled via a boundary

CodeCogsEqn (4)


CodeCogsEqn (7)

The problem is solved for the same eigenvalue. Conceptually, the domains are the same, i.e. both equations are formulated on the same unit square mesh and then coupled via the boundary. However, if this is not possible I would define the equations on separate meshes and couple them via an interface.

So far the multiphenics functionality seems to be my only option, which could allow me to implement a coupling and work with separate meshes.

Is there a way to use standard FEniCS functionality?

Has someone come across a similar problem and knows of a way to re-formulate this?

Thank you for taking the time. Any help or links are much appreciated.

There is a mixed-dimensional functionality (look at , she answered a lot of questions regarding this), but I have no expertise in this.

However, you can use standard fenics functionality to solve this problem via a Picard Iteration, i.e., solve the first PDE with a frozen value for u_2, and use the obtained result as guess for the boundary condition of the second equation, which is then plugged into the first equation again, and so on.