couple subdomains

[gdmcbain]

Coupling domains defined on different meshes by an InterfaceMesh1D is demonstrated in ex04, but since a boundary value problem can be solved in a subdomain #159 #160, can problems on subdomains be coupled too?

What are the relative advantages of multiple subdomains and multiple domains?

• It might be simpler to construct the mesh and bases on the interface between subdomains.
  • An InterfaceMesh1D requires a rule: Callable[[float, float], bool] which might be awkward for more complicated shapes.
• Multiple domains needn't have matching meshes. This is particularly important for problems with large deformation.
• The ease of either solving ‘monolithically’ versus ‘partitioned’, ‘staggered’, ‘segregrated’ (see, e.g. Piperno & Farhat 2001 for the terminology) is probably about the same for subdomains and domains, given the flexibility of scikit-fem; e.g. ex04 demonstrates the use of scipy.sparse.bmat for monolithic solution across mismatched meshes.

This issue is to demonstrate the possibility of coupling subdomains; comparison can follow.

An obvious choice for a first example is one following the insulated wire ex17 #89 #90 and the solution for the wire with the coating kept cold ex26 #159 #160, extending the latter with a Laplace equation in the annulus to recover the solution of the former. This belongs to the class of Poisson problems with discontinuous coefficients treated by Gustafsson, Stenberg, & Videman (2018).

The motivation is multiphysics like conjugate heat transfer, in which different subdomains are governed by different partial differential equations; however, this doesn't intrinsically distinguish between multiple subdomains and multiple domains, the difference will come down to practical implementation details, which is what is to be explored here.

[gdmcbain]

Monolithic conjugate heat transfer is demonstrated in #218.

[gdmcbain]

ex04 is broken without InterfaceMesh1D #215.

[kinnala]

I'm now convinced that the simplest way to do this is to use a global Lagrange multiplier together with a stabilized method. I'll try to make an example at some point because I want to try out this approach.