> What you describe makes perfect sense in case of Dirichlet BCs. However, I 
> have no-flux (natural) BCs at the boundary between domains. I don't 
> understand 
> what to do with the cells in domain #2 to satisfy these?

Hm, I must admit that I don't quite know what the right solution is then. It's 
*possible* that just adding the Neumann boundary terms on the interfaces is 
enough. But I'm not entirely sure and it would seem to me that what happens in 
domain #2 then still influences what is going on in domain #1 -- the Neumann 
terms are just another source term, and the ellipticity of the problem in that 
case implies that whatever sources you have in domain #2 (or don't have) have 
an effect on the solution in domain #1. You might want to think this through 
some more.

This would suggest that using FE_Nothing in domain #2 might be the better 
choice after all. step-46 will be your friend understanding how to use it.

I'll add another thought: If you have non-zero Neumann boundary conditions, 
and you approximate the original domain by one that approximates the boundary 
by a stair-step function as you seem to do, then the numerical solution does 
not converge to the exact one. That's because the stair-step boundary has a 
different length than the exact boundary (and doesn't converge to the exact 
length), and the total flux into the domain equals the average flux on the 
boundary times the length of the boundary.

(Fun fact: Ivo Babuska explained this to me at a conference dinner at the 
first conference I ever went to, in 1998 or 1999.)


> Additionally, would it be possible to further reduce the solution effort by 
> constraining dofs inside domain #2 to zero?

Yes, but that's equivalent to using a diagonal mass matrix. I suspect it's not 
worth it.

Best
  W.


-- 
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Wolfgang Bangerth          email:                 [email protected]
                            www: http://www.math.colostate.edu/~bangerth/

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