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[Martin Sandve Alnæs <martinal@xxxxxxxxx>]

In addition to integration over the entire domain, explained in another
post, we have introduced a concept called 'regions' in ufl. For this post,
assume a single mesh. Support for multiple meshes will get some attention
in the future.

The ufl language changes for expressing domain relations is work in
progress, but some useful features are already possible in trunk. We keep
the concept of disjoint numbered subdomains, which can be easily
represented as a single meshfunction in dolfin, having a single integer
label for each cell. In addition we introduce the concept of a region,
which is the union of any number of subdomains. Thus regions of one domain
can be overlapping by including some of the same subdomain(s). With some
automatic symbolic manipulation in ufl, we can now express integrals over
overlapping regions with zero additional cost at assembly time.

Here's the currently simplest syntax for integrals over regions:

# Define regions as tuples of subdomain labels
DL, DM, DR = (1,2), (2,), (2,3) # ***

# Define new measures associated with the interior domains
dx = Measure("dx")[domains]

# Make forms for equation
a = u*v*dx() + alpha*dot(grad(u), grad(v))*dx() # Integrals over entire
domain
L = f*v*dx(DR) + g*v*dx(DL) - (f+g)/2*v*dx(DM) # Integrals over regions

For the full running code, see this demo in the latest dolfin:
  demo/undocumented/overlapping-regions/python/demo_overlapping-regions.py

Martin