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I am solving an elliptic boundary value problem on a subset of the rectangle [-1,1]x[-1,1]. The domain contains the line segment x=0, however this does not need to be a part of the boundary, so it is not aligned with the facets by default.

I want to compute the flux of a solution function across the line segment x=0. I know from the tutorial how to compute fluxes over boundary elements. However, I am not sure whether making the line a boundary element is necessary in this case. Nor am I sure how to insert it, since simply inserting the line into the mesh will cause degrees of freedom to conflict, as would naively gluing rectangles together. Can you offer any suggestions on how to compute this flux?

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Make sure that the mesh is aligned with x = 0. Then mark the facets along that line and compute the flux using an "interior facet integral". You can do this by supplying a facet function markers that (e.g.) marks those facets by 1 and the remaining facets by 0. Then compute the flux by

line_segment = AutoSubDomain(lambda x: near(x[0], 0))
markers = FacetFunction("size_t", mesh)
markers.set_all(0)
line_segment.mark(markers, 1)
dS = dS[markers]
flux = assemble(<expression for flux>*dS(1))

If you use a built-in mesh like RectangleMesh, it is easy to make sure that the mesh is aligned with the line segment. If you use an external mesh generator, it should have support for respecting your line segment when meshing.

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  • $\begingroup$ Thanks, that is quite helpful! However, at present I am using DOLFIN's CSG for my mesh generation, and my domain is slightly more complicated than just a rectangle, being defined by: Rectangle(-2,-2,2,2)-Ellipse(-1,0,0.12,0.25)-Ellipse(1,0,0.12,0.25) Can I adjust this to get the CSG to respect the line segment, or will I need to use an external mesh generator? $\endgroup$
    – Ian
    Commented Jun 3, 2013 at 21:59
  • $\begingroup$ Yes, you can do that with DOLFIN CSG (insert subdomains into your domain which are respected by the meshing). However, this is part of a separate branch which has not yet been merged. I'll ask its author to comment on the status of this feature. $\endgroup$
    – Anders Logg
    Commented Jun 4, 2013 at 6:33
  • $\begingroup$ What if the mesh doesn't align with the line segment? Is there a work-around for that scenario? $\endgroup$
    – Paul
    Commented Nov 1, 2017 at 20:42
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As Anders said, the CSG functionality in Dolfin will be able to handle this soon. If you want to try it out now, get the code here: https://bitbucket.org/benjamik/dolfin and check out the branch called benjamik/csg-2d. Take a look at demo/undocumented/csg/2d/python/demo_csg_2d.py. The functionality is not documented yet except from this demo.

There are a couple of remaining issues that need to be sorted out before merging this into the official dolfin repo, but none of them should affect your case.

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