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For a static Laplace's equation problem, is it possible to specify boundary
conditions for an element edge rather than just at the nodes?

* No

The best way to specify Neumann boundary conditions is to integrate (probably
analytically), over the Neumann portion of the boundary, the product of the
corresponding weight (test) functions and the flux per unit boundary.

There may be some code that allows specification of fluxes per unit boundaries
at nodes, which are then interpolated (probably linearly) and integrated over
adjacent faces, but I don't know how well this has been tested.