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I know that in FVM, it is possible to show that a discretisation scheme is conservative by adding the discrete terms over a few control volumes and showing that all terms cancel apart from those relating to the flux in and out of the entire domain.

I'm not sure what happens in FEM? My thoughts are: for instance for a 1D case where there are two neighbouring elements sharing a node, and hence 4 equations exists for a 3-element grid, one adds the relevant terms from the 4 equations to see which terms cancel out and which remain in the end. Am I right?

Thank you.

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The way this is usually proven in the finite element context is different, but many finite element schemes satisfy conservation properties. For example, if you think of the Stokes equations, as long as the pressure space contains the piecewise constant functions, then mass is conserved. Similar properties can often be shown for the mixed Laplace equation typically used for porous media flow.

It is typically more complicated to show such properties for first order conservation equations, but even there it is sometimes possible if finite element spaces are appropriately chosen.

However, the ways you would show this results from the weak form of the equations, with particularly chosen test functions. It doesn't quite work as you suggest in your question.

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  • $\begingroup$ Thanks again. Can I ask if the same hold for a DGFEM case where the test functions are shape functions located on the roots of a Legendre polynomial? $\endgroup$
    – melody
    Dec 1 '15 at 15:05
  • $\begingroup$ @melody, it is, but it's because the functions are discontinuous (like in FVMs), and you can do the proof over one element at a time. $\endgroup$
    – Bill Barth
    Dec 1 '15 at 18:05
  • $\begingroup$ Let me add that even for almost any Stokes discretization with continuous pressure spaces there is some sort of local conservation preserved. Understanding this requires a bit of a different viewpoint, e.g. dual meshes instead of primal mesh elements. We did some work on this recently. $\endgroup$ Dec 1 '15 at 18:29
  • $\begingroup$ @Bill, thank you. just to clarify do you mean that it is possible to prove conservation as in FVM or is it not or do you mean that only local conservation can be proved at an element level? $\endgroup$
    – melody
    Dec 1 '15 at 23:23
  • $\begingroup$ @Christian, I had a quick looked papers, I will have to read it more carefully. For the time being I am just concerned with a simple 1D structured grid and inviscid laminar flow equations. $\endgroup$
    – melody
    Dec 1 '15 at 23:23
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Why don't you include constant function $\psi(x)=1,x\in\Omega$ in your test function space? Then if you integrate it with conservative form of your differential equation. You can recover the conservation laws.

I think for continuous Galerkin, because the test function space must be zero at boundary, it cannot include constant function. So conservation might be in question. But for DG, the function space includes constant function and the BC is imposed weakly. Conservation is no longer a problem.

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