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I know that FVM is based on the integral form of conservation laws, and FDS is based on PDEs. What I'm confused by, is whether FEM and DGFEM formulations are based on integral or pde form of conservation laws.

I have developed FEM formulations starting from integral form of transport equations but I've also found literature on that FEM approximates PDEs, so slightly confused there.

I am also not sure about DGFEM methods, with this method again, I start from the integral form of the equations, not sure if they are also be based on PDEs.

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    $\begingroup$ It depends on what you mean by "integral form". FEM and DGFEM are based on variational (or weak) forms of PDEs, so they involve both partial derivatives and integrals. (An integral equation would only involve integrals, but no derivatives.) $\endgroup$ – Christian Clason Nov 28 '15 at 10:13
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Basically, you have the following set of derivations:

  • Strong form of the PDE -> Finite differences
  • Integral form of the PDE -> Finite volumes
  • Weak (variational) form of the PDE -> Galerkin methods (including FEM and DGFEM)
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  • $\begingroup$ Thanks very much. Are the integral forms used in FVM not a weak form of PDEs also? $\endgroup$ – melody Nov 28 '15 at 19:46
  • $\begingroup$ @melody: in some sense, yes. As long your test function space consists of piecewise constant functions, you can consider FVM as a type of FEM. $\endgroup$ – Paul Nov 28 '15 at 21:18
  • $\begingroup$ @melody -- as Paul says, you can under some assumptions understand it that way. Specifically, this is true for first order equations and if you assume the solution to be sufficiently smooth to allow for integration by parts. But it is easier to understand the integral form used for the FVM as arising from the conservation property described by these PDEs, rather than arising from the weak form by using a particular choice of test functions. The conservation property is really what FVM is based on. $\endgroup$ – Wolfgang Bangerth Nov 29 '15 at 21:37
  • $\begingroup$ Thanks very much this helped a lot. Do you know of a simple enough reference available which explains test functions, weak formulations, etc related to FEM for a non-mathematician? thanks. $\endgroup$ – melody Dec 1 '15 at 0:14
  • $\begingroup$ I just so happen to know one: lecture 4 at math.tamu.edu/~bangerth/videos.html :-) $\endgroup$ – Wolfgang Bangerth Dec 1 '15 at 4:47

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