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I do understand the meanning of "conservative discretization" within the FVM/FDM framework, indeed it is well explained in this post.

Now, according to the table in this slide (pp.8), it concludes:

  • FEM suffers from "Stability for conservation laws", while Discontinuous Galerkin (DG) is just fine

I interpret this statement as, FEM(Galerkin) is a non-conservative discretization, while Discontinuous Galerkin is conservative. Am I right ?

Could anybody elaborate in mathematical sense the meaning of conservative/non-conservative in FEM/DG ? ( I do have some experience with conventional FEM but not DG :-) )

Perhaps I should ask simply: Is it possible to demonstrate whether it's a conservative discretization based upon any weak formulation(from conventional FEM/SUPG/DG)?

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No. The statement means that the finite element method, without modifications, will lead to numerical solutions for conservation laws that violate many of the physically reasonable constraints (e.g., that solutions have certain monotony properties, that entropy can only grow, of that the density can not become negative -- in mathematical language, the discrete solution at every time step does not satisfy stability estimates with the solution of the previous time step and boundary values as data).

The reasons are complicated and beyond what can be explained on this forum. But, if you allow me the conflict of interest, you can find more material in lecture 31 at http://www.math.tamu.edu/~bangerth/videos.html .

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  • $\begingroup$ Thanks for your video, watched it and learned sth. Perhaps I should ask more specifically, e.g. could you demonstrate the weak form of FEM/SUPG/DG is indeed a conservative discretisation, in the way FVM is explained? $\endgroup$
    – lorniper
    Apr 25, 2015 at 10:19
  • $\begingroup$ It all depends on the equation and which stabilization exactly you choose. You can make the DG form conservative for some equations. I don't think you can make SUPG conservative for any equation. But there are other approaches than SUPG for continuous FEM as well. $\endgroup$
    – Wolfgang Bangerth
    Apr 25, 2015 at 13:13
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Long story short.

The example the simplest diffusion equation $-\nabla\cdot(A\nabla u) = f$, with certain boundary condition to form up a BVP.

Suppose a finite dimensional approximation $u_h$ to $u$, is obtained by a discretization scheme, continuous Galerkin (CG), non-conforming (Crouzeix-Raviart), DG (IPDG, LDG, HDG, DPG, etc), or mixed formulation.

Then we call this scheme conservative if the numerical flux $\boldsymbol{\sigma}_h:= -A\nabla u_h$ on each element satisfy the equilibrium equation $\nabla\cdot \boldsymbol{\sigma}_h = f_h$ "in some sense".

The CG and non-conforming are non-conservative.

All the DG schemes are conservative if the vector flux $\boldsymbol{\sigma}_h$ is formulated in an appropriate way. The conservativity means the equilibrium equation is satisfied in the integral sense on each element. But for DG, there are more things to consider like the consistency of $u_h$ and its "scalar flux" $\hat{u}_h$ defined on the skeleton of the mesh. For the summary, please refer to the paper by DN Arnold, F Brezzi, B Cockburn, LD Marini on SINUM, Unified analysis of discontinuous Galerkin methods for elliptic problems, page 1761-1762.

Mixed methods are "naturally" conservative because the equilibrium equation is discretized at the same time with the constitutive equation.

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  • $\begingroup$ Could you elaborate on "The CG and non-conforming are non-conservative", taking CG discretization of the diffusion equation for example.. $\endgroup$
    – lorniper
    May 28, 2015 at 13:39
  • $\begingroup$ Note that the CG method is conservative. This has been shown several times in the literature, notably in The continuous Galerkin method is locally conservative by Hughes in 2000. You simply have to do a post-processing routine to recover those fluxes, similar to static condensation. $\endgroup$
    – James Wright
    Oct 29, 2022 at 17:20

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