There are many flavors of FEM and FVM. If by FEM you mean the Continuous Galerkin FEM, then yes it is a far from ideal method for solving advection-diffusion equations, especially when advection is the dominating effect. The Discontinuous Galerkin (DG) FEM on the other hand is much better for advection-diffusion equations (though far from perfect). It has properties very similar to the FVM as it also ensures local/element-wise mass conservation. Ferziger's book is quite biased in favor of FVM and doesn't discuss other numerical methods in depth.
I don't think there are any PETSc DMPlex DG examples at the moment (though I could be wrong). But this paper gives a nice introduction/overview to DG: http://epubs.siam.org/doi/abs/10.1137/S0036142901384162 Now if you are interested in a really sophisticated "PETSc example" of solving the advection-diffusion equation using the two-point flux FVM, there's PFLOTRAN: http://www.pflotran.org Justin PS - Is there any particular reason why PFLOTRAN is not listed as on the PETSc homepage? It seems to be a pretty major "Related packages that use PETSc." On Thu, Apr 6, 2017 at 1:16 AM, Ingo Gaertner <[email protected]> wrote: > > > 2017-04-05 19:56 GMT+02:00 Jed Brown <[email protected]>: > >> Ingo Gaertner <[email protected]> writes: >> >> > Hi Matt, >> > I don't care if FV is suboptimal to solve the Poisson equation. I only >> want >> > to better understand the method by getting my hands dirty, and also >> > implement the general transport equation later. We were told that FVM is >> > far more efficient for the transport equation than FEM, and this is why >> > most CFD codes would use FVM. Do you contradict? Do you have benchmarks >> > that show bad performance for the (parabolic) transport equation >> >> What is the "parabolic transport equation"? Advection-dominated >> diffusion? The hyperbolic part is usually the hard part. FEM can solve >> these problems, but FV is a good method, particularly if you want local >> conservation and monotonicity. >> > > By transport equation I mean the advection-diffusion equation. This is > always parabolic, independent of whether it is advection dominated or > diffusion dominated. And the elliptic Poisson equation can be solved by > making it timedependent and converge to steady state, again solving a > parabolic equation. At least this is how I learned the terms. > My impression is that everybody has his hammer, be it FEM or FVM, so that > every problem looks like a nail. You can also hammer a screw into the wall > if the wall isn't too hard. > > Ingo > > > <https://www.avast.com/sig-email?utm_medium=email&utm_source=link&utm_campaign=sig-email&utm_content=webmail> > Virenfrei. > www.avast.com > <https://www.avast.com/sig-email?utm_medium=email&utm_source=link&utm_campaign=sig-email&utm_content=webmail> > <#m_-6291878642536978062_DAB4FAD8-2DD7-40BB-A1B8-4E2AA1F9FDF2> >
