<div dir="ltr"><div><div><div><div>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.<br><br></div>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: <a href="http://epubs.siam.org/doi/abs/10.1137/S0036142901384162">http://epubs.siam.org/doi/abs/10.1137/S0036142901384162</a><br><br></div>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: <a href="http://www.pflotran.org">http://www.pflotran.org</a><br><br></div>Justin<br><br></div>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."<br></div><div class="gmail_extra"><br><div class="gmail_quote">On Thu, Apr 6, 2017 at 1:16 AM, Ingo Gaertner <span dir="ltr"><<a href="mailto:ingogaertner.tus@gmail.com" target="_blank">ingogaertner.tus@gmail.com</a>></span> wrote:<br><blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex"><div dir="ltr"><br><div class="gmail_extra"><br><div class="gmail_quote"><span class="">2017-04-05 19:56 GMT+02:00 Jed Brown <span dir="ltr"><<a href="mailto:jed@jedbrown.org" target="_blank">jed@jedbrown.org</a>></span>:<br></span><span class=""><blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex"><span>Ingo Gaertner <<a href="mailto:ingogaertner.tus@gmail.com" target="_blank">ingogaertner.tus@gmail.com</a>> writes:<br>
<br>
> Hi Matt,<br>
> I don't care if FV is suboptimal to solve the Poisson equation. I only want<br>
> to better understand the method by getting my hands dirty, and also<br>
> implement the general transport equation later. We were told that FVM is<br>
> far more efficient for the transport equation than FEM, and this is why<br>
> most CFD codes would use FVM. Do you contradict? Do you have benchmarks<br>
> that show bad performance for the (parabolic) transport equation<br>
<br>
</span>What is the "parabolic transport equation"? Advection-dominated<br>
diffusion? The hyperbolic part is usually the hard part. FEM can solve<br>
these problems, but FV is a good method, particularly if you want local<br>
conservation and monotonicity.<br></blockquote><div> </div></span><div>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.<br></div><div>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.<br><br></div><div>Ingo<br></div></div></div></div><div id="m_-6291878642536978062DAB4FAD8-2DD7-40BB-A1B8-4E2AA1F9FDF2"><br> <table style="border-top:1px solid #d3d4de">
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