[petsc-users] examples of DMPlex*FVM methods

[Justin Chang]

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 <ingogaertner.tus at gmail.com>
wrote:

>
>
> 2017-04-05 19:56 GMT+02:00 Jed Brown <jed at jedbrown.org>:
>
>> Ingo Gaertner <ingogaertner.tus at gmail.com> 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
>
>
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