[petsc-users] examples of DMPlex*FVM methods

[Jed Brown]

Matthew Knepley <knepley at gmail.com> writes:

> On Wed, Apr 5, 2017 at 12:03 PM, Jed Brown <jed at jedbrown.org> wrote:
>
>> Matthew Knepley <knepley at gmail.com> writes:
>> > As a side note, I think using FV to solve an elliptic equation should be
>> a
>> > felony. Continuous FEM is excellent for this, whereas FV needs
>> > a variety of twisted hacks and is always worse in terms of computation
>> and
>> > accuracy.
>>
>> Unless you need exact (no discretization error) local conservation,
>> e.g., for a projection in a staggered grid incompressible flow problem,
>> in which case you can use either FV or mixed FEM (algebraically
>> equivalent to FV in some cases).
>>
>
> Okay, the words are getting in the way of me understanding. I want to see
> if I can pull something I can use out of the above explanation.
>
> First, "locally conservative" bothers me. It does not seem to indicate what
> it really does. I start with the Poisson equation
>
>   \Delta p = f
>
> So the setup is then that I discretize both the quantity and its derivative
> (I will use mixed FEM style since I know it better)
>
>   div  v = f
>   grad p = v
>
> Now, you might expect that "local conservation" would give me the exact
> result for
>
>   \int_T p
>
> everywhere, meaning the integral of p over every cell T. 

Since when is pressure a conserved quantity?

In your notation above, local conservation means

  \int_T (div v - f) = 0

I.e., if you have a tracer moving in a source-free velocity field v
solving the above equation, its concentration satisfies

  c_t + div(c v) = 0

and it will be conserved element-wise.
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