[petsc-users] examples of DMPlex*FVM methods

[Jed Brown]

Matthew Knepley <knepley at gmail.com> writes:

> On Wed, Apr 5, 2017 at 9:57 PM, Jed Brown <jed at jedbrown.org> wrote:
>
>> Matthew Knepley <knepley at gmail.com> writes:
>>
>> > On Wed, Apr 5, 2017 at 1:13 PM, Jed Brown <jed at jedbrown.org> wrote:
>> >
>> >> 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.
>> >>
>> >
>> > But again that seems like a terrible term. What that statement above
>> means
>> > is that globally
>> > I will have no loss, but the individual amounts in each cell are not
>> > accurate to machine error,
>> > they are accurate to discretization error because the flux is only
>> accurate
>> > to discretization error.
>>
>> No. The velocity field is divergence-free up to solver tolerance.  Since
>> the piecewise constants are in the test space, there is a literal
>> equation that reads
>>
>>   \int_T (div v - f) = 0.
>>
>> That holds up to solver tolerance, not just up to discretization error.
>> That's what local conservation means.
>>
>> If you use continuous FEM, you don't have a statement like the above.
>>
>
> Okay, that is what you mean by local conservation. That state is still only
> accurate to discretization error.
> Why do I care about satisfying that equation to machine precision?

I swear we've had this discussion before.  If you have a tracer moving
in a velocity field that is not discrete divergence-free (i.e.,
satisfying the element-wise equation above), you'll get artifacts in the
concentration (possibly violating positivity or a maximum principle).
The (normal component of) velocity is also more accurate when you solve
in mixed H(div) form (or an equivalent FV method) than if you solve in
H^1.
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