[petsc-users] examples of DMPlex*FVM methods

[Matthew Knepley]

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.

   Matt

-- 
What most experimenters take for granted before they begin their
experiments is infinitely more interesting than any results to which their
experiments lead.
-- Norbert Wiener
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