[petsc-users] examples of DMPlex*FVM methods

[Matthew Knepley]

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. However, there is
discretization error in the fluxes v, and then I determine p by adding them
up. So the thing that is exact is the fact that anything going out of one
cell T, goes into another cell T'. Thus

  \int_\Omega p

is exact IF my boundary conditions for p are also exact. Otherwise they
cannot be more accurate than the inflows/outflows. So suppose you have
exact BC on your Poisson equation, then you can get exact global
conservation. Why is it "locally conservative"? Moreover, why would exact
global conservation of p be necessary? That might be equivalent to mass
loss, but are you telling me that 10^-8 mass loss is distinguishable from
10^-14? I do not understand that.

  Thanks,

    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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