<div dir="ltr"><div class="gmail_extra"><div class="gmail_quote">On Thu, Apr 6, 2017 at 7:04 AM, Jed Brown <span dir="ltr"><<a href="mailto:jed@jedbrown.org" target="_blank">jed@jedbrown.org</a>></span> wrote:<br><blockquote class="gmail_quote" style="margin:0px 0px 0px 0.8ex;border-left:1px solid rgb(204,204,204);padding-left:1ex"><div class="gmail-HOEnZb"><div class="gmail-h5">Matthew Knepley <<a href="mailto:knepley@gmail.com">knepley@gmail.com</a>> writes:<br>
<br>
> On Wed, Apr 5, 2017 at 9:57 PM, Jed Brown <<a href="mailto:jed@jedbrown.org">jed@jedbrown.org</a>> wrote:<br>
><br>
>> Matthew Knepley <<a href="mailto:knepley@gmail.com">knepley@gmail.com</a>> writes:<br>
>><br>
>> > On Wed, Apr 5, 2017 at 1:13 PM, Jed Brown <<a href="mailto:jed@jedbrown.org">jed@jedbrown.org</a>> wrote:<br>
>> ><br>
>> >> Matthew Knepley <<a href="mailto:knepley@gmail.com">knepley@gmail.com</a>> writes:<br>
>> >><br>
>> >> > On Wed, Apr 5, 2017 at 12:03 PM, Jed Brown <<a href="mailto:jed@jedbrown.org">jed@jedbrown.org</a>> wrote:<br>
>> >> ><br>
>> >> >> Matthew Knepley <<a href="mailto:knepley@gmail.com">knepley@gmail.com</a>> writes:<br>
>> >> >> > As a side note, I think using FV to solve an elliptic equation<br>
>> should<br>
>> >> be<br>
>> >> >> a<br>
>> >> >> > felony. Continuous FEM is excellent for this, whereas FV needs<br>
>> >> >> > a variety of twisted hacks and is always worse in terms of<br>
>> computation<br>
>> >> >> and<br>
>> >> >> > accuracy.<br>
>> >> >><br>
>> >> >> Unless you need exact (no discretization error) local conservation,<br>
>> >> >> e.g., for a projection in a staggered grid incompressible flow<br>
>> problem,<br>
>> >> >> in which case you can use either FV or mixed FEM (algebraically<br>
>> >> >> equivalent to FV in some cases).<br>
>> >> >><br>
>> >> ><br>
>> >> > Okay, the words are getting in the way of me understanding. I want to<br>
>> see<br>
>> >> > if I can pull something I can use out of the above explanation.<br>
>> >> ><br>
>> >> > First, "locally conservative" bothers me. It does not seem to indicate<br>
>> >> what<br>
>> >> > it really does. I start with the Poisson equation<br>
>> >> ><br>
>> >> > \Delta p = f<br>
>> >> ><br>
>> >> > So the setup is then that I discretize both the quantity and its<br>
>> >> derivative<br>
>> >> > (I will use mixed FEM style since I know it better)<br>
>> >> ><br>
>> >> > div v = f<br>
>> >> > grad p = v<br>
>> >> ><br>
>> >> > Now, you might expect that "local conservation" would give me the<br>
>> exact<br>
>> >> > result for<br>
>> >> ><br>
>> >> > \int_T p<br>
>> >> ><br>
>> >> > everywhere, meaning the integral of p over every cell T.<br>
>> >><br>
>> >> Since when is pressure a conserved quantity?<br>
>> >><br>
>> >> In your notation above, local conservation means<br>
>> >><br>
>> >> \int_T (div v - f) = 0<br>
>> >><br>
>> >> I.e., if you have a tracer moving in a source-free velocity field v<br>
>> >> solving the above equation, its concentration satisfies<br>
>> >><br>
>> >> c_t + div(c v) = 0<br>
>> >><br>
>> >> and it will be conserved element-wise.<br>
>> >><br>
>> ><br>
>> > But again that seems like a terrible term. What that statement above<br>
>> means<br>
>> > is that globally<br>
>> > I will have no loss, but the individual amounts in each cell are not<br>
>> > accurate to machine error,<br>
>> > they are accurate to discretization error because the flux is only<br>
>> accurate<br>
>> > to discretization error.<br>
>><br>
>> No. The velocity field is divergence-free up to solver tolerance. Since<br>
>> the piecewise constants are in the test space, there is a literal<br>
>> equation that reads<br>
>><br>
>> \int_T (div v - f) = 0.<br>
>><br>
>> That holds up to solver tolerance, not just up to discretization error.<br>
>> That's what local conservation means.<br>
>><br>
>> If you use continuous FEM, you don't have a statement like the above.<br>
>><br>
><br>
> Okay, that is what you mean by local conservation. That state is still only<br>
> accurate to discretization error.<br>
> Why do I care about satisfying that equation to machine precision?<br>
<br>
</div></div>I swear we've had this discussion before. If you have a tracer moving<br>
in a velocity field that is not discrete divergence-free (i.e.,<br>
satisfying the element-wise equation above), you'll get artifacts in the<br>
concentration (possibly violating positivity or a maximum principle).<br>
The (normal component of) velocity is also more accurate when you solve<br>
in mixed H(div) form (or an equivalent FV method) than if you solve in<br>
H^1.<br>
</blockquote></div><br>Okay, that makes sense. If I do not have fluxes matching the sources, I do not</div><div class="gmail_extra">preserve montonicity for an advected field. I might need this to machine precision</div><div class="gmail_extra">because some other equations cannot tolerate a negative number there. I will</div><div class="gmail_extra">write this one down.</div><div class="gmail_extra"><br></div><div class="gmail_extra"><span style="color:rgb(80,0,80)">Why do I need it "for a projection in a staggered grid incompressible flow </span><span style="color:rgb(80,0,80)">problem".</span></div><div class="gmail_extra"><font color="#500050">This would mean I satisfy (I think)</font></div><div class="gmail_extra"><font color="#500050"><br></font></div><div class="gmail_extra"><font color="#500050"> \int_T div p = 0</font></div><div class="gmail_extra"><font color="#500050"><br></font></div><div class="gmail_extra"><font color="#500050">meaning that there is a force balance on each cell to machine precision. If I just care</font></div><div class="gmail_extra"><font color="#500050">about the fluid flow, this does not seem important.</font></div><div class="gmail_extra"><font color="#500050"><br></font></div><div class="gmail_extra"><font color="#500050"> Thanks,</font></div><div class="gmail_extra"><font color="#500050"><br></font></div><div class="gmail_extra"><font color="#500050"> Matt<br clear="all"></font><div><br></div>-- <br><div class="gmail_signature">What most experimenters take for granted before they begin their experiments is infinitely more interesting than any results to which their experiments lead.<br>-- Norbert Wiener</div>
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