On Wed, Apr 5, 2017 at 9:57 PM, Jed Brown <[email protected]> wrote: > Matthew Knepley <[email protected]> writes: > > > On Wed, Apr 5, 2017 at 1:13 PM, Jed Brown <[email protected]> wrote: > > > >> Matthew Knepley <[email protected]> writes: > >> > >> > On Wed, Apr 5, 2017 at 12:03 PM, Jed Brown <[email protected]> wrote: > >> > > >> >> Matthew Knepley <[email protected]> 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? 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
