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.
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