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.
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