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. 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
