[petsc-users] Neumann BC with non-symmetric matrix

[Matthew Knepley]

On Tue, Mar 1, 2016 at 1:48 PM, Mohammad Mirzadeh <mirzadeh at gmail.com>
wrote:

> On Tue, Mar 1, 2016 at 1:15 PM, Jed Brown <jed at jedbrown.org> wrote:
>
>> Mohammad Mirzadeh <mirzadeh at gmail.com> writes:
>>
>> > I am not familiar with the terminology used here. What does the
>> refluxing
>> > mean?
>>
>> The Chombo/BoxLib family of methods evaluate fluxes between coarse grid
>> cells overlaying refined grids, then later visit the fine grids and
>> reevaluate those fluxes.  The correction needs to be propagated back to
>> the adjoining coarse grid cell to maintain conservation.  It's an
>> implementation detail that they call refluxing.
>>
>
> Thanks for clarification.
>
>
>>
>> > Right. I think if the discretization is conservative, i.e. discretizing
>> div
>> > of grad, and is compact, i.e. only involves neighboring cells sharing a
>> > common face, then it is possible to construct symmetric discretization.
>> An
>> > example, that I have used before in other contexts, is described here:
>> > http://physbam.stanford.edu/~fedkiw/papers/stanford2004-02.pdf
>>
>> It's unfortunate that this paper repeats some unfounded multigrid
>> slander and then basically claims to have uniform convergence using
>> incomplete Cholesky with CG.  In reality, incomplete Cholesky is
>> asymptotically no better than Jacobi.
>>
>> > An interesting observation is although the fluxes are only first order
>> > accurate, the final solution to the linear system exhibits super
>> > convergence, i.e. second-order accurate, even in L_inf.
>>
>> Perhaps for aligned coefficients; definitely not for unaligned
>> coefficients.
>>
>
> Could you elaborate what you mean by aligned/unaligned coefficients? Do
> you mean anisotropic diffusion coefficient?
>

Jed (I think) means coefficients where the variation is aligned to the
grid. For example, where coefficient jumps or large variation
happens on cell boundaries.

   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
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