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

[Mohammad Mirzadeh]

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?
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