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

[Jed Brown]

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.

> 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.
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