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

[Barry Smith]

> On Feb 24, 2016, at 12:07 AM, Mohammad Mirzadeh <mirzadeh at gmail.com> wrote:
> 
> 
> Barry,
> On Wednesday, February 24, 2016, Barry Smith <bsmith at mcs.anl.gov> wrote:
> 
> > On Feb 23, 2016, at 11:35 PM, Mohammad Mirzadeh <mirzadeh at gmail.com> wrote:
> >
> > Dear all,
> >
> > I am dealing with a situation I was hoping to get some suggestions here. Suppose after discretizing a poisson equation with purely neumann (or periodic) bc I end up with a matrix that is *almost* symmetric, i.e. it is symmetric for almost all grid points with the exception of a few points.
> 
>   How come it is not purely symmetric? The usual finite elements with pure Neumann or periodic bc will give a completely symmetric matrix.
> 
>   Barry
> 
> 
> So this is a finite difference discretization on adaptive Cartesian grids. It turns out that the discretization is non-symmetric at the corse-fine interface. It's actually not because of the BC itself. 

  Oh yes, I remember that issue vaguely now.

> >
> > The correct way of handling this problem is by specifying the nullspace to MatSetNullSpace. However, since the matrix is non-symmetric in general I would need to pass the nullspace of A^T. Now it turns out that if A is *sufficiently close to being symmetric*, I can get away with the constant vector, which is the nullspace of A and not A^T, but obviously this does not always work. Sometimes the KSP converges and in other situations the residual stagnates which is to be expected.
> >
> > Now, here are my questions (sorry if they are too many!):
> >
> > 1) Is there any efficient way of calculating nullspace of A^T in this case?

  Not that I know of.

> Is SVD the only way?

  I think if you make sure you only use right preconditioning such as with -ksp_type gmres -ksp_pc_side right AND you know that the right hand side is automatically in the range of A then the null space of A^T is never needed in the computation. Is your right hand side in the range of A?

  Barry

> >
> > 2) I have tried fixing the solution at an arbitrary point, and while it generally works, for some problems I get numerical artifacts, e.g. slight asymmetry in the solution and/or increased error close to the point where I fix the solution. Is this, more or less, expected as a known artifact?

  Yeah this approach is not good. We never recommend it.
> >
> > 3) An alternative to 2 is to enforce some global constraint on the solution, e.g. to require that the average be zero. My question here is two-fold:
> 
>   Requiring the average be zero is exactly the same as providing a null space of the constant function. Saying the average is zero is the same as saying the solution is orthogonal to the constant function. I don't see any reason to introduce the Lagrange multiplier and all its complications inside of just providing the constant null space.
> 
> Is this also true at the discrete level when the matrix is non-symmetric? I have always viewed this as just a constraint that could really be anything.  
> 
> >
> > 3-1) Is this generally any better than solution 2, in terms of not messing too much with the condition number of the matrix?
> >
> > 3-2) I don't quite know how to implement this using PETSc. Generally speaking I'd like to solve
> >
> > | A        U |   | X |   | B |
> > |            | * |   | = |   |
> > | U^T      0 |   | s |   | 0 |
> >
> >
> > where U is a constant vector (of ones) and s is effectively a Lagrange multiplier. I suspect I need to use MatCreateSchurComplement and pass that to the KSP? Or do I need create my own matrix type from scratch through MatCreateShell?
> >
> > Any help is appreciated!
> >
> > Thanks,
> > Mohammad
> >
> >
> 
> 
> 
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