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

[Mohammad Mirzadeh]

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.

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?
Is SVD the only way?

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?

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:

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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