> On Feb 23, 2016, at 11:35 PM, Mohammad Mirzadeh <[email protected]> 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 > > 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: 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. > > 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 > >
