If someone would like to condense this thread, it would make a very nice
'question' on the Q&A page which could then be answered by whoever posts
the question.
Garth
On 2014-01-22 19:23, Nico Schlömer wrote:
As far as I can tell, I'm trying to solve the extended system, and it
has no nullspace.
There are two separate approaches discussed here:
* extending the system such that the nullspace is trivial (Langrange)
* using the Krylov solver on the original system with the nontrivial
nullspace, paying a little attention to the details.
--Nico
On Wed, Jan 22, 2014 at 8:17 PM, Nikolaus Rath
<[email protected]> wrote:
On 01/22/2014 11:08 AM, Nico Schlömer wrote:
As Garth was mentioning, this problem is delicate for iterative
solver, not only because
its indefiniteness, but because the Lagrangian constraint you're
imposing yields
a column (the last one) of the full matrix that belongs to the
kernel of the top-left block.
Since the nullspace is at hands, I would provide it to the solver
and then use CG+AMG,
with Jacobi relaxation at coarser scale instead Gauss elimination
(at least with petsc boomeramg).
Why is there a nullspace? Doesn't the \int u = 0 constraint remove
the
remaining degree of freedom resulting from the pure Neumann
boundary
conditions?
It does not remove any DOF. It adds just one DOF - the Lagrange
multiplier and an equation which makes the system regular.
I'm probably just using different terminology, but how can the
system be
regular if it has a nullspace? If there is u such that A.u = 0, I
would
say that A is singular, not regular.
The original problem is singular indeed. What Jan did is add a row
and
a column (Lagrange multiplier) such that the new extended system is
regular.
This is what I thought - the extended system does *not* have a
nullspace. But the extended system is the system we're trying to
solve.
So I'm not sure how to understand Simone's suggestion above:
Since the nullspace is at hands, I would provide it to the solver
and then use CG+AMG,
with Jacobi relaxation at coarser scale instead Gauss elimination
(at least with petsc boomeramg).
As far as I can tell, I'm trying to solve the extended system, and it
has no nullspace.
Best,
Nikolaus
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