On 29/10/14 20:06, Jed Brown wrote:
Stephan Kramer <[email protected]> writes:
On 18/09/14 15:41, Mark Adams wrote:
Thanks for updating me Stephan,
This is a common problem. I'm surprised we have not seen it more.
Note, the problem with the solver dying when MatSetBlockSize is set
(correctly). Is some sort of bug. You have Stokes and use FieldSplit,
so GAMG is working on a 2D elasticity like problem. Is that correct?
And the equations are ordered node first: [v0_x, v0_y, v1_x, v1_y,
v2_x, ....] correct?
Correct - although we're not using FieldSplit because of historic
reasons (the code predates fieldsplit) - but yes, it's the velocity
block K of a Stokes system (K G; G^T 0) with variable viscosity.
I finally figured out why the convergence was so much worse if we do set
the block size: It turns out to do with the scaling of the number I put
on the diagonal for the eliminated DOFs associated with strong bcs. This
problem has free slip bcs and I eliminate the DOF associated with the
normal component by zeroing out the row and column and putting an
arbitrary number on the diagonal (I chose 1.0).
Are the slip surfaces always aligned with the coordinate axes or are you
rotating the vertex coordinate system to line up? If the latter, is
that rotation accounted for when defining the near-null space?
The boundaries are non-aligned and the boundary conditions are rotated
indeed. Both our physical coordinate space and the velocity components
that we solve for are x,y aligned. After the assembly of the matrix
(with "natural" bcs only) the dofs at the boundary nodes are rotated
with a PTAP to align with the normal and tangential directions. After
that the strong bc is applied with a call to MatZeroRowsColumns().
The current problem I'm looking at is in a cylindrical (annulus) domain
with free slip on all sides. Thus we have a real null space containing
the rotational mode, and a near null space containing all three of the
usual modes. We have indeed applied the same rotation to all these null
space modes
Normally the number doesn't really matter as the row is completely
decoupled from the rest of the system. However in this case, when
using gamg with blocks, it affects the strong coupling criterion for
*both* velocity components associated with the boundary node. This
criterion is scaled with respect to the 1-norm of the diagonal block,
thus if the diagonal for the normal component is chosen too big it
also declares all connections to the dof associated with the
tangential component weak. In this way all the boundary nodes (the
pairs of velocity components at these nodes) became isolated clusters
and were disregarded at the coarse level. After choosing a smaller
value for the diagonals the solve now converges fine in roughly the
same number of iterations as when not setting the block size.
The tricky thing here is the choice of the dummy diagonal number becomes
very sensitive - with a typical gamg threshold of 0.01 it can't be more
than an order out. Currently I use MatZeroRowsColumns() with a constant
value for the diagonal. Because I have a variable viscosity it seems I
can no longer do that. Now I can just go in and look at what the
diagonal is for the other component at each boundary node and use that
value instead (or at least a value that's smaller) - but it all seems a
bit hackish to me. Perhaps we're following the wrong approach altogether
and should be dealing with the bcs in an other way? Any thoughts would
be more than welcome...
I think we should improve the strength of connection measure. For
example, the (2 bs)×(2 bs) matrix
A B
C D
is strongly coupled if
A^{-1} (A - B D^{-1} C)
is far from the identity. It would normally be measured as
sqrt(|| A^{-1} B D^{-1} C ||)
In the case of scalars, this reduces to the standard measure
sqrt(|bc / ad|).
Note that this still doesn't work in more exotic cases like unaligned
anisotropy with certain discretizations, but it should fix the problem
you're seeing.
That sounds like a nice solution as indeed it falls back to the scalar
case when one of the two dofs is lifted.
Cheers
Stephan
