Forming a sparse approximation of a MatShell

[Jed Brown]

I'm trying to improve the preconditioning of my spectral collocation method for
non-Newtonian incompressible Stokes flow.  My current algorithm uses MatShell
for the full Jacobian as well as each of its blocks [A B1'; B2 0] and the Schur
complement S = -B2*A*B1'.  I needed a preconditioner for A so I thought I'd
solve the same problem using finite differences on the Chebyshev nodes.  In
reality, the stencil is really ugly in 3D so I just used a simpler elliptic
operator.  This works okay, but it's performance decays significantly as I
increase the continuation parameter.  Also, dealing with general boundary
conditions is rather tricky and it seems to be a much weaker preconditioner when
I have mixed boundary conditions.  To rectify this, I tried a finite element
discretization on the Chebyshev nodes (using Q1 elements).  This must be scaled
by the inverse (lumped) mass matrix due to the collocation nature of the
spectral method.  Strangely, even though it captures all the terms in the
Jacobian, it is slightly weaker than the finite difference version.  At least it
is less error-prone and boundary conditions are easier to get right.
Regardless, forming the explicit matrix separately from the spectral matrix
causes a duplication of concepts that have to be kept in sync.  So I started
thinking, the spectral matrix is pretty cheap to apply a few times, so perhaps I
can use a coloring to compute a sparse approximation.  However, the
documentation I found is using the function from the SNES context to form the
matrix.  In my case, the entire Jacobian doesn't help, I just want an
approximation of A.  (A itself is full, but implemented via FFT.)  What is the
correct way to do this?  Should I just stick with finite differences or finite
elements?

Also, any ideas for preconditioning S?  It's condition number also grows
significantly with the continuation parameter.

Thanks,

Jed
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