Add kronpc for stage-coupled Kronecker product preconditioners

[436ahsan]

Summary

This PR adds a kronpc framework for applying stage-coupled Kronecker product preconditioners of the form

[
L \otimes K^{-1},
]

where (L) is a small dense stage matrix, typically derived from the Runge--Kutta Butcher matrix, and (K^{-1}) is supplied by a single-stage PETSc/Firedrake subsolver.

The main motivation is to support Runge-Kutta stage preconditioners appearing in augmented Lagrangian Schur-complement approximations for time-dependent Navier-Stokes problems.

This allows users to build preconditioners involving Kronecker products of Runge-Kutta stage matrices with pressure, mass, and stiffness inverses. This structure is motivated by the augmented Lagrangian Runge-Kutta preconditioner of Leveque, He, and Olshanskii, arXiv:2506.04451.

Changes

• Adds irksome/kronpc.py.
• Provides:
  • KronPC
  • MassKronPC
  • StiffnessKronPC
  • SIPGStiffnessKronPC
• Re-exports these classes from irksome.__init__.
• Adds a test for MassKronPC against a full monolithic LU solve.
• Updates the implementation for the current petsc4py KSP.setDMActive API.

Testing

OMP_NUM_THREADS=1 python -m pytest tests/test_kronpc.py

[rckirby]

Thanks for the feature. We might want to add a demo of the Leveque et al preconditioner using this for flow later, but for now we probably want to just make sure both mass & stiffness get tested.

[rckirby]

Is it possible to get this to test the stiffness PCs as well with a parameterization? You'd need to parametrize over the PC paired with a form that did mass vs stiffness.

[436ahsan]

I parameterized the KronPC test over both MassKronPC and StiffnessKronPC. The stiffness case is paired with the corresponding shifted stiffness form A \otimes (K + shift M). I use a larger shift only in the test to keep the reference operator well-conditioned; the default remains 1e-12. I also tried adding SIPGStiffnessKronPC, but the current SIPG implementation projects out a constant nullspace in apply(), while the SIPG form includes boundary penalty terms. I think for this reason there are still some issues regarding passing the test with SIPG, and I need to think more about how to resolve this issue.