Use iterative solver with preconditioning

[agzimmerman]

e.g. https://fenicsproject.org/pub/tutorial/sphinx1/._ftut1006.html

[agzimmerman]

Since a Krylov solver could save us a lot of memory, perhaps this will allow us to get useful 3D results without MPI.

[agzimmerman]

fenics.list_linear_solver_methods()

Solver method | Description

bicgstab | Biconjugate gradient stabilized method cg | Conjugate gradient method default | default linear solver gmres | Generalized minimal residual method minres | Minimal residual method mumps | MUMPS (MUltifrontal Massively Parallel Sparse direct Solver) petsc | PETSc built in LU solver richardson | Richardson method tfqmr | Transpose-free quasi-minimal residual method umfpack | UMFPACK (Unsymmetric MultiFrontal sparse LU factorization)

[agzimmerman]

Make sure to also set the dual solver, as shown in https://www.allanswered.com/post/omvxz/adaptivenonlinearvariationalsolver-does-not-pass-solver-parameters/

[agzimmerman]

So far Krylov solvers are only working for the Stefan problem test.

Preconditioners are usually needed to make Krylov solvers efficient. Maybe in our case they're needed to get a solution at all. Now I'm looking into how to design a preconditioner. I think we'll need a block preconditioner, which I've seen done for the incompressible Navier-Stokes equations. There was a talk at SIAM-CSE a year ago where a guy might have also done something like this with a coupled energy equation; but I don't think he coupled the energy equation monolithically, i.e. he only needed to precondition the NS system.

Here's a nice answer from W. Bangerth about preconditioning in general, advocating block preconditioning: https://scicomp.stackexchange.com/questions/1129/what-guidelines-should-i-use-when-searching-for-good-preconditioning-methods-for

[agzimmerman]

This paper might use an approach similar to what we need: https://arxiv.org/pdf/1610.03991.pdf

Edit: More useful links: https://www.researchgate.net/publication/309131739_Preconditioning_of_a_coupled_Cahn--Hilliard_Navier--Stokes_system , https://www.researchgate.net/publication/260393812_A_stable_and_linear_time_discretization_for_a_thermodynamically_consistent_model_for_two-phase_incompressible_flow

[agzimmerman]

Here's more info about exporting the sparse matrix: https://www.allanswered.com/post/nqzoo/how-to-get-sparse-matrix/

[agzimmerman]

Here's a discussion related to block solving: https://www.allanswered.com/post/bevkr/using-fieldsplit-on-a-petsc-matnest-object-via-cbc-block/

[agzimmerman]

Try block preconditioning only the Navier-Stokes part and not preconditioning Stefan problem part

[agzimmerman]

Here is a package for preconditioning incompressible Navier Stokes using FEniCS : https://github.com/blechta/fenapack

[agzimmerman]

Thomas Gibson from the FEniCS conference pointed me towards Rayleigh–Bénard preconditioners. Here's an example I Googled: https://onlinelibrary.wiley.com/doi/full/10.1002/nla.2096

[agzimmerman]

Consider using the BDDC preconditioner, e.g. https://arxiv.org/abs/1101.1775

I just stumbled on some of my old notes from a conference where I wrote down that BDDC is a very robust preconditioner, and something related to "low rank Schur approximations", and "hierarchical low rank operators". "Taskification", "Meet the curse of dimensionality with the blessing of low rank". I seemed to think it was important to look in to at the time.

[agzimmerman]

Using HDiv conforming FEM for the Navier-Stokes sub problem could allow us to derive a SPD matrix for the Navier-Stokes block. See issue #252

[agzimmerman]

Here's a demo from Firedrake: https://www.firedrakeproject.org/demos/rayleigh-benard.py.html