[petsc-users] A question about solving a saddle point system with a direct solver

[Jau-Uei Chen]

To whom it may concern,

I am writing to ask about using PETSc with a direct solver to solve a
linear system where a single zero-value eigenvalue exists.

Currently, I am working on developing a finite-element solver for a
linearized incompressible MHD equation. The code is based on an open-source
library called MFEM which has its own wrapper for PETSc and is used in my
code. From analysis, I already know that the linear system (Ax=b) to be
solved is a saddle point system. By using the flags "solver_pc_type svd"
and "solver_pc_svd_monitor", I indeed observe it. Here is an example of an
output:

    SVD: condition number 3.271390119581e+18, 1 of 66 singular values are
(nearly) zero
    SVD: smallest singular values: 3.236925932523e-17 3.108788619412e-04
3.840514506502e-04 4.599292003910e-04 4.909419974671e-04
    SVD: largest singular values : 4.007319935079e+00 4.027759008411e+00
4.817755760754e+00 4.176127583956e+01 1.058924751347e+02


However, What surprises me is that the numerical solutions are still
relatively accurate by comparing to the exact ones (i.e. manufactured
solutions) when I perform convergence tests even if I am using a direct
solver (i.e. -solver_ksp_type preonly -solver_pc_type lu
-solver_pc_factor_mat_solver_type
mumps). My question is: Why the direct solver won't break down in this
context? I understand that it won't be an issue for iterative solvers such
as GMRES [1][2] but not really sure why it won't cause trouble in direct
solvers.

Any comments or suggestions are greatly appreciated.

Best Regards,
Jau-Uei Chen

Reference:
[1] Benzi, Michele, et al. “Numerical Solution of Saddle Point Problems.”
Acta Numerica, vol. 14, May 2005, pp. 1–137. DOI.org (Crossref),
https://doi.org/10.1017/S0962492904000212.
[2] Elman, Howard C., et al. Finite Elements and Fast Iterative Solvers:
With Applications in Incompressible Fluid Dynamics. Second edition, Oxford
University Press, 2014.
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