[petsc-users] Solving a Singular System with PETSc

[Jose E. Roman]

A correction: it is B=A+sigma*I when you *add* 1e-6 to the diagonal entries. but if you "multiply each diagonal of the system matrix with (1.0 + 1.0e-6)" you are doing a different thing.

> El 27 feb 2022, a las 9:21, Jose E. Roman <jroman at dsic.upv.es> escribió:
> 
> In both cases, it is like you are solving a nonsingular system with a matrix B. With MatNullSpace, B=A-e*e' where e=ones(n,1) normalized, and with your approach it is B=A+sigma*I with sigma=1e-6. The first approach shifts the zero eigenvalue, while in the second approach all eigenvalues are shifted.
> 
> Jose
> 
> 
>> El 27 feb 2022, a las 8:36, Bojan Niceno <bojan.niceno.scientist at gmail.com> escribió:
>> 
>> Dear all,
>> 
>> I have coupled PETSc with my computational fluid dynamics (CFD) solver for incompressible flows where the most computationally intensive part is a solution of the linear system for pressure - which is singular.
>> 
>> A simple call to PETSc solvers resulted in divergence, as expected, but things work when I set the null space for the pressure matrix as demonstrated in src/ksp/ksp/tutorials/ex29.c:
>>    MatNullSpace nullspace;
>>    ierr = MatNullSpaceCreate(PETSC_COMM_WORLD,PETSC_TRUE,0,0,&nullspace);CHKERRQ(ierr);
>>    ierr = MatSetNullSpace(J,nullspace);CHKERRQ(ierr);
>>    ierr = MatNullSpaceDestroy(&nullspace);CHKERRQ(ierr);
>> 
>> However, the effect of setting the null space as described above, has almost the same effect (convergence history is almost the same) as if when I multiply each diagonal of the system matrix with (1.0 + 1.0e-6), i.e., desingularize the matrix by making it slightly diagonally dominant.
>> 
>> I prefer the former solution as the latter one seems a bit like an ad-hoc patch and I am not sure how general it is, but I wonder, from a mathematical point of view, is it the same thing?  Any thoughts on that?
>> 
>> 
>>    Cheers,
>> 
>>    Bojan Niceno
>