[petsc-users] MINRES and PCFIELDSPLIT

[Adriano Côrtes]

Hi all,

I'm still trying to make MINRES and PCFIELDSPLIT working together in
my code, but I am not understanding how things are going.
Ok, first of all, I'm trying to solve a Stokes problem with an inf-sup
stable element with all boundary of Dirichlet type, that is, my system
is

       [ A   B^t ]
M = [            ]
       [ B    0   ]

and its kernel is the column vector [zeros(ndof_vel,1) ;
ones(ndof_press,1) ] (using Matlab notation)

Lets say some facts that I'm sure of, since I damped the matrices
using MatView to a Matlab file and inspected them:

a. M is symmetric for sure and singular, with the kernel mentioned above;
b. The block A is SPD (all eigenvalues are positive);
c. S = - B * A^{-1} * B^t negative semi-definite (with only one zero
eigenvalue and all others negative); that is -S is positive
semi-definite;

Now I will talk about my tests using MINRES and PCFIELDSPLIT to solve
this problem:

1. MINRES without preconditioner converges (very slowly) to any
specified tolerance;

2. MINRES without preconditioner AND using KSPSetNullspace to set the
null space of the matrix as [zeros(ndof_vel,1) ; ones(ndof_press,1) ]
converges to any specified tolerance, but the number of iterations are
almost the same of the test above;

3. MINRES with PCFIELDSPLIT, using SPLIT = SCHUR, and SCHURFACT =
DIAG, and letting Petsc managing the other options with default values
(GMRES for the KPS and ILU(0) for PC). In this case I think Petsc is
using

       [  A    0 ]
P  = [           ]
       [  0   -S ]

that in my case is symmetric positive semi-definite (by just one zero
eigenvalue), but approximating with ksp(A) and ksp(-S). Before someone
as I am also setting  KSPSetNullspace as above for the MINRES
iterations, plus passing the option
-fieldsplit_p_ksp_constant_null_space for the Schur block, and what I
obtain (with -ksp_rtol 1e-7) is

    0 KSP preconditioned resid norm 4.632440509011e-01 true resid norm
2.987970930100e-02 ||r(i)||/||b|| 1.000000000000e+00
    1 KSP preconditioned resid norm 1.484245122855e-01 true resid norm
3.127586084436e-03 ||r(i)||/||b|| 1.046725740512e-01
    2 KSP preconditioned resid norm 1.474500219584e-01 true resid norm
3.303438991154e-03 ||r(i)||/||b|| 1.105579360855e-01
    3 KSP preconditioned resid norm 3.685525320417e-06 true resid norm
2.519862718263e-07 ||r(i)||/||b|| 8.433357543339e-06
    4 KSP preconditioned resid norm 3.665488598091e-06 true resid norm
2.739588290212e-07 ||r(i)||/||b|| 9.168724710853e-06
    5 KSP preconditioned resid norm 9.845910914885e-07 true resid norm
2.763504619849e-07 ||r(i)||/||b|| 9.248766753419e-06
    6 KSP preconditioned resid norm 9.519073659560e-08 true resid norm
1.490709049851e-07 ||r(i)||/||b|| 4.989034648343e-06
    7 KSP preconditioned resid norm 9.490881783501e-08 true resid norm
1.489003256545e-07 ||r(i)||/||b|| 4.983325779864e-06
  Linear solve did not converge due to DIVERGED_INDEFINITE_MAT iterations 8

So I ask: why this last line? What is computed inside Petsc that
forces it to be printed? My guess is that o preconditioned norm used.
Am I right?

Then I decide to change some options for PCFIELDSPLIT, passing the options

#Velocity
-fieldsplit_u_ksp_type cg
-fieldsplit_u_pc_type bjacobi
-fieldsplit_u_ksp_rtol 1e-03

#Pressure
-fieldsplit_p_ksp_type cg
-fieldsplit_p_pc_type bjacobi
-fieldsplit_p_ksp_rtol 1e-03
-fieldsplit_p_ksp_constant_null_space

Then I obtained the -ksp_monitor (again with -ksp_rtol 1e-7):

    0 KSP preconditioned resid norm 4.632522584339e-01 true resid norm
2.987970930100e-02 ||r(i)||/||b|| 1.000000000000e+00
    1 KSP preconditioned resid norm 1.485092747354e-01 true resid norm
3.133707103392e-03 ||r(i)||/||b|| 1.048774294229e-01
    2 KSP preconditioned resid norm 1.475544621957e-01 true resid norm
3.307719603712e-03 ||r(i)||/||b|| 1.107011976051e-01
    3 KSP preconditioned resid norm 3.647747431146e-04 true resid norm
3.795610417928e-05 ||r(i)||/||b|| 1.270296969657e-03
    4 KSP preconditioned resid norm 3.614914908198e-04 true resid norm
4.114945598888e-05 ||r(i)||/||b|| 1.377170559939e-03
    5 KSP preconditioned resid norm 1.289144645113e-04 true resid norm
4.067177049739e-05 ||r(i)||/||b|| 1.361183607500e-03
    6 KSP preconditioned resid norm 5.143993331403e-05 true resid norm
2.794092581519e-05 ||r(i)||/||b|| 9.351137099001e-04
    7 KSP preconditioned resid norm 2.361302329725e-05 true resid norm
2.715477815463e-05 ||r(i)||/||b|| 9.088032912597e-04
    8 KSP preconditioned resid norm 2.352126784416e-05 true resid norm
2.732009930616e-05 ||r(i)||/||b|| 9.143361814852e-04
  Linear solve did not converge due to DIVERGED_INDEFINITE_MAT iterations 9

And again the same line is printed. Can somebody explain it for me?
Thanks.

Adriano Cortes