[petsc-users] Non-matching KSP and SNES norms during SNES solve

[Jean-Arthur Louis Olive]

Hi all,
we are using PETSc to solve the steady state Stokes equations with non-linear viscosities using finite difference. Recently we have realized that our true residual norm after the last KSP solve did not match next SNES function norm when solving the linear Stokes equations.

So to understand this better, we set up two extremely simple linear residuals, one with no coupling between variables (vx, vy, P and T), the other with one coupling term (shown below).

RESIDUAL 1 (NO COUPLING):
  for (j=info->ys; j<info->ys+info->ym; j++) {
    for (i=info->xs; i<info->xs+info->xm; i++) {
      f[j][i].P = x[j][i].P - 3000000;
      f[j][i].vx= 2*x[j][i].vx;
      f[j][i].vy= 3*x[j][i].vy - 2;
      f[j][i].T = x[j][i].T;
 }

RESIDUAL 2 (ONE COUPLING TERM):
  for (j=info->ys; j<info->ys+info->ym; j++) {
    for (i=info->xs; i<info->xs+info->xm; i++) {
      f[j][i].P = x[j][i].P - 3;
      f[j][i].vx= x[j][i].vx - 3*x[j][i].vy;
      f[j][i].vy= x[j][i].vy - 2;
      f[j][i].T = x[j][i].T;     
    }
  }


and our default set of options is:


OPTIONS:  mpiexec -np $np ../Stokes -snes_max_it 4 -ksp_atol 2.0e+2 -ksp_max_it 20 -ksp_rtol 9.0e-1 -ksp_type fgmres -snes_monitor -snes_converged_reason -snes_view -log_summary -options_left 1 -ksp_monitor_true_residual -pc_type none -snes_linesearch_type cp 


With the uncoupled residual (Residual 1), we get matching KSP and SNES norm, highlighted below:


Result from Solve - RESIDUAL 1 
  0 SNES Function norm 8.485281374240e+07 
    0 KSP unpreconditioned resid norm 8.485281374240e+07 true resid norm 8.485281374240e+07 ||r(i)||/||b|| 1.000000000000e+00
    1 KSP unpreconditioned resid norm 1.131370849896e+02 true resid norm 1.131370849896e+02 ||r(i)||/||b|| 1.333333333330e-06
  1 SNES Function norm 1.131370849896e+02 
    0 KSP unpreconditioned resid norm 1.131370849896e+02 true resid norm 1.131370849896e+02 ||r(i)||/||b|| 1.000000000000e+00
  2 SNES Function norm 1.131370849896e+02 
Nonlinear solve converged due to CONVERGED_SNORM_RELATIVE iterations 2


With the coupled residual (Residual 2), the norms do not match, see below


Result from Solve - RESIDUAL 2:
  0 SNES Function norm 1.019803902719e+02 
    0 KSP unpreconditioned resid norm 1.019803902719e+02 true resid norm 1.019803902719e+02 ||r(i)||/||b|| 1.000000000000e+00
    1 KSP unpreconditioned resid norm 8.741176309016e+01 true resid norm 8.741176309016e+01 ||r(i)||/||b|| 8.571428571429e-01
  1 SNES Function norm 1.697056274848e+02 
    0 KSP unpreconditioned resid norm 1.697056274848e+02 true resid norm 1.697056274848e+02 ||r(i)||/||b|| 1.000000000000e+00
    1 KSP unpreconditioned resid norm 5.828670868165e-12 true resid norm 5.777940247956e-12 ||r(i)||/||b|| 3.404683942184e-14
  2 SNES Function norm 3.236770473841e-07 
Nonlinear solve converged due to CONVERGED_FNORM_RELATIVE iterations 2


Lastly, if we add -snes_fd to our options, the norms for residual 2 get better - they match after the first iteration but not after the second.


Result from Solve with -snes_fd - RESIDUAL 2
 0 SNES Function norm 8.485281374240e+07 
    0 KSP unpreconditioned resid norm 8.485281374240e+07 true resid norm 8.485281374240e+07 ||r(i)||/||b|| 1.000000000000e+00
    1 KSP unpreconditioned resid norm 2.039607805429e+02 true resid norm 2.039607805429e+02 ||r(i)||/||b|| 2.403700850300e-06
  1 SNES Function norm 2.039607805429e+02 
    0 KSP unpreconditioned resid norm 2.039607805429e+02 true resid norm 2.039607805429e+02 ||r(i)||/||b|| 1.000000000000e+00
    1 KSP unpreconditioned resid norm 2.529822128436e+01 true resid norm 2.529822128436e+01 ||r(i)||/||b|| 1.240347346045e-01
  2 SNES Function norm 2.549509757105e+01 [SLIGHTLY DIFFERENT]
    0 KSP unpreconditioned resid norm 2.549509757105e+01 true resid norm 2.549509757105e+01 ||r(i)||/||b|| 1.000000000000e+00
  3 SNES Function norm 2.549509757105e+01 
Nonlinear solve converged due to CONVERGED_SNORM_RELATIVE iterations 3


Does this mean that our Jacobian is not approximated properly by the default “coloring” method when it has off-diagonal terms?

Thanks a lot,
Arthur and Eric
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