[petsc-users] shell preconditioner for Schur complement

[Matteo Semplice]

Dear PETSc users,
     we are trying to program a preconditioner for the Schur complement 
of a Stokes system, but it seems that the r.h.s. for the Schur 
complement system differs from what we expect by a scale factor, which 
we don't understand.

Our setup has a system matrix A divided in 2x2 blocks for velocity and 
pressure variables. We have programmed our preconditioner in a routine 
PrecondSchur and in the main program we do

PC pc;
KSPGetPC(kspA,&pc);
PCSetFromOptions(pc);
KSPSetOperators(kspA, A, A);
KSPSetInitialGuessNonzero(kspA,PETSC_FALSE);
KSPSetFromOptions(kspA);
KSP *subksp;
PetscInt nfield;
PCSetUp(pc);
PCFieldSplitGetSubKSP(pc, &nfield, &subksp);
PC pcSchur;
KSPGetPC(subksp[1],&pcSchur);
PCSetType(pcSchur,PCSHELL);
PCShellSetApply(pcSchur,PrecondSchur);
KSPSetFromOptions(subksp[1]);

and eventually

KSPSolve(A,b,solution);

We run the code with options

  -ksp_type fgmres \
  -pc_type fieldsplit -pc_fieldsplit_type schur \
  -pc_fieldsplit_schur_fact_type full \

and, from reading section 2.3.5 of the PETSc manual, we'd expect that 
the first r.h.s. passed to PrecondSchur be exactly
     b_1-A_10*inv(A_00)*b_0

Instead (from a monitor function attached to the subksp[1] solver), the 
first r.h.s. appears to be scalar multiple of the above vector; we are 
guessing that we should take into account this multiplicative factor in 
our preconditioner routine, but we cannot understand where it comes from 
and how its value is determined.

Could you explain us what is going on in the PC_SCHUR exactly, or point 
us to some working code example?

Thanks in advance!

     Matteo